Minimal Isometric Immersions of Flat $n$ -tori into Spheres

Ying Lü¹, Peng Wang², Zhenxiao Xie ^🖂 3

Abstract

In 1985, Bryant established that a flat $2$ -torus admits a minimal isometric immersion into some round sphere if and only if a certain rationality condition is satisfied. We show that when $n\geq 3$ , the rationality criterion is no longer a necessary, but a sufficient condition for a flat $n$ -torus to admit minimal isometric immersions into spheres. We also derive an upper bound for the algebraic irrationality degree of such immersions. When $n=3$ , this bound is sharp and explicit embedded examples are provided respectively for each possible degree. Moreover, by constructing a family of non-homogeneous minimal flat $3$ -tori, we show that minimal isometric immersions (embeddings) of flat $n$ -tori are not necessarily homogeneous when $n\geq 3$ . In addition, we establish a deformation theorem that every flat $n$ -torus admitting a minimal isometric spherical immersion can be isometrically, minimally and homogeneously immersed into a sphere of dimension at most $n^{2}+n-1$ .

¹¹footnotetext: School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P. R. China. Email: lueying@xmu.edu.cn²²footnotetext: School of Mathematics and Statistics, Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), FJKLAMA, Fujian Normal University, 350117 Fuzhou, P.R. China. Email: pengwang@fjnu.edu.cn³³footnotetext: School of Mathematical Sciences, Beihang University, Beijing 100191, P. R. China. Email: xiezhenxiao@buaa.edu.cn

Keywords: Minimal immersions; flat tori; lattices; isometric immersions

MSC(2020): 53C42, 53C40, 11H06

1 Introduction

The investigation of minimal isometric immersions of space forms into round spheres has been a subject of significant interest in differential geometry, owing to its deep connections with representation theory and spectral geometry.

The $2$ -dimensional case has been completely resolved through fundamental contributions by Calabi [3], Kenmotsu [18], and Bryant [2]. The only minimal surfaces of constant positive Gaussian curvature in $\mathbb{S}^{n}$ are the round $2$ -spheres and the Veronese-Borúvka $2$ -spheres. Surfaces of constant negative Gaussian curvature cannot be minimally and isometrically immersed into any sphere. While minimal isometric immersions of $\mathbb{R}^{2}$ into $\mathbb{S}^{n}$ can be explicitly parameterized, Bryant [2, page 270] remarked that a flat $2$ -torus $T^{2}=\mathbb{R}^{2}/\Lambda_{2}$ admits a minimal isometric immersion into some sphere $\mathbb{S}^{n}$ if and only if some rational condition satisfies. In terms of the Gram matrix $Q$ of $\Lambda_{2}$ , the rational condition can be stated as $Q$ has exclusively rational entries up to a dilation. For recent progress in this direction, we refer to the classification in [4] of minimal surfaces of constant negative curvature in a Hilbert sphere.

For higher-dimensional space forms, only the case of positive sectional curvature has been investigated very well, primarily through the foundational work of do Carmo and Wallach [9]. Their work established a deep connection between minimal isometric immersions of spheres into spheres and representation theory, leading to what is now known as the do Carmo-Wallach theory. They showed there are in general many minimal isometric immersions of $\mathbb{S}^{m}$ into $\mathbb{S}^{n}$ , and they can be parameterized by a compact convex body of finite dimension. These results were subsequently extended to general isotropy-irreducible Riemannian homogeneous spaces in [20, 45, 41]. The extension builds on Takahashi’s foundational work [34] establishing standard minimal isometric immersions through the isotropy representation. There are also many works devoted to investigating the moduli space of minimal isometric immersions of $\mathbb{S}^{m}$ into spheres and related problems, such as the determination of the exact dimension of this moduli space [39], and the study of minimal target dimension [25, 8, 42, 11]. For further developments and additional references on related topics, we refer the reader to [15, 10, 40, 35] and the references therein.

Since flat space forms are not isotropy-irreducible, the do Carmo-Wallach theory has limited applicability in this context. While it can be employed to study eigenmaps (i.e., harmonic maps with constant energy density) from flat $n$ -tori into spheres [31, 30], it does not extend in general to the investigation of those minimal isometric immersions. A natural question in this field is to determine which flat $n$ -tori, aside from the well-understood case of $2$ -tori, admit minimal isometric immersions into spheres. According to Moore’s result [27], any flat $n$ -torus admitting a minimal isometric immersion into $\mathbb{S}^{2n-1}$ must be either the Clifford $n$ -torus or one of its coverings. For further results concerning isometric immersions of space forms into space forms, we refer to the survey articles [5, 1] and the book [7].

Recently, the authors [22] classified all minimal isometric immersions of flat $3,4$ -tori into spheres by the first eigenfunctions (called $\lambda_{1}$ -minimal isometric immersions for short). The $\lambda_{1}$ -minimal immersions of Riemannian manifolds into spheres are highly related to the concept of conformal volume in submanifold geometry and the first conformal spectrum in spectral geometry, see [21, 26, 13, 28, 36, 37, 29, 17] and references therein. Our classification is based on establishing new variational characterizations for minimal flat $n$ -tori, formulated through their underlying lattice structures (see Theorem 2.4 in Section 2). After publishing our paper [22], we realized that $\lambda_{1}$ -minimal isometric immersions of flat $n$ -tori is deeply connected to the semi-eutatic lattices of rank $n$ in geometry of numbers [24].

In this paper, we employ the aforementioned variational characterizations to investigate the problem of which flat $n$ -tori admit minimal isometric immersions into spheres. An immediate consequence is that a generic flat $n$ -torus admits no minimal isometric immersion into any sphere, which was shown by Bryant [2] for the $2$ -dimensional case. We first extend Bryant’s criterion for flat 2-tori to arbitrary dimensions, providing a sufficient condition for minimal isometric immersions. We call a flat $n$ -torus $\mathbb{R}^{n}/\Lambda_{n}$ rational, if up to a dilation, the Gram matrix $Q$ in (1) of the lattice $\Lambda_{n}$ is rational (i.e., $Q\in GL(n,\mathbb{Q})$ ).

Theorem 1.

Up to a dilation, every flat rational $n$ -torus admits a minimal isometric immersion into some sphere.

The proof of Theorem 1 employs a classical result of Voronoï [43] in geometry of numbers. Unlike the $2$ -dimensional case, our explicit constructions demonstrate that the rationality condition is not necessary for the existence of minimal isometric immersions in higher dimensions. Example 4.1 and Examples 4.4 $\sim$ 4.6 demonstrate that minimal flat $3$ -tori in spheres may be quartic, cubic, or quadratic irrational (see Section 4 for precise definitions). In fact, these examples exhaust all possible degrees of field extensions over $\mathbb{Q}$ that can occur in the minimal isometric immersions of flat $3$ -tori into spheres.

Theorem 2.

Let $T^{n}=\mathbb{R}^{n}/\Lambda_{n}$ be a flat $n$ -torus, admitting a minimal isometric immersion into some sphere $\mathbb{S}^{m}$ . Denote by $K$ the field extension over $\mathbb{Q}$ generated by all ratios $q_{ij}/q_{11}$ of the Gram matrix $(q_{ij})$ of $\Lambda_{n}$ . Then the extension degree satisfies

[K:\mathbb{Q}]\leq(n-1)^{\lfloor{\frac{(n-1)(n+2)}{4}}\rfloor},

where $\lfloor\cdot\rfloor$ represents the integer part of a real number.

For the $3$ -dimensional case, the upper bound of extension degree established in Theorem 2 is equal to $4$ , which is sharp. Moreover, irrational minimal flat $3$ -tori exhibit a certain rigidity: they must be homogeneous, and when $[K:\mathbb{Q}]$ equals $3$ or $4$ , the minimal homothetic immersion of such a $3$ -torus is unique up to congruence. In particular, if such a $3$ -torus admits a minimal immersion into spheres via eigenfunctions corresponding to the $k$ -th eigenvalue, then it cannot admit a minimal homothetic immersion via eigenfunctions belonging to any other eigenvalues. Note that this property fails to hold for minimal homothetic immersions of spheres, flat $2$ -tori, or irrational flat $n$ -tori into spheres.

In contrast to the homogeneous nature of minimal flat $2$ -tori and minimal irrational flat $3$ -tori, we construct a family of rational flat $3$ -tori indexed by primes congruent to $1$ modulo $4$ , each of which admits a $13$ -parameter family of non-homogeneous minimal isometric immersions into spheres of dimension at most $23$ . Any such minimal immersion that is full in $\mathbb{S}^{23}$ is in fact an embedding. We also construct a non-homogeneous minimal isometric immersion into $\mathbb{S}^{7}$ whose image has self-intersections.

We further investigate the minimal target dimension for minimal isometric immersions of flat $n$ -tori, and obtain the following deformation theorem.

Theorem 3.

Let $x:T^{n}\triangleq\mathbb{R}^{n}/\Lambda_{n}\rightarrow{\mathbb{S}^{2N-1}}$ be a minimal flat $n$ -torus, where $2N$ is the dimension of the eigenspace of $x$ with respect to the eigenvalue $n$ . Then $x$ can be deformed by a homotopy of minimal isometric immersions into a homogeneous immersion in $\mathbb{S}^{p}\subset\mathbb{S}^{m}$ with $p<n^{2}+n$ .

Setting $n={2}$ in Theorem 3, we see that the minimal target dimension is $5$ for all flat $2$ -tori admitting minimal isometric immersions into spheres, except for the Clifford $2$ -torus and its covers.

The paper is organized as follows. In Section 2, we begin by reviewing the spectral theory of flat $n$ -tori and subsequently introduce our variational characterizations of their minimal isometric immersions. It is also shown in this section that the images of homogeneous minimal flat $n$ -tori in spheres are embedded. Section 3 focuses on minimal isometric immersions of rational flat $n$ -tori, including the proof of Theorem 1. In Section 4, we first construct several examples of irrational minimal flat $3$ -tori and establish Theorem 2 in the case of $n=3$ . Then we address the rigidity and homogeneity of irrational minimal flat $3$ -tori. Section 5 is devoted to the existence of non-homogeneous rational minimal flat $3$ -tori and to the study of their embeddedness. Finally, we establish Theorem 3, and give a proof to Theorem 2 for general $n$ in Section 6.

2 Preliminaries

In this section, we will recall the basic theory of flat $n$ -tori and the setup developed by us for minimal isometric immersions of flat $n$ -tori in [22], with particular emphasis on a variational characterization.

2.1 Flat tori and lattices

A flat torus $T^{n}$ of dimension $n$ can be described as

T^{n}=\mathbb{R}^{n}/{\Lambda_{n}},

where $\Lambda_{n}$ is a lattice of rank $n$ in $\mathbb{R}^{n}$ . Set $L_{n}$ to be a generator matrix of $\Lambda_{n}$ , which means $\Lambda_{n}$ can be generated by row vectors of $L_{n}$ . The Gram matrix of $\Lambda_{n}$ is then defined as

Q:=L_{n}L_{n}^{t}.

(1)

Two tori $T^{n}=\mathbb{R}^{n}/{\Lambda_{n}}$ and $\widetilde{T}^{n}=\mathbb{R}^{n}/{\widetilde{\Lambda}_{n}}$ are isometric if and only if $\Lambda_{n}$ and $\widetilde{\Lambda}_{n}$ are isometric, i.e., there exists an orthogonal matrix $O$ and a unimodular matrix $U\in SL(n,\mathbb{Z})$ , such that $L_{n}=U\,\widetilde{L}_{n}\,O$ , where $L_{n}$ (w.r.t. $\widetilde{L}_{n}$ ) is a generator matrix of lattice $\Lambda_{n}$ (w.r.t. $\widetilde{\Lambda}_{n}$ ). It follows that the moduli space of flat $n$ -tori is

SL(n,\mathbb{Z})\setminus GL(n,\mathbb{R})\,/\,O(n).

The dual lattice of $\Lambda_{n}$ is defined as the lattice $\Lambda_{n}^{*}$ generated by row vectors of the matrix ${L}_{n}^{*}=(L_{n}^{-1})^{t}$ . It is well known that the spectrum of $T^{n}=\mathbb{R}^{n}/{\Lambda_{n}}$ is given by

\mathrm{Spec}(T^{n})=\Big\{4\pi^{2}|\xi|^{2}\,\Big|\,\xi\in\Lambda_{n}^{*}\Big\},

and $e^{2\pi\langle\xi,u\rangle i}$ is an eigenfunction corresponding to the eigenvalue $4\pi^{2}|\xi|^{2}$ , where $u=(u_{1},u_{2},\cdots,u_{n})$ is the coordinates of $\mathbb{R}^{n}$ , such that the flat metric on $\mathbb{R}^{n}$ ( $T^{n}$ ) can be expressed as $du_{1}^{2}+du_{2}^{2}+\cdots+du_{n}^{2}$ .

For convenience, we introduce the following definition.

Definition 2.1.

Let $\mathbb{R}^{n}/\Lambda_{n}$ be a flat $n$ -torus with $Q=(q_{ij})$ the Gram matrix of the lattice $\Lambda_{n}$ . Consider the field extension $K$ of $\mathbb{Q}$ generated by the ratios $\left\{\frac{q_{ij}}{q_{11}}\mid 1\leq i,j\leq n\right\}$ .

(1)

If $K=\mathbb{Q}$ , we call $\mathbb{R}^{n}/\Lambda_{n}$ a rational flat $n$ -torus.
(2)

If the extension degree $[K:\mathbb{Q}]>1$ , we call $\mathbb{R}^{n}/\Lambda_{n}$ an irrational flat $n$ -torus.

2.2 Algebraic characterizations of minimal isometric immersions of flat $n$ -tori

Let $T^{n}=\mathbb{R}^{n}/{\Lambda_{n}}$ be a flat $n$ -torus, which admits a minimal isometric immersion in some sphere. Then $n$ is an eigenvalue of $T^{n}$ , whose eigenspace is assumed to be of dimension $2N$ . We choose the coordinates $u=(u_{1},u_{2},\cdots,u_{n})$ on $\mathbb{R}^{n}$ ( $T^{n}$ ) such that the induced metric can be expressed as

\frac{4\pi^{2}}{n}(du_{1}^{2}+du_{2}^{2}+\cdots+du_{n}^{2}).

It follows that there are exactly $N$ distinct lattice vectors (up to $\pm 1$ ) having the length $1$ in the dual lattice $\Lambda_{n}^{*}$ . We denote them by

\xi_{1},\xi_{2},\cdots,\xi_{N}.

By the theorem of Takahashi [34], a minimal isometric immersion of $T^{n}$ in spheres can be determined by some $2N\times 2N$ matrix $A$ as follows:

x=\begin{pmatrix}\Theta_{1}&\Theta_{2}&\cdots&\Theta_{N}\end{pmatrix}A:T^{n}=\mathbb{R}^{n}/{\Lambda_{n}}\longrightarrow\mathbb{S}^{2N-1},

(2)

where $\Theta_{r}=\begin{pmatrix}\cos\theta_{r}&\sin\theta_{r}\end{pmatrix},$ $\theta_{r}=、2\pi\langle\xi_{r},u\rangle$ for $~1\leq r\leq N$ .

As in [22], we write

AA^{t}=\begin{pmatrix}A_{11}&A_{12}&\cdots&A_{1N}\\ A_{21}&A_{22}&\cdots&A_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ A_{N1}&A_{N2}&\cdots&A_{NN}\end{pmatrix},~~~A_{rs}=\begin{pmatrix}A_{rs}^{11}&A_{rs}^{12}\\ A_{rs}^{21}&A_{rs}^{22}\end{pmatrix},

(3)

and define $\mathcal{E}=\{\xi_{r}\pm\xi_{s},1\leq r<s\leq{N}\}$ .

Definition 2.2.

Given $\eta\in\mathcal{E}$ , we call the set of pairs

\{(\xi_{r_{1}},\xi_{s_{1}}),\cdots,(\xi_{r_{p}},\xi_{s_{p}}),(\xi_{r_{p+1}},-\xi_{s_{p+1}}),\cdots,(\xi_{r_{q}},-\xi_{s_{q}})\}

an $\eta$ -set if $r_{i}<s_{i}$ and the relations

\xi_{r_{1}}+\xi_{s_{1}}=\cdots=\xi_{r_{p}}+\xi_{s_{p}}=\xi_{r_{p+1}}-\xi_{s_{p+1}}=\cdots=\xi_{r_{q}}-\xi_{s_{q}}=\eta

collectively exhaust all possible realizations of $\eta$ .

It is straightforward to verify that for a given $\eta$ -set, the vectors $\pm\xi_{r_{1}},\cdots,\pm\xi_{r_{q}},\pm\xi_{s_{1}},\cdots,\pm\xi_{s_{q}}$ involved in it are distinct with each other.

It follows from the proof of Lemma 2.2 in [22] that

A_{rr}=\begin{pmatrix}a_{r}&\\ &a_{r}\end{pmatrix},

(4)

and the condition $|x|=1$ is equivalent to

	$\displaystyle\sum_{\xi_{r}+\xi_{s}=\eta}(A_{rs}^{11}-A_{rs}^{22})+\sum_{\xi_{r}-\xi_{s}=\eta}(A_{rs}^{11}+A_{rs}^{22})=0,~~~\forall\eta\in\mathcal{E},$		(5)
	$\displaystyle\sum_{\xi_{r}+\xi_{s}=\eta}(A_{rs}^{12}+A_{rs}^{21})+\sum_{\xi_{r}-\xi_{s}=\eta}(A_{rs}^{12}-A_{rs}^{21})=0,~~~\forall\eta\in\mathcal{E}.$		(6)

Furthermore, the proof of Lemma 2.3 in [22] implies that $x$ is an isometric immersion if and only if

\sum_{r=1}^{N}a_{r}\xi_{r}\xi_{r}^{t}=\frac{I_{n}}{n},

(7)

	$\displaystyle\sum_{\xi_{r}+\xi_{s}=\eta}(A_{rs}^{11}-A_{rs}^{22})(\xi_{r}\xi_{s}^{t}+\xi_{s}\xi_{r}^{t})+\sum_{\xi_{r}-\xi_{s}=\eta}(A_{rs}^{11}+A_{rs}^{22})(\xi_{r}\xi_{s}^{t}+\xi_{s}\xi_{r}^{t})=O,~~~\forall\eta\in\mathcal{E},$		(8)
	$\displaystyle\sum_{\xi_{r}+\xi_{s}=\eta}(A_{rs}^{12}+A_{rs}^{21})(\xi_{r}\xi_{s}^{t}+\xi_{s}\xi_{r}^{t})+\sum_{\xi_{r}-\xi_{s}=\eta}(A_{rs}^{12}-A_{rs}^{21})(\xi_{r}\xi_{s}^{t}+\xi_{s}\xi_{r}^{t})=O,~~~\forall\eta\in\mathcal{E},$		(9)

where $O$ denotes the $n\times n$ null matrix.

Note that for a given $n$ -torus $T^{n}=\mathbb{R}^{n}/\Lambda_{n}$ , a solution of (5) $\sim$ (9) will provide a matrix of $2N\times 2N$ . If it is positive semi-definite, then its square root gives us a minimal isometric immersion of $T^{n}$ in $\mathbb{S}^{2N-1}$ as in (2). In fact, solutions of (5) $\sim$ (9) parameterize the moduli space $\mathcal{M}(T^{n})$ of all minimal isometric immersions of $T^{n}$ in $\mathbb{S}^{2N-1}$ , which is obviously a convex set in $\mathrm{Sym}_{2N}$ if it is nonempty. Similar description for the minimal isometric immersions of isotropic irreducible Riemannian homogeneous space into spheres has been obtained (see do Carmo-Wallach [9], Li [20], Wang-Ziller [45], Toth [40] and references theirin), so has been for the eigenmaps of $n$ -tori into spheres (see Park-Urakawa [31]). However, the difficulty here is that for general $n$ -tori, the equations (5) $\sim$ (9) may have no solutions, as shown in the case of dimension $2$ by Bryant [2].

It is easy to see that (5),(6),(8) and (9) form a homogeneous linear system. So any non-trivial solution can be deformed continuously into the trivial solution. This indicates that the key equation is (7), which characterizes the existence of homogeneous minimal isometric immersions of $T^{n}$ into $\mathbb{S}^{2N-1}$ . This yields the following conclusion.

Proposition 2.3.

A flat $n$ -torus can be minimally and isometrically immersed into spheres if and only if it admits a homogeneous, minimal isometric immersion into some sphere.

At the end of this subsection, we fix a notational convention: for a full minimal isometric immersion $x:\mathbb{R}^{n}/\Lambda_{n}\rightarrow\mathbb{S}^{m}$ expressed in the form (2), the lattice vectors appearing in its coordinates may correspond only to a subset (not necessarily all) of the lattice vectors of length $1$ . These vectors will still be denoted by

\xi_{1},\xi_{2},\dots,\xi_{N}

throughout the subsequent discussion.

2.3 A variational characterization

In [22], the solving of (7) is analyzed by us from a variational perspective. We provide a brief overview of it here. Let $\{\eta_{1},\eta_{2},\cdots,\eta_{n}\}$ be a generator of the dual lattice $\Lambda_{n}^{*}$ , and $Q$ be the Gram matrix of $\Lambda_{n}^{*}$ with respect to this generator. Then there exist integers $a_{j_{k}}$ such that

\xi_{j}=a_{j_{1}}\eta_{1}+a_{j_{2}}\eta_{2}+\cdots\cdots+a_{j_{n}}\eta_{n},\quad 1\leq j\leq N.

Set

Y_{j}=(a_{j_{1}},a_{j_{2}},\cdots\cdots,a_{j_{n}})^{t},\quad Y=(Y_{1},\cdots,Y_{N}).\quad 1\leq j\leq N.

We call $\{Q,Y\}$ a matrix data of $x$ .

In terms of the matrix data, the condition $|\xi_{j}|=1$ is equivalent to

Y_{j}^{t}QY_{j}=1,

(10)

i.e., $Y_{j}$ lies on the hyper-ellipsoid $\mathcal{Q}$ determined by

\begin{pmatrix}u_{1}\!&u_{2}\!&\cdots\!&u_{n}\end{pmatrix}Q\begin{pmatrix}u_{1}\\ u_{2}\\ \vdots\\ u_{n}\end{pmatrix}=1.

The equation (7) is equivalent to

c_{1}^{2}Y_{1}Y_{1}^{t}+c_{2}^{2}Y_{2}Y_{2}^{t}+\cdots\cdots+c_{N}^{2}Y_{N}Y_{N}^{t}=\frac{1}{n}Q^{-1},

(11)

i.e., $\frac{Q^{-1}}{n}$ lies in the convex hull spanned by $\{Y_{j}Y_{j}^{t}\}_{j=1}^{N}$ . We consider the space $\mathrm{Sym}_{n}$ of $n\times n$ symmetric matrices over $\mathbb{R}$ , and equip it with the inner product:

\langle S_{1},S_{2}\rangle=\operatorname{tr}(S_{1}S_{2}),\quad S_{1},S_{2}\in\mathrm{Sym}_{n}.

Let $\Sigma$ (resp. $\Sigma_{+}$ ) be the set of semi-positive (resp. positive) definite matrices, which forms a close (resp. open) cone in $\mathrm{Sym}_{n}$ . Given a subset $X\subset\mathbb{Z}^{n}$ , we denote by $C_{X}$ the convex hull spanned by $AA^{t}$ for all $A\in X$ , and define

W_{X}\triangleq\{M\,|\,M\in\Sigma_{+},\langle M,AA^{t}\rangle=1\},

whose geometric meaning is the set of all hyper-ellipsoids passing through every point in $X$ . With the notations established, we present the variational characterization obtained in [22].

Theorem 2.4.

Let $x:T^{n}=\mathbb{R}^{n}/\Lambda_{n}\rightarrow\mathbb{S}^{m}$ be a minimal flat torus, and $\{Q,Y\}$ be the matrix data of $x$ . Then $\frac{{Q}^{-1}}{n}$ is a maximum point of the determinant function restricted on $C_{Y}$ , and $Q$ is a maximum point of the determinant function restricted on $W_{Y}$ .

Conversely, given a finite set $X\subset\mathbb{Z}^{n}$ such that $C_{X}\cap\Sigma_{+}\neq\varnothing$ , if $P\in\overset{\circ}{C_{X}}$ is a critical point of the determinant function restricted on $C_{X}$ , then the torus $\mathbb{R}^{n}/\Lambda_{n}$ determined by $nP$ (as the Gram matrix of $\Lambda_{n}$ ) admits a minimal isometric immersion in some sphere.

Remark 2.5.

For any given integer set $X$ , the critical point of the determinant function restricted on $C_{X}$ is determined by a system of algebraic equations with integer coefficients, which implies flat $n$ -tori that admit minimal isometric immersions into spheres have the property that the entries of their Gram matrices are algebraic numbers. Thus, a generic flat $n$ -torus cannot be minimally, isometrically immersed in any spheres. When $n=2$ , the aforementioned algebraic equations are all linear. This implies, as asserted by Bryant in [2], that only rational flat $2$ -tori could admit minimal isometric immersions into spheres.

Remark 2.6.

Let $\{\eta_{1}^{*},\eta_{2}^{*},\cdots,\eta_{n}^{*}\}$ be the dual frame of $\{\eta_{1},\eta_{2},\cdots,\eta_{n}\}$ . It forms a generator of $\Lambda_{n}$ . Let $\mathbf{u}=(\mathrm{u}_{1},\mathrm{u}_{2},\cdots,\mathrm{u}_{n})$ be the coordinate of $u\in\mathbb{R}^{n}$ with respect to $\{\eta_{1}^{*},\eta_{2}^{*},\cdots,\eta_{n}^{*}\}$ . Then the function $\theta_{i}$ defined in (2) has an explicit expression

\theta_{i}=2\pi\langle\xi_{i},u\rangle=2\pi\langle(\eta_{1},\cdots,\eta_{n})Y_{i},(\eta_{1}^{*},\cdots,\eta_{n}^{*})\mathbf{u}^{t}\rangle=2\pi\langle Y_{i},\mathbf{u}^{t}\rangle.

In terms of the matrix data $\{Q,Y\}$ , a homogeneous minimal immersion $x$ is embedded if and only if the image of the map

\begin{split}\mathcal{Y}:\Omega\triangleq\{&(\mathrm{u}_{1},\mathrm{u}_{2},\cdots,\mathrm{u}_{n})\;|\;0\leq\mathrm{u}_{i}<1\}\longrightarrow\mathbb{R}^{N}\\ &~~~~~~~\mathbf{u}=(\mathrm{u}_{1},\mathrm{u}_{2},\cdots,\mathrm{u}_{n})~~~~~\mapsto~~\frac{1}{2\pi}(\theta_{1},\theta_{2},\cdots,\theta_{N})=(\mathrm{u}_{1},\mathrm{u}_{2},\cdots,\mathrm{u}_{n})Y\end{split}

(12)

intersects $\mathbb{Z}^{N}$ solely at the origin, or equivalently, $\mathcal{Y}(\mathbf{u})$ is never a nonzero integer vector for any $\mathbf{u}\in\{(\mathrm{u}_{1},\mathrm{u}_{2},\cdots,\mathrm{u}_{n})\;|\;|\mathrm{u}_{i}|<1\}$ . When $N=n$ , by the first Minkowski’s Convex Body Theorem [33], this condition is equivalent to the determinant constraint $\det(Y)=\pm 1$ (a similar characterization appeared in [30, 31]). When $N>n$ , if one of the $n\times n$ minor of $Y$ is equal to $\pm 1$ , then $x$ is embedded. In fact, without loss of generality, we may assume the minors given by $\{Y_{1},Y_{2},\cdots,Y_{n}\}$ is equal to $\pm 1$ . Then the first Minkowski’s Convex Body Theorem [33] implies that the origin is the unique point in $\Omega$ such that $\frac{\theta_{1}}{2\pi},\frac{\theta_{2}}{2\pi},\cdots,\frac{\theta_{n}}{2\pi}$ are integers; hence the origin is the unique integer point in $\mathcal{Y}(\Omega)$ .

Proposition 2.7.

Any homogeneous minimal isometric immersion of a flat $n$ -torus $T^{n}$ into a sphere can be realized as a minimal isometric embedding of a flat $n$ -torus $\widetilde{T}^{n}$ which is covered by $T^{n}$ .

Proof.

Let $x:T^{n}=\mathbb{R}^{n}/\Lambda_{n}\rightarrow\mathbb{S}^{N}$ be a homogeneous minimal flat $n$ -torus with $\{Q,Y\}$ as its matrix data.

Note that the map defined in (12) can be extended to a linear map, still denoted by $\mathcal{Y}$ , from $\mathbb{R}^{n}$ to $\mathbb{R}^{N}$ . Since $Y$ is an integer matrix with $\operatorname{rank}(Y)=n$ , the map $\mathcal{Y}:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{N}$ is an injective group homomorphism with respect to the addition group structure of $\mathbb{R}^{n}$ and $\mathbb{R}^{N}$ , and its image $\mathcal{Y}(\mathbb{Z}^{n})$ forms a sublattice of $\mathbb{Z}^{N}$ of rank $n$ . Consequently, the preimage

L_{n}\triangleq\mathcal{Y}^{-1}(\mathbb{Z}^{N})

is itself a lattice of rank $n$ , with $\mathbb{Z}^{n}$ as its sublattice.

If $\Omega\cap(L_{n}\setminus\{\mathbf{0}\})=\emptyset$ , then by Remark 2.6 the map $x\colon T^{n}\to\mathbb{S}^{N}$ is already a minimal embedding, completing the proof. Otherwise, there exists a generator $\{\mathbf{v}_{1},\mathbf{v}_{2},\dots,\mathbf{v}_{n}\}$ of $L_{n}$ such that each $\mathbf{v}_{i}$ lies in $\overline{\Omega}$ and at least one belongs to $\Omega$ . Consider the generator matrix $V$ of $L_{n}$ formed by $\{\mathbf{v}_{1},\mathbf{v}_{2},\dots,\mathbf{v}_{n}\}$ as row vectors. Since $\mathbb{Z}^{n}$ is a sublattice of $L_{n}$ , the dual lattice $L_{n}^{*}$ is a sublattice of $(\mathbb{Z}^{n})^{*}=\mathbb{Z}^{n}$ . Therefore, $\mathrm{V}^{-1}$ is an integer matrix.

Set

\widetilde{Y}\triangleq\mathrm{V}Y,\quad\quad\quad\widetilde{Q}\triangleq(\mathrm{V}^{-1})^{t}\,Q\,\mathrm{V}^{-1}.

By definition, every column of $\widetilde{Y}$ is an integer vector. Denote by $\widetilde{\Lambda}_{n}$ the lattice with $\widetilde{Q}^{-1}$ as its Gram matrix, and let $\widetilde{T}^{n}\triangleq\mathbb{R}^{n}/\widetilde{\Lambda}_{n}$ be the associated flat $n$ -torus. It follows that $T^{n}$ is a Riemannian covering of $\widetilde{T}^{n}$ .

Set $\widetilde{\mathbf{u}}=(\widetilde{\mathrm{u}}_{1},\widetilde{\mathrm{u}}_{2},\cdots,\widetilde{\mathrm{u}}_{n})\triangleq\mathbf{u}\mathrm{V}^{-1}$ . By (12), we have

\theta_{i}=2\pi\langle Y_{i},\mathbf{u}^{t}\rangle=2\pi\langle\widetilde{Y_{i}},\widetilde{\mathbf{u}}^{t}\rangle,

which implies that the map $x$ induces a minimal isometric immersion of $\widetilde{T}^{n}$ into $\mathbb{S}^{N}$ . Note that the image of the map

\begin{split}\widetilde{\mathcal{Y}}:\widetilde{\Omega}\triangleq\{&(\widetilde{\mathrm{u}}_{1},\widetilde{\mathrm{u}}_{2},\cdots,\widetilde{\mathrm{u}}_{n})\;|\;0\leq\widetilde{\mathrm{u}}_{i}<1\}\longrightarrow\mathbb{R}^{N}\\ &~~~~~~~\widetilde{\mathbf{u}}=(\widetilde{\mathrm{u}}_{1},\widetilde{\mathrm{u}}_{2},\cdots,\widetilde{\mathrm{u}}_{n})~~~~~\mapsto~~\frac{1}{2\pi}(\theta_{1},\theta_{2},\cdots,\theta_{N})=(\widetilde{\mathrm{u}}_{1},\widetilde{\mathrm{u}}_{2},\cdots,\widetilde{\mathrm{u}}_{n})\widetilde{Y}\end{split}

is exactly that of $\mathcal{Y}$ restricted on the parallel polyhedron spanned by $\{\mathbf{v}_{1},\mathbf{v}_{2},\dots,\mathbf{v}_{n}\}$ . Then, the generator property of $\{\mathbf{v}_{1},\mathbf{v}_{2},\dots,\mathbf{v}_{n}\}$ ensures that $\widetilde{\mathcal{Y}}(\widetilde{\Omega})$ contains no integer vector but zero. Consequently, the minimal isometric immersion $x:\widetilde{T}^{n}\rightarrow\mathbb{S}^{N}$ is an embedding by Remark 2.6. ∎

Because every minimal flat $2$ -torus is homogeneous, the following holds.

Corollary 2.8.

Every minimal isometric immersion of a flat $2$ -torus $T^{2}$ into a sphere factors through a minimal isometric embedding of a flat $2$ -torus $\widetilde{T}^{2}$ that is finitely covered by $T^{2}$ .

3 Minimal isometric immersions of rational flat $n$ -tori into spheres

Let $\mathbb{R}^{n}/\Lambda_{n}$ be a rational flat $n$ -torus (see Definition 2.1). Up to a dilation, we may assume that the Gram matrix of $\Lambda_{n}$ lies in $GL(n,\mathbb{Q})$ , i.e., all its entries are rational numbers. Note that this condition is equivalent to requiring that the Gram matrix of the dual lattice $\Lambda_{n}^{*}$ also belongs to $GL(n,\mathbb{Q})$ .

Given a positive symmetric matrix $Q\in GL(n,\mathbb{R})$ , we use the notation $\mathcal{Q}$ to denote the hyper-ellipsoid determined by $u^{t}\,Q\,u=1$ .

Lemma 3.1.

Let $Q\in GL(n,\mathbb{Q})$ be a positive definite symmetric matrix. Then up to a dilation, the rational points on the hyper-ellipsoid $\mathcal{Q}$ are dense.

Proof.

First, note that there exists a point $u_{0}$ on the hyper-ellipsoid $\mathcal{Q}$ and a nonzero real number $\lambda$ such that $\frac{u_{0}}{\lambda}$ is a rational point. In fact, one can consider a line passing through the origin with a direction determined by a rational vector. The intersection of this line with $\mathcal{Q}$ is nonempty.

Let $\Pi$ be a coordinate hyperplane that not containing the point $u_{0}$ . For any given point $u^{\prime}\in\Pi$ , the line passing through $u_{0}$ and $u^{\prime}$ intersects the hyper-ellipsoid $\mathcal{Q}$ at the point

u=u_{0}-2\frac{u_{0}Q(u^{\prime}-u_{0})^{t}}{(u^{\prime}-u_{0})Q(u^{\prime}-u_{0})^{t}}\,(u^{\prime}-u_{0}).

Then the conclusion follows from the fact that the points set $\{u^{\prime}\in\Pi\;|\;\frac{u^{\prime}}{\lambda}\text{is~rational}\}$ is dense in $\Pi$ . ∎

Lemma 3.2.

Let $Q$ be a positive definite symmetric matrix. Then there exist $N=\frac{n(n+1)}{2}$ points $\{Y_{1},\cdots,Y_{N}\}$ on the hyper-ellipsoid $\mathcal{Q}$ such that $\{Y_{1}Y_{1}^{t},Y_{2}Y_{2}^{t},\cdots Y_{N}Y_{N}^{t}\}$ spans the whole space $\mathrm{Sym}_{n}$ .

Proof.

Such points can be easily found by considering the intersection of rays directed by $\{e_{j}+e_{k}\;|\;1\leq j\leq k\leq n\}$ with the hyper-ellipsoid $\mathcal{Q}$ . ∎

In geometry of number, for a given matrix (quadratic form) $Q\in\Sigma_{+}$ the Voronoï domain $\mathcal{V}(Q)$ is defined as

\mathcal{V}(Q):=\hbox{ the cone spanned by }\{uu^{t}\;|\;u\in\mathrm{Min}(Q)\},

where $\mathrm{Min}(Q)$ denotes the set of shortest vectors of the lattice determined by $Q$ . $Q$ is called a perfect form if the rank of $\{uu^{t}\;|\;u\in\mathrm{Min}(Q)\}$ equals $\frac{n(n+1)}{2}$ . In [43], Voronoï established the finiteness of $n$ -dimensional perfect forms modulo $GL(n,\mathbb{Z})$ -equivalence and derived the following result.

Lemma 3.3.

The cone $\Sigma_{+}\subset\mathrm{Sym}_{n}$ of positive definite matrices can be covered by the Voronoï domains of the $n$ -dimensional perfect forms, i.e.,

\Sigma_{+}\subset\bigcup_{\begin{array}[]{c}Q_{p}\;\rm{is~perfect}\\ \lambda_{1}(Q_{p})=1\end{array}}\mathcal{V}(Q_{p}),

where $\lambda_{1}(Q_{p})$ denotes the shortest length of vectors in the lattice determined by $Q_{p}$ .

Theorem 3.4.

Up to a dilation, every rational flat $n$ -torus admits a minimal isometric immersion into some spheres.

Proof.

Let $\mathbb{R}^{n}/\Lambda_{n}^{*}$ be a rational flat $n$ -torus, and $Q\in GL(n,\mathbb{Q})$ be a Gram matrix of $\Lambda_{n}$ with respect to some generator. Assume that $Y_{1},\cdots,Y_{N}$ are points on the hyper-ellipsoid $\mathcal{Q}$ satisfying the property of Lemma 3.2.

In the space $\mathrm{Sym}_{n}$ , we consider the affine hyperplane $\Pi_{Q}$ defined by

\Pi_{Q}\triangleq\{P\;|\;\langle P,Q\rangle=1\}.

Note that all of $Y_{1}Y_{1}^{t},Y_{2}Y_{2}^{t},\cdots,Y_{N}Y_{N}^{t}$ belong to $\Pi_{Q}$ , and from Lemma 3.2, we have

\Pi_{Q}=\{\lambda_{1}Y_{1}Y_{1}^{t}+\lambda_{2}Y_{2}Y_{2}^{t}+\cdots+\lambda_{N}Y_{N}Y_{N}^{t}\;|\;\lambda_{1},\lambda_{2},\cdots,\lambda_{N}\in\mathbb{R},\lambda_{1}+\lambda_{2}+\cdots+\lambda_{N}=1\}.

Obviously, $\frac{Q^{-1}}{n}$ is contained in $\Pi_{Q}$ , so $\Pi_{Q}\cap\Sigma_{+}\not=\emptyset$ . Since $\ln\circ\det$ on $\Sigma_{+}$ is strictly concave, the maximum point exists and is unique on $\Pi_{Q}\cap\Sigma_{+}$ for the determinant function $\det$ . We denote it by $Q_{1}$ . It follows from the proof of Theorem 3.6 in [22] that

\langle\frac{Q_{1}^{-1}}{n},Y_{i}Y_{i}^{t}\rangle=1,~~~1\leq i\leq N.

Using Lemma 3.2 again, we obtain that $Q_{1}=\frac{Q^{-1}}{n}$ .

By Lemma 3.3, $Q^{-1}/n$ is contained in some Voronoï domain $\mathcal{V}(Q_{p})$ with vertices $u_{i}u_{i}^{t}$ ( $1\leq i\leq N)$ . However, it’s location could be on the boundary of $\mathcal{V}(Q_{p})$ . To proceed, we need to find an open cone containing $Q^{-1}/n$ . Because $Q^{-1}/n$ is positive-definite, there exists $\epsilon>0$ so small that $Q^{*}=Q^{-1}/n-\epsilon\sum_{i=1}^{N}u_{i}u_{i}^{t}$ is still positive-definite. Analogously, $Q^{*}$ is also contained in some Voronoï domain. We may assume that

Q^{*}=\sum_{i=1}^{k}c_{i}v_{i}{v_{i}}^{t},\quad c_{i}>0,~~k\leq N,

which leads to

\frac{Q^{-1}}{n}=\sum_{i=1}^{k}c_{i}v_{i}{v_{i}}^{t}+\epsilon\sum_{i=1}^{N}u_{i}u_{i}^{t},

i.e., $\frac{Q^{-1}}{n}$ is contained inside the polytope with vertices $v_{i}{v_{i}}^{t}$ and $u_{j}u_{j}^{t}$ . Since $v_{i}^{t}Qv_{i}$ and $u_{j}^{t}Qu_{j}$ are all positive, we may assume that all of them equal $1$ after performing the necessary rescaling on $v_{i}$ and $u_{j}$ , which implies that all these points lie on the hyper-ellipsoid $\mathcal{Q}$ . From Lemma 3.1 and the rationality assumption of $Q$ , after applying a certain dilation if necessary, there are finite rational points $R_{1},R_{2},\cdots,R_{N+k}$ on $\mathcal{Q}$ so close to $v_{i}$ and $u_{j}$ that $\frac{Q^{-1}}{n}$ can be expressed as a convex combination of

\{R_{1}R_{1}^{t},R_{2}R_{2}^{t},\cdots,R_{N+k}R_{N+k}^{t}\}.

Multiplying a suitable integer $\mu$ to $R_{1},\cdots,R_{N+k}$ , we obtain a set

Y=\{\mu R_{1},\mu R_{2},\cdots,\mu{R}_{N+k}\}\subset\mathbb{Z}^{n}.

Then $\{Q/\mu^{2},Y\}$ forms a matrix data satisfying the assumption of Theorem 2.4; hence it provides a minimal isometric immersion of $\mathbb{R}^{n}/\mu\Lambda_{n}^{*}$ .

∎

In the above proof, the density property implies there are infinitely many choices of rational points $\{R_{1},\cdots,R_{N+k}\}$ , and so there exist infinitely many choices of $\mu$ and $Y$ in the construction of matrix data. As a consequence, we derive the following proposition, which extends another result previously asserted by Bryant for the $2$ -torus in [2].

Proposition 3.5.

For every rational flat $n$ -torus $T^{n}$ , there exist infinitely many positive integers $k$ so that $T^{n}$ can be minimally immersed into spheres using the $k$ -th eigenfunctions.

Remark 3.6.

Given a minimal rational flat $n$ -torus $x:T_{1}^{n}=\mathbb{R}^{n}/\Lambda_{n}\rightarrow\mathbb{S}^{m}$ with $\{Q,Y\}$ as its matrix data. The standard congruence transformation shows the existence of a rational matrix $B$ , such that $\det B=\pm 1$ and $D\triangleq B^{t}QB$ is diagonal. Decompose $B^{-1}$ as $B^{-1}=F^{-1}\widetilde{B}$ , where $F$ is diagonal and $\widetilde{B},F\in M(n,\mathbb{Z})$ . Consider the flat $n$ -torus $T_{2}^{n}\triangleq\mathbb{R}^{n}/\tilde{\Lambda}_{n}$ , where $\tilde{\Lambda}_{n}$ is an orthotope lattice taking $D^{-1}F^{2}$ as its Gram matrix. Then the matrix data $\{D(F^{-1})^{2},Y\}$ gives a minimal immersion of $\tilde{x}:T_{2}^{n}\rightarrow\mathbb{S}^{m}$ . It is easy to see that $\tilde{\Lambda}_{n}$ is a sublattice of ${\Lambda}_{n}$ , which implies $\tilde{x}$ is a covering of $x$ . Therefore, for every minimal rational flat $n$ -torus in spheres, its image can be viewed as the result of a minimal immersion of an orthotope flat $n$ -torus.

4 Minimal isometric immersions of flat $3$ -tori

In this section, we focus on the minimal isometric immersions (embeddings) of flat $3$ -tori into spheres. For irrational cases, we establish an upper bound on the algebraic irrationality degree and demonstrate its optimality through explicit examples. Furthermore, we show all irrational minimal flat 3-tori must be homogeneous, while an explicit non-homogeneous example in the rational case is provided.

4.1 Irrational minimal flat $3$ -tori

Note that no irrational flat $2$ -torus admits minimal isometric immersions into spheres. Our work in [22] showed that this remains true for $\lambda_{1}$ -minimal isometric immersions of irrational $3$ -tori. In this subsection, we consider the general minimal isometric immersions of irrational flat $3$ -tori.

For an irrational flat $n$ -torus $\mathbb{R}^{n}/\Lambda_{n}$ , we call it quadratic irrational (resp. cubic irrational, quartic irrational) if the field extension degree $[K:\mathbb{Q}]$ (see Definition 2.1) equals $2$ (resp. $3$ , $4$ ).

We first introduce an approach to construct minimal isometric immersions of quadratic irrational flat $3$ -tori via Theorem 2.4.

We begin by selecting a set $Y=\{Y_{1},Y_{2},\cdots,Y_{N}\}\subset\mathbb{Z}^{3}$ such that

\mathrm{rank}\,{Y}=3,~~~~~\mathrm{rank}\{Y_{1}Y_{1}^{t},Y_{2}Y_{2}^{t},\cdots,Y_{N}Y_{N}^{t}\}=5,

Note that in $\mathrm{Sym}_{3}$ , the line defined by the constraints

\langle Q,Y_{j}Y_{j}^{t}\rangle=1,~~~1\leq j\leq N

intersects the hyperplane $\mathrm{Span}\{Y_{1}Y_{1}^{t},Y_{2}Y_{2}^{t},\cdots,Y_{N}Y_{N}^{t}\}$ at a unique point, which is denoted by $Q_{0}$ . Since such $Q_{0}$ is completely determined by the linear equations

\langle\sum_{k=1}^{N}c_{k}Y_{k}Y_{k}^{t},Y_{j}Y_{j}^{t}\rangle=1,~~~1\leq j\leq N,

it is obviously rational.

Next, we choose a rational matrix on the line

\langle Q,Y_{j}Y_{j}^{t}\rangle=0,~~~1\leq j\leq N,

and denote it by $Q_{1}$ . Then the hyper-ellipsoids passing through all points in $Y$ constitute a line segment, which can be parameterized by $W_{Y}=\{Q_{0}+t\,Q_{1}\;|\;t\in(a,b)\}$ . If necessary, we rechoose $Q_{0}$ as $Q_{0}+t_{0}Q_{1}$ for some rational $t_{0}\in(a,b)$ so that $Q_{0}$ is non-degenerate.

Then, we consider the maximal point of $\det$ restricted on $W_{Y}$ . Observe that $\det(Q_{0}+t\,Q_{1})$ , as a polynomial in $t$ , has degree at most $3$ . Consequently, the point $t_{0}$ maximizing the determinant function $\det$ on $W_{Y}$ belongs to $GL(n,\mathbb{Q}(w))$ , where $\mathbb{Q}(w)$ is a quadratic extension of $\mathbb{Q}$ . This maximal point is not rational if and only if the derivative of

\det(Q_{0}+t\,Q_{1})=\det Q_{0}\left(1+\operatorname{tr}(Q_{0}^{-1}Q_{1})\,t+\frac{\operatorname{tr}(Q_{0}^{-1}Q_{1})^{2}-\operatorname{tr}(Q_{0}^{-1}Q_{1}Q_{0}^{-1}Q_{1})}{2}\,t^{2}+\det(Q_{0}^{-1}Q_{1})\,t^{3}\right)

with respect to $t$ has no rational zeros, which is equivalent to say that

\left(\operatorname{tr}(Q_{0}^{-1}Q_{1})^{2}-\operatorname{tr}(Q_{0}^{-1}Q_{1}Q_{0}^{-1}Q_{1})\right)^{2}-12\det(Q_{0}^{-1}Q_{1})\operatorname{tr}(Q_{0}^{-1}Q_{1})

is not the square of some rational number.

Finally, to obtain the desired minimal isometric immersions into spheres we need to verify whether the inverse of $3(Q_{0}+t_{0}\,Q_{1})$ lies in the convex hull $C_{Y}$ ; that is, to studying the existence of non-negative coefficients $\{c_{1}^{2},\cdots,c_{N}^{2}\}$ such that

\frac{(Q_{0}+t_{0}Q_{1})^{-1}}{3}=c_{1}^{2}Y_{1}Y_{1}^{t}+c_{2}^{2}Y_{2}Y_{2}^{t}+\cdots\cdots+c_{N}^{2}Y_{N}Y_{N}^{t},

(13)

and

c_{1}^{2}+c_{2}^{2}+\cdots+c_{N}^{2}=1.

Example 4.1.

A quadratic irrational minimal flat $3$ -torus in $\mathbb{S}^{9}$ .

Consider the integer set

Y=\left(\begin{array}[]{ccccc}1&0&0&6&6\\ 0&1&0&12&9\\ 0&0&1&-15&-12\\ \end{array}\right).

By the above approach, it is straightforward to calculate that

Q_{0}=\left(\begin{array}[]{ccc}1&-\frac{343}{1233}&\frac{397}{1233}\\ -\frac{343}{1233}&1&\frac{1048}{1233}\\ \frac{397}{1233}&\frac{1048}{1233}&1\\ \end{array}\right),~~~Q_{1}=\left(\begin{array}[]{ccc}0&10&6\\ 10&0&1\\ 6&1&0\\ \end{array}\right),

and the maximal point of $\det$ on $W_{Y}=\{Q_{0}+t\,Q_{1}\;|\;t\in\mathbb{R}\}\cap\Sigma_{+}$ is given by

t_{0}=\frac{39337-137\sqrt{10801}}{443880}.

Moreover, the equation (13) can be solved by choosing $(c_{1}^{2},c_{2}^{2},c_{3}^{2},c_{4}^{2},c_{5}^{2})$ to be

\left(\frac{3\left(12773-107\sqrt{10801}\right)}{48040},\frac{27\left(1105-7\sqrt{10801}\right)}{38432},\frac{3\left(541\sqrt{10801}-52459\right)}{192160},\frac{121721+481\sqrt{10801}}{576480},\frac{7\left(191\sqrt{10801}+2791\right)}{576480}\right).

Then, we obtain a minimal embedding of a flat irrational $3$ -torus in $\mathbb{S}^{9}$ , whose matrix data is given by

Q=\left(\begin{array}[]{ccc}1&\frac{39337-137\sqrt{10801}}{44388}-\frac{343}{1233}&\frac{39337-137\sqrt{10801}}{73980}+\frac{397}{1233}\\ \frac{39337-137\sqrt{10801}}{44388}-\frac{343}{1233}&1&\frac{39337-137\sqrt{10801}}{443880}+\frac{1048}{1233}\\ \frac{39337-137\sqrt{10801}}{73980}+\frac{397}{1233}&\frac{39337-137\sqrt{10801}}{443880}+\frac{1048}{1233}&1\\ \end{array}\right),

Y=\left(\begin{array}[]{ccccc}1&0&0&6&6\\ 0&1&0&12&9\\ 0&0&1&-15&-12\\ \end{array}\right).

The embeddedness property follows directly from Remark 2.6.

Next, we establish an upper bound for the algebraic irrationality degree of minimal flat $3$ -tori.

Theorem 4.2.

Let $T^{3}=\mathbb{R}^{3}/\Lambda_{3}$ be a flat $3$ -torus. If it admits a minimal isometric immersion into some sphere $\mathbb{S}^{m}$ , then the Gram matrix of $\Lambda_{3}$ lies in $GL({3},\mathbb{Q}(w))$ , and the extension degree of $\mathbb{Q}(w)/\mathbb{Q}$ is at most $4$ . Moreover, if the minimal isometric immersion of $T^{3}$ in $\mathbb{S}^{m}$ is full and $[\mathbb{Q}(w):\mathbb{Q}]=3$ or $4$ , then $m=7$ and the immersion is homogeneous.

Proof.

Suppose the matrix data associated to the minimal immersion of $T^{3}$ is $\{Q,Y\}$ . Then we have

\mathrm{rank}\,Y=3,~~~3\leq\mathrm{rank}\{Y_{j}Y_{j}^{t}\,|\,Y_{j}\in Y\}\leq 6.

If $\mathrm{rank}\{Y_{j}Y_{j}^{t}\,|\,Y_{j}\in Y\}=6$ , then $Q$ can be determined uniquely from (10), and thus it must be rational.

If $\mathrm{rank}\{Y_{j}Y_{j}^{t}\,|\,Y_{j}\in Y\}=5$ , then it follows from the approach stated at the beginning of this section that $Q$ is at most quadratic irrational.

If $\mathrm{rank}\{Y_{j}Y_{j}^{t}\,|\,Y_{j}\in Y\}=4$ , without loss of generality, we assume that

\mathrm{rank}\{Y_{1},Y_{2},Y_{3}\}=3,~~~\mathrm{rank}\{Y_{1}Y_{1}^{t},Y_{2}Y_{2}^{t},Y_{3}Y_{3}^{t},Y_{4}Y_{4}^{t}\}=4.

(14)

Let $P$ be the matrix formed by the column vectors $\{Y_{1},Y_{2},Y_{3}\}$ . Set $\widetilde{Y}\triangleq P^{-1}Y$ and $\widetilde{Q}\triangleq P^{t}QP$ . Note that $\widetilde{Q}$ achieves the maximum of the determinant function on the set $\Sigma_{\widetilde{Y}}$ of all positive definite matrices such that

\widetilde{Y}_{j}^{t}\widetilde{Q}\widetilde{Y}_{j}=1,~~~1\leq j\leq\sharp(Y).

(15)

We assume that $\widetilde{Y}_{4}=(r_{1},r_{2},r_{3})^{t}$ and $\widetilde{Y}_{j}=(s_{j_{1}},s_{j_{2}},s_{j_{3}})^{t}$ for $5\leq j\leq\sharp(Y)$ . Then it follows from

\widetilde{Y}_{j}\widetilde{Y}_{j}^{t}=\lambda_{j_{1}}\widetilde{Y}_{1}\widetilde{Y}_{1}^{t}+\lambda_{j_{2}}\widetilde{Y}_{2}\widetilde{Y}_{2}^{t}+\lambda_{j_{3}}\widetilde{Y}_{3}\widetilde{Y}_{3}^{t}+\lambda_{j_{4}}\widetilde{Y}_{4}\widetilde{Y}_{4}^{t}

that

s_{j_{1}}s_{j_{2}}=\lambda_{j_{4}}r_{1}r_{2},~~~s_{j_{1}}s_{j_{3}}=\lambda_{j_{4}}r_{1}r_{3},~~s_{j_{2}}s_{j_{3}}=\lambda_{j_{4}}r_{2}r_{3}.

(16)

Due to (14), we observe that $\{r_{1},r_{2},r_{3}\}$ contains at least two nonzero elements. Without loss of generality, we assume that $r_{1}\neq 0$ and $r_{3}\neq 0$ . If $r_{2}=0$ , then it is straightforward to verify that all matrices in $C_{\widetilde{Y}}$ must necessarily exhibit a block-diagonal structure, which implies the maximum point $\widetilde{Q}$ must be rational.

Next, we consider the case $r_{2}\neq 0$ . If $\lambda_{j_{4}}=0$ , then $\widetilde{Y}_{j}$ is parallel to one of $\{\widetilde{Y}_{1},\widetilde{Y}_{2},\widetilde{Y}_{3}\}$ ; hence it must equal one of $\{\pm\widetilde{Y}_{1},\pm\widetilde{Y}_{2},\pm\widetilde{Y}_{3}\}$ by (15). If $\lambda_{j_{4}}\neq 0$ , then using (16) we obtain that $s_{j_{2}}r_{3}=r_{2}s_{j_{3}},\,s_{j_{1}}r_{2}=r_{1}s_{j_{2}},$ and $s_{j_{1}}r_{3}=r_{1}s_{j_{3}}$ , which implies $\widetilde{Y}_{j}//\widetilde{Y}_{4}$ ; hence $\widetilde{Y}_{j}=\pm\widetilde{Y}_{4}$ by (15). Therefore, we derive that $\sharp(Y)=4$ in this case. We parameterize the matrices in $W_{\widetilde{Y}}$ as

\left(\begin{array}[]{ccc}1&a&b\\ a&1&c\\ b&c&1\\ \end{array}\right),

(17)

with $a,b,c$ satisfying

2a\,r_{1}r_{2}+2b\,r_{1}r_{3}+2c\,r_{2}r_{3}+r_{1}^{2}+r_{2}^{2}+r_{3}^{2}=1,~~|a|<1,~~~|b|<1,~~~|c|<1.

The determinant of such matrices can be expressed as

2abc-a^{2}-b^{2}-c^{2}+1.

Using the method of Lagrange multipliers, the critical point of the determinant function on $W_{\widetilde{Y}}$ satisfies:

b=-\frac{(ar_{2}+r_{1})\left(2ar_{1}r_{2}+r_{1}^{2}+r_{2}^{2}+r_{3}^{2}-1\right)}{2r_{3}\left(2ar_{1}r_{2}+r_{1}^{2}+r_{2}^{2}\right)},\quad c=-\frac{(ar_{1}+r_{2})\left(2ar_{1}r_{2}+r_{1}^{2}+r_{2}^{2}+r_{3}^{2}-1\right)}{2r_{3}\left(2ar_{1}r_{2}+r_{1}^{2}+r_{2}^{2}\right)},

(18)

	$\displaystyle\!\!\!\!\!r_{1}r_{2}\left((r_{1}^{2}+r_{2}^{2})^{2}-\left(r_{3}^{2}-1\right)^{2}\right)-\left(r_{1}^{2}+r_{2}^{2}\right)\left(r_{1}^{4}-2r_{1}^{2}\left(r_{2}^{2}+r_{3}^{2}+1\right)+r_{2}^{4}-2r_{2}^{2}\left(r_{3}^{2}+1\right)+\left(r_{3}^{2}-1\right)^{2}\right)a$		(19)
	$\displaystyle-r_{1}r_{2}\left(7r_{1}^{4}+2r_{1}^{2}\left(5r_{2}^{2}-4\left(r_{3}^{2}+1\right)\right)+7r_{2}^{4}-8r_{2}^{2}\left(r_{3}^{2}+1\right)+\left(r_{3}^{2}-1\right)^{2}\right)a^{2}$
	$\displaystyle-8r_{1}^{2}r_{2}^{2}\left(2r_{1}^{2}+2r_{2}^{2}-r_{3}^{2}-1\right)\,a^{3}-12r_{1}^{3}r_{2}^{3}\,a^{4}=0$

Since $r_{1},r_{2},r_{3}\in\mathbb{Q}$ , it follows that there exists a field extension $\mathbb{Q}(w)$ of $\mathbb{Q}$ such that the extension degree $[\mathbb{Q}(w):\mathbb{Q}]$ is at most $4$ and $a,b,c\in\mathbb{Q}(w)$ .

If $\mathrm{rank}\{Y_{j}Y_{j}^{t}\,|\,Y_{j}\in Y\}=3$ , then following the same argument as above, up to multiplying a rational matrix on the left, we may assume that

Y_{1}=(1,0,0),~~~Y_{2}=(0,1,0),~~~Y_{3}=(0,0,1).

Therefore, in this case the maximizer of $\det$ on $W_{Y}$ is $GL(n,\mathbb{Q})$ -congruent to the identity matrix.

For the second part of the theorem, it follows from the above argument that $\mathrm{rank}\{Y_{j}Y_{j}^{t}\,|\,Y_{j}\in Y\}={4}$ . It is easy to verify that any $\eta$ -set in $Y$ satisfies Lemma 2.3 in [22], which implies the minimal isometric immersion must be homogeneous. ∎

Remark 4.3.

The proof of Theorem 4.2 provides a method to construct examples in $\mathbb{S}^{7}$ . First, choose a rational set $Y$ of the following form

Y_{1}=(1,0,0),\qquad Y_{2}=(0,1,0),\qquad Y_{3}=(0,0,1),\qquad Y_{4}=(r_{1},r_{2},r_{3}).

Next, compute the roots of equation (19) and construct matrices $Q$ of the form (17), where the parameters $b$ and $c$ are given in (18). For such a $Q$ , if $Q$ is positive definite and there exist positive coefficients $c_{1}^{2},c_{2}^{2},c_{3}^{2},c_{4}^{2}$ satisfying

c_{1}^{2}+c_{2}^{2}+c_{3}^{2}+c_{4}^{2}=1,~~~c_{1}^{2}Y_{1}Y_{1}^{t}+c_{2}^{2}Y_{2}Y_{2}^{t}+c_{3}^{2}Y_{3}Y_{3}^{t}+c_{4}^{2}Y_{4}Y_{4}^{t}\;=\;\frac{1}{3}\,Q^{-1},

(20)

then, after a dilation, the pair $\{Q,Y\}$ yields the matrix data of a minimal flat $3$ -torus in $\mathbb{S}^{7}$ .

By this method, we construct examples of irrational minimal flat $3$ -tori in $\mathbb{S}^{7}$ with all possible algebraic irrationality degree.

Example 4.4.

A quadratic irrational flat $3$ -torus into $\mathbb{S}^{7}$ .

Consider the integer set

Y=\left(\begin{array}[]{cccc}1&0&0&-3\\ 0&1&0&4\\ 0&0&1&-3\\ \end{array}\right).

Now the equation (19) is equivalent to

f(x)\triangleq-9\left(32x^{2}-20x-17\right)\left(72x^{2}-115x+44\right)=0.

One of its roots is given by $a=\frac{1}{144}\bigl(115-\sqrt{553}\bigr)$ , which determines the following positive definite matrix by (18),

{Q=\left(\begin{array}[]{ccc}1&\left(115-\sqrt{553}\right)/144&\left(16-\sqrt{553}\right)/54\\ \left(115-\sqrt{553}\right)/144&1&\left(115-\sqrt{553}\right)/144\\ \left(16-\sqrt{553}\right)/{54}&\left(115-\sqrt{553}\right)/144&1\\ \end{array}\right)}.

It is straightforward to verify that

(c_{1}^{2},\,c_{2}^{2},\,c_{3}^{2},\,c_{4}^{2})=(38-\sqrt{553})\left(\frac{2}{99},\frac{\sqrt{553}-13}{1782},\frac{2}{99},\frac{\sqrt{553}+17}{1782}\right)

solve the equations in (20). Due to Remark 4.3, the matrix data $\{Q,Y\}$ gives a minimal embedding of a quadratic irrational flat $3$ -torus into $\mathbb{S}^{7}$ , where the embeddedness property follows directly from Remark 2.6.

This example shows that $\mathrm{rank}\{Y_{j}Y_{j}^{t}\,|\,Y_{j}\in Y\}=5$ is not a necessary condition to produce a quadratic irrational flat $3$ -torus allowing minimal immersion into $\mathbb{S}^{m}$ .

Example 4.5.

A cubic irrational minimal flat $3$ -torus in $\mathbb{S}^{7}$ .

Consider the integer set

Y=\left(\begin{array}[]{cccc}4&0&0&-5\\ 0&4&0&2\\ 0&0&4&-3\\ \end{array}\right).

Now the equation (19) is equivalent to

f(x)\triangleq(50x^{3}-160x^{2}+149x-33)(x+1)=0.

Note that $f(0)f(\frac{1}{2})<0$ , so there exists $a\in[0,\frac{1}{2}]$ as a real root of $f(x)$ . By (18), set

{Q\triangleq\frac{1}{16}\left(\begin{array}[]{ccc}1&a&\frac{(2a-5)(10a-11)}{3(20a-29)}\\ a&1&-\frac{(5a-2)(10a-11)}{3(20a-29)}\\ \frac{(2a-5)(10a-11)}{3(20a-29)}&-\frac{(5a-2)(10a-11)}{3(20a-29)}&1\\ \end{array}\right)}.

It is straightforward to check that at $a\in[0,\frac{1}{2}]$

\det(Q)=\frac{20(a+1)^{2}(5a-7)(a-1)}{9(29-20a)}>0,

hence $Q$ is positive definite. Furthermore, one can also verify that the following coefficients

		$\displaystyle c_{1}^{2}=\frac{57-(10a-19)^{2}}{24(a^{2}-1)(5a-7)(20a-29)},$		$\displaystyle c_{2}^{2}=\frac{-(430a^{2}-1235a+883)}{15(a^{2}-1)(5a-7)(20a-29)},$
		$\displaystyle c_{3}^{2}=\frac{3(5a-9)}{10(a+1)(5a-7)},$		$\displaystyle c_{4}^{2}=\frac{4(10a-11)}{15(a+1)(5a-7)}$

are positive and the equations in (20) hold true.

Due to Remark 4.3, we obtain a matrix data $\{Q,Y\}$ of minimal flat $3$ -torus in $\mathbb{S}^{7}$ . Note that $a$ is a root of

50x^{3}-160x^{2}+149x-33,

which is irreducible¹¹ 1 It can be verified by the command “IrreduciblePolynomialQ[ $\cdot$ ]” on WolframAlpha: www.wolfram.com in $\mathbb{Q}$ . Therefore, the minimal flat $3$ -torus we obtained is cubic irrational.

Unfortunately, this immersion fails to be an embedding. In fact, the following points

(0,\frac{1}{4},\frac{1}{2}),~~~(0,\frac{1}{2},0),~~~(\frac{1}{4},0,\frac{1}{4})

are all mapped to nonzero integer points under (12). Regarding them as row vectors, we obtain a matrix

\widetilde{Y}=\left(\begin{array}[]{cccc}0&1&2&-1\\ 0&2&0&1\\ 1&0&1&-2\\ \end{array}\right).

(21)

Set

\widetilde{Q}\triangleq\left(\begin{array}[]{ccc}-2&0&2\\ 1&2&-1\\ 4&0&0\\ \end{array}\right)Q\left(\begin{array}[]{ccc}-2&0&2\\ 1&2&-1\\ 4&0&0\\ \end{array}\right)^{t}={\frac{1}{16}}\left(\begin{array}[]{ccc}\frac{16\left(10a^{2}-66a+71\right)}{87-60a}&-\frac{40\left(3a^{2}-a-4\right)}{20a-29}&\frac{16\left(10a^{2}-66a+71\right)}{60a-87}\\ -\frac{40\left(3a^{2}-a-4\right)}{20a-29}&\frac{16\left(25a^{2}-9a-34\right)}{60a-87}&\frac{8\left(50a^{2}-21a-71\right)}{60a-87}\\ \frac{16\left(10a^{2}-66a+71\right)}{60a-87}&\frac{8\left(50a^{2}-21a-71\right)}{60a-87}&16\\ \end{array}\right).

Then the matrix data $\{\widetilde{Q},\widetilde{Y}\}$ gives a minimal embedding of a cubic irrational flat 3-torus, which is covered by the aforementioned immersed example. Here the embeddedness property follows directly from an analysis of (12) and a verification that $\mathcal{Y}(\Omega)$ contains no nonzero integer points.

Example 4.6.

A quartic irrational minimal flat $3$ -torus in $\mathbb{S}^{7}$ .

Consider the matrix data

Y=\left(\begin{array}[]{cccc}1&0&0&5\\ 0&1&0&7\\ 0&0&1&8\\ \end{array}\right).

Now the equation (19) is equivalent to

f(x)\triangleq-14700x^{4}-23240x^{3}+1079x^{2}+10730x+1507=0.

Note that $f(-\frac{3}{20})f(-\frac{1}{7})<0$ , so $f(x)$ has a real root $a\approx-0.149201\in[-\frac{3}{20},-\frac{1}{7}]$ . By (18), set

{Q=\begin{pmatrix}1&a&{-\frac{(7a+5)(70a+137)}{32(35a+37)}}\\ a&1&{-\frac{(5a+7)(70a+137)}{32(35a+37)}}\\ {-\frac{(7a+5)(70a+137)}{32(35a+37)}}&{-\frac{(5a+7)(70a+137)}{32(35a+37)}}&1\\ \end{pmatrix}}.

It is straightforward to check that at $a\in[-\frac{3}{20},-\frac{1}{7}]$ ,

\det(Q)=\frac{35(a^{2}-1)(10a-1)(14a+5)}{512(37+35a)}>0,

hence $Q$ is positive definite. Furthermore, one can also verify that the following coefficients

		$\displaystyle c_{1}^{2}={\frac{24500a^{4}-16800a^{3}-107125a^{2}-67998a-4945}{42(a-1)(a+1)(10a-1)(14a+5)(35a+37)}},$
		$\displaystyle c_{2}^{2}={\frac{34300a^{4}+23520a^{3}-88151a^{2}-84090a-8699}{30(a-1)(a+1)(10a-1)(14a+5)(35a+37)}},$
		$\displaystyle c_{3}^{2}=\frac{-128(70a+11)}{105(10a-1)(14a+5)},$
		$\displaystyle c_{4}^{2}=\frac{-2(70a+137)}{105(10a-1)(14a+5)}$

satisfy the equations in (20). We claim that they are all positive at $a$ .

First, consider the polynomials, and define

	$\displaystyle g_{1}(x)$	$\displaystyle\triangleq 24500x^{4}-16800x^{3}-107125x^{2}-67998x-4945,$
	$\displaystyle g_{2}(x)$	$\displaystyle\triangleq 34300x^{4}+23520x^{3}-88151x^{2}-84090x-8699.$

A straightforward computation yields that, for both $i=1,2$ ,

\lim_{x\to-\infty}g_{i}(x)=+\infty,\quad g_{i}(-1)<0,\quad g_{i}\left(-\frac{1}{2}\right)>0,\quad g_{i}\left(-\frac{1}{7}\right)>0,\quad g_{i}(0)<0,\quad\lim_{x\to+\infty}g_{i}(x)=+\infty.

By the Intermediate Value Theorem, the alternating signs indicate that the four real roots of each degree-4 polynomial $g_{i}(x)$ are completely isolated within the intervals $(-\infty,-1)$ , $\left(-1,-\frac{1}{2}\right)$ , $\left(-\frac{1}{7},0\right)$ , and $(0,+\infty)$ . Consequently, no real roots exist within the interval $\left[-\frac{1}{2},-\frac{1}{7}\right]$ . Since the endpoint values are positive, both $g_{1}(x)$ and $g_{2}(x)$ are strictly positive in $\left[-\frac{1}{2},-\frac{1}{7}\right]$ . Therefore, we have $g_{i}(a)>0$ , which yields that both $c_{1}^{2}$ and $c_{2}^{2}$ are positive. Finally, one can readily verify that $c_{3}^{2}$ , $c_{4}^{2}$ are also positive. By Remark 4.3, we obtain a matrix data $\{Q,Y\}$ of minimal flat $3$ -torus in $\mathbb{S}^{7}$ .

Note that $f(x)$ is irreducible²² 2 It can be verified by the command “IrreduciblePolynomialQ[ $\cdot$ ]” on WolframAlpha: www.wolfram.com in $\mathbb{Q}$ . Therefore, this example is a minimal immersion of a quartic irrational flat $3$ -torus. Again by Remark 2.6,this immersion is actually an embedding.

4.2 The uniqueness of minimal immersion for irrational flat $3$ -torus

As shown in Proposition 3.5, for any given rational flat $n$ -torus $T^{n}$ , there exist infinitely many $k\in\mathbb{Z}^{+}$ such that $T^{n}$ can be homothetically and minimally immersed into spheres by the $k$ -th eigenfunctions. It is a natural question to ask, how many such immersions an irrational torus can admit into spheres? In the $3$ -dimensional case, we show that such an immersion is unique if the algebraic irrationality degree exceeds $2$ .

Theorem 4.7.

For every cubic and quartic irrational flat $3$ -torus, if a minimal homothetic immersion into spheres exists, then it is unique up to congruence.

Proof.

Suppose $\{Q,Y\}$ is the matrix data for an immersion of such torus. Then it follows from Theorem 4.2 that $Y=\{Y_{1},Y_{2},Y_{3},Y_{4}\}$ . Furthermore, there exist positive real numbers $c_{1},c_{2},c_{3},c_{4}$ such that

c_{1}Y_{1}Y_{1}^{t}+c_{2}Y_{2}Y_{2}^{t}+c_{3}Y_{3}Y_{3}^{t}+c_{4}Y_{4}Y_{4}^{t}=\frac{Q^{-1}}{3},\quad\sum_{i=1}^{4}c_{i}=1.

(22)

Suppose a matrix data $\{\frac{Q}{\lambda^{2}},X\}$ gives another minimal immersion of this $3$ -torus, where $X=\{X_{i},1\leq i\leq 4\}$ . Let $4+d=\mathrm{rank}\{X_{j}X_{j}^{t},Y_{i}Y_{i}^{t}|1\leq i,j\leq 4\}$ , then

\mathrm{dim}(\mathrm{Span}\{X_{j}X_{j}^{t}\}\cap\mathrm{Span}\{Y_{i}Y_{i}^{t}\})=4-d.

If $\lambda^{2}$ is irrational, we assume $X_{1}X_{1}^{t},\cdots,X_{d}X_{d}^{t},Y_{1}Y_{1}^{t},\cdots,Y_{4}Y_{4}^{t}$ are $4+d$ linearly independent vectors in $\mathrm{Sym}_{3}$ . For any $d<\alpha\leq 4$ , we have

X_{\alpha}X_{\alpha}^{t}=\sum_{i=1}^{4}k_{i\alpha}Y_{i}Y_{i}^{t}+\sum_{j=1}^{d}h_{j\alpha}X_{j}X_{j}^{t},

where all of $k_{i\alpha}$ and $h_{j\alpha}$ are obviously rational. Then from $\langle Q,Y_{i}Y_{i}^{t}\rangle=1$ , $\langle Q,X_{j}X_{j}^{t}\rangle=\lambda^{2}$ ,

\sum_{i=1}^{4}k_{i\alpha}=1-\sum_{j=1}^{d}h_{j\alpha}=0.

(23)

X_{\alpha}X_{\alpha}^{t}-\sum_{j=1}^{d}h_{j\alpha}X_{j}X_{j}^{t}=\sum_{i=1}^{4}k_{i\alpha}Y_{i}Y_{i}^{t},\quad d+1\leq\alpha\leq 4

are exactly $4-d$ linearly independent vectors in $\mathrm{Span}\{X_{j}X_{j}^{t}\}\cap\mathrm{Span}\{Y_{i}Y_{i}^{t}\}$ . Due to (23), $Q$ is orthogonal to $\mathrm{Span}\{X_{j}X_{j}^{t}\}\cap\mathrm{Span}\{Y_{i}Y_{i}^{t}\}$ . However, (22) shows $Q^{-1}\in\mathrm{Span}\{Y_{i}Y_{i}^{t}\}$ and $\langle Q,Q^{-1}\rangle=3\not=0$ , which suggests $Q^{-1}\not\in\mathrm{Span}\{X_{j}X_{j}^{t}\}$ . So $\{\frac{Q}{\lambda^{2}},X\}$ cannot give a minimal immersion into spheres, yielding a contradiction.

Hence, $\lambda^{2}$ is rational. In this case, applying the similar argument as the beginning of subsection 4.1, we conclude that the maximum of the determinant function on

\{\widehat{Q}\in\Sigma_{+}\subset\mathrm{Sym}_{3}\mid\langle\widehat{Q},Y_{i}Y_{i}^{t}\rangle=\langle\widehat{Q},\frac{1}{\lambda^{2}}X_{j}X_{j}^{t}\rangle=1,1\leq i,j\leq 4\}

has extension degree at most $2$ provided $d>0$ . Since this maximum cannot exceed the maximal value of the determinant on $W_{Y}$ (which is attained by $Q$ ), we conclude that $Q$ itself must be the maximizer. By our initial assumption, $Q$ is either a cubic or quartic form. Hence, we derive that $d=0$ and then $\mathrm{Span}\{X_{j}X_{j}^{t}\}=\mathrm{Span}\{Y_{i}Y_{i}^{t}\}$ . We choose $Q_{1},Q_{2}\in\mathrm{Sym}_{3}$ such that $\mathrm{Span}\{Q_{1},Q_{2}\}\perp\mathrm{Span}\{Y_{i}Y_{i}^{t},1\leq i\leq 4\}$ . Note that $x^{t}Q_{1}x=0$ and $x^{t}Q_{2}x=0$ define two quadratic cones whose intersection is composed by $4$ non-coplanar lines (spanned by $Y_{1},Y_{2},Y_{3},Y_{4}$ , respectively). Therefore, $\{\pm Y_{1},\pm Y_{2},\pm Y_{3},\pm Y_{4}\}$ constitute the whole intersection of the hyper-ellipsoid defined by $Q$ and the aforementioned two quadratic cones. Consequently, we deduce that $X\subset\{\pm\lambda Y_{i}\}$ , and thus $\{\frac{Q}{\lambda^{2}},X\}$ is conformally equivalent to $\{Q,Y\}$ . Then the uniqueness follows from the homogeneity established in Theorem 4.2 and

{\mathrm{rank}\{Y_{1}Y_{1}^{t},Y_{2}Y_{2}^{t},Y_{3}Y_{3}^{t},Y_{4}Y_{4}^{t}\}=4.}

∎

Remark 4.8.

As for the quadratic case, it seems to be very subtle for the uniqueness and the rigidity of minimal immersions. Let $\{Q,Y\}$ be the matrix data for a minimal immersion of a quadratic irrational flat 3-torus with $\mathrm{rank}\{Y_{j}Y_{j}^{t}\mid Y_{j}\in Y\}=5$ . It turns out that there may exist many rational points on the hyper-ellipsoid $\mathcal{Q}$ determined by $u^{t}Qu=1$ so that new minimal immersions can be constructed. In fact, given a new rational point $Z$ , let $l$ be a common multiple of the denominators of its coordinates. Note that $ZZ^{t}\in{\mathrm{Span}}\{Y_{j}Y_{j}^{t}\mid Y_{j}\in Y\}$ , since $Q$ is irrational. For each face $F$ of the convex hull of $\{Y_{j}Y_{j}^{t}\mid Y_{j}\in Y\}$ , consider the inverse cone of $F$ with respect to $\frac{Q^{-1}}{3}$ . Then there exists at least one such inverse cone containing $ZZ^{t}$ , and we denote its corresponding face by $\widetilde{F}$ . Define $\widetilde{Y}$ to be the set consisting of the vertices of $\widetilde{F}$ together with $Z$ . It follows that $\{\frac{Q}{l^{2}},l\widetilde{Y}\}$ provides new minimal immersions.

To show how many rational points lie on $\mathcal{Q}$ , we write $Q=Q_{1}+\alpha Q_{2}$ , where $Q_{1},Q_{2}\in GL(n,\mathbb{Q})$ are two symmetric matrices and $[\mathbb{Q}(\alpha):\mathbb{Q}]=2$ . Then a rational point $Z_{1}$ lies on $\mathcal{Q}$ if and only if

Z_{1}^{t}Q_{1}Z_{1}=1,~~~Z_{1}^{t}Q_{2}Z_{1}=0,

i.e., it lies on the intersection of two quadratic surfaces. In algebraic geometry, it is known (cf. [16, Chap. 4, Ex. 3.6]) that such an intersection is either a rational curve (possibly singular and reducible) or a quartic elliptic curve. In the rational case, the curve admits infinitely many rational points. In the elliptic case, the set of rational points forms a finitely generated abelian group (by the Mordell-Weil theorem, cf. [32]), meaning it may be either finite (e.g., torsion points alone) or infinite (if the rank is positive).

4.3 Homogeneity of irrational minimal flat $3$ -torus

It follows from Theorem 4.2 that the cubic and quartic irrational minimal flat $3$ -tori in spheres are obviously homogeneous. In this subsection, we show that this homogeneity also holds in the quadratic case.

Theorem 4.9.

The minimal isometric immersions of irrational flat $3$ -tori into spheres are homogeneous.

Proof.

Assume that the matrix data associated to this minimal isometric immersion is $\{Q,Y\}$ . By the homogeneity proof in Theorem 4.2, it needs only to prove for the case of $\mathrm{rank}\{Y_{j}Y_{j}^{t}\,|\,Y_{j}\in Y\}=5$ . If there is a non-trivial $\eta$ -set:

Y_{1}+Y_{2}=Y_{3}+Y_{4}=Y_{5}+Y_{6}=\eta,

then the plane $\{c_{1}Y_{1}+c_{2}Y_{2}+c_{3}Y_{3}|c_{1}+c_{2}+c_{3}=1\}$ intersects the ellipsoid $x^{t}Qx=1$ at an ellipse with center $\eta/2$ .

Set $P\triangleq(\frac{\eta}{2},Y_{1}-\frac{\eta}{2},Y_{3}-\frac{\eta}{2})$ , $\widetilde{Q}\triangleq P^{t}QP$ , and $\widetilde{Y}_{i}\triangleq P^{-1}Y_{i}~(1\leq i\leq 6)$ . Then we have

\widetilde{Y}_{i}^{t}\widetilde{Q}\widetilde{Y}_{i}=1,\quad\widetilde{Y}_{i}=\begin{pmatrix}1\\ y_{i}\end{pmatrix},

in which $y_{i}\in\mathbb{Q}^{2}$ for $1\leq i\leq 6$ , and

y_{1}=-y_{2}=\begin{pmatrix}1\\ 0\end{pmatrix},\quad y_{3}=-y_{4}=\begin{pmatrix}0\\ 1\end{pmatrix},\quad y_{5}=-y_{6}.

Note that the matrix $R\in\mathrm{Sym}_{2}$ solving $y_{i}^{t}Ry_{i}=1$ for all $1\leq i\leq 6$ is uniquely determined and must be rational. Moreover, there exists a number $a\in\mathbb{R}$ such that

\widetilde{Q}=\begin{pmatrix}a&\\ &(1-a)R\end{pmatrix}.

By assumption, $Q$ is irrational. It follows that $\widetilde{Q}$ is irrational, and hence $a$ is irrational. This implies that other $\widetilde{Y}_{j}(j>6)$ if existing, must be of the form $\begin{pmatrix}1\\ y_{j}\end{pmatrix}$ with $y_{j}^{t}Ry_{j}=1$ . Consequently, we have

W_{\widetilde{Y}}=\left\{\begin{pmatrix}t&\\ &(1-t)R\end{pmatrix}\,\bigg|\,t\in\mathbb{R}\right\}.

By Theorem 2.4, we derive that $a$ achieves the maximum of determinant function $t(1-t)^{2}\mathrm{det}(R)$ , which is obviously rational. This contradiction implies that the non-trivial $\eta$ -set consists of exactly $2$ pairs and thus the immersion is homogeneous by Lemma 2.3 in [22]. ∎

5 Examples of non-homogeneous minimal flat $3$ -tori in spheres

In this section, we show the existence of non-homogeneous minimal isometric immersions (even embeddings) of rational flat $3$ -tori with $\mathrm{rank}\{{Y_{i}Y_{i}^{t}}\mid 1\leq i\leq\sharp(Y)\}=5$ . Moreover, by minimal products [6, 38, 44] with the Clifford $(n-3)$ -torus, one can show there exists non-homogeneous minimal flat $n$ -tori in spheres for every $n\geq 3$ . From (2), it is easy to see that a minimal flat $n$ -tori is homogeneous if and only if the matrix $A$ is diagonal up to multiplying an orthogonal matrix on the right, or equivalently, the symmetric matrix $AA^{t}$ is diagonal. We begin by introducing a family of interesting $3$ -tori.

Let

\mathfrak{P}=\{\mathrm{p}\mid\mathrm{p}\ \text{is a prime number congruent to }1\text{ modulo }4\}.

By Fermat’s theorem on the sum of two squares [19], for every $r\in\mathfrak{P}$ there exists a unique ordered pair of positive integers $(m_{1},m_{2})$ such that

r=m_{1}^{2}+m_{2}^{2},\qquad\gcd(m_{1},m_{2})=1,\qquad m_{1}>m_{2}.

For each given $r\in\mathfrak{P}$ and associated pair $(m_{1},m_{2})$ , set

p\triangleq m_{1}^{2}-m_{2}^{2},~~~~~~q\triangleq 2m_{1}m_{2}.

Then $\{p,q,r\}$ forms a primitive Pythagorean triple, i.e., they satisfy

p^{2}+q^{2}=r^{2},~~~(p,q,r)=1.

Conversely, given a primitive Pythagorean triple $\{p,q,r\}$ , we claim that $r\in\mathfrak{P}$ provided that it is not the hypotenuse of any other Pythagorean triple. Note that $r$ is odd and it is also a sum of two squares. Suppose that $r$ admits a prime factorization

r=\prod_{i\in\Gamma}(4k_{i}+1)^{f_{i}}\cdot\prod_{i\in\Gamma^{\prime}}(4k_{i}+3)^{g_{i}}.

Since any primitive divisor of a sum of two squares is itself a sum of two squares, we obtain $g_{i}=0$ for all $i\in\Gamma^{\prime}$ because a number of the form $4k_{i}+3$ can never be expressed as a sum of two nonzero squares. By assumption, we have $4k_{1}+1\in\mathfrak{P}$ ; hence there exists a unique ordered pair $(a_{1},b_{1})$ of positive integers satisfying $4k_{1}+1=a_{1}^{2}+b_{1}^{2}$ . Set $r^{\prime}=r/(4k_{1}+1)\in\mathbb{Z}$ , then one can verify that

\{r^{\prime}(a_{1}^{2}-b_{1}^{2}),r^{\prime}(2a_{1}b_{1}),r\}

is also a Pythagorean triple with $r$ as the hypotenuse. This is a new Pythagorean triple if $r^{\prime}\neq 1$ . So we show the claim holds.

Given $r\in\mathfrak{P}$ , consider the lattice $\Lambda_{3,r}^{*}$ generated by

\eta_{1}=\begin{pmatrix}2/\sqrt{3}\\ 0\\ 0\end{pmatrix},\quad\eta_{2}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}-1/\sqrt{2}\\ 1/r\\ 0\end{pmatrix},\quad\eta_{3}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}-1/\sqrt{2}\\ 0\\ 1/r\end{pmatrix}.

(24)

We denote by $\mathbb{T}^{3}_{r}=\mathbb{R}^{3}/\Lambda_{3,r}$ the flat $3$ -torus determined by its dual lattice $\Lambda_{3,r}$ .

5.1 A family of non-homogeneous minimal isometric immersions of $\mathbb{T}^{3}_{r}$ into spheres

Let $\mathbb{T}^{3}_{r}$ be a flat $3$ -torus determined by an integer $r\in\mathfrak{P}$ . By the uniqueness of the primitive Pythagorean triple determined by $r$ , one can verify directly that

\xi_{1}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}1/\sqrt{2}\\ 1\\ 0\end{pmatrix},\quad\xi_{2}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}1/\sqrt{2}\\ -1\\ 0\end{pmatrix},\quad\xi_{3}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}1/\sqrt{2}\\ 0\\ 1\end{pmatrix},\quad\xi_{4}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}1/\sqrt{2}\\ 0\\ -1\end{pmatrix},

\xi_{5}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}1/\sqrt{2}\\ p/r\\ q/r\end{pmatrix},\quad\xi_{6}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}1/\sqrt{2}\\ -p/r\\ -q/r\end{pmatrix},\quad\xi_{7}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}1/\sqrt{2}\\ -q/r\\ p/r\end{pmatrix},\quad\xi_{8}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}1/\sqrt{2}\\ q/r\\ -p/r\end{pmatrix},

\xi_{9}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}1/\sqrt{2}\\ p/r\\ -q/r\end{pmatrix},\quad\xi_{10}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}1/\sqrt{2}\\ -p/r\\ q/r\end{pmatrix},\quad\xi_{11}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}1/\sqrt{2}\\ q/r\\ p/r\end{pmatrix},\quad\xi_{12}=\frac{\sqrt{2}}{\sqrt{3}}\begin{pmatrix}1/\sqrt{2}\\ -q/r\\ -p/r\end{pmatrix}

constitute all the vectors (up to $\pm 1$ ) of norm $1$ in the lattice $\Lambda_{3,r}^{*}$ .

To construct the minimal isometric immersions of $\mathbb{T}^{3}_{r}$ into spheres, particularly those that are non-homogeneous, we first analyze $\{\xi_{i}\}_{j=1}^{12}$ to identify all $\eta$ -sets (see Definition 2.2). Then, by solving equations (5), (6), (8), and (9) for these $\eta$ -sets, together with (7), we obtain the expression for $AA^{t}$ . It will be shown that $AA^{t}$ can be non-diagonal, implying that the minimal immersion of the form (2) is non-homogeneous.

Define $\eta_{0}=(\frac{2}{\sqrt{3}},0,0)^{t}$ . The following $\eta_{0}$ -set is obvious:

\eta_{0}=\xi_{1}+\xi_{2}=\xi_{3}+\xi_{4}=\cdots=\xi_{11}+\xi_{12}.

In order to find all the possible $\eta$ -sets, we define a projection $P:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}$ by

Pv=\begin{pmatrix}0&\frac{\sqrt{3}}{\sqrt{2}}&0\\ 0&0&\frac{\sqrt{3}}{\sqrt{2}}\end{pmatrix}v,\quad v\in\mathbb{R}^{3}.

So $\{P\xi_{i},1\leq i\leq 12\}$ consists of all the unit normal vectors of lattice generated by $(\frac{1}{r},0)^{t}$ and $(0,\frac{1}{r})^{t}$ .

Lemma 5.1.

The aforementioned $\eta_{0}$ -set is the only non-trivial $\eta$ -set.

This lemma will be established through the following two lemmas. Note that for any given $i<j$ and $k<l$ , the pairs $(\xi_{i},\xi_{j})$ and $(\xi_{k},-\xi_{l})$ cannot belong to the same $\eta$ -set.

Lemma 5.2.

If $(a,b)\not=(2k-1,2k)$ for some $k$ then the $(\xi_{a}+\xi_{b})$ -set contains only $(\xi_{a},\xi_{b})$ .

Proof.

Obviously, $v\triangleq P(\xi_{a}+\xi_{b})=P\xi_{a}+P\xi_{b}\neq 0$ . For any pair of unit normal vectors whose sum equals $v$ , they must correspond to the two intersection points of the unit circle centered at the origin $o$ and the perpendicular bisector of the line segment $ov$ . The conclusion comes from the uniqueness of such intersection. ∎

Lemma 5.3.

For all $1\leq a<b\leq 12$ , the $(\xi_{a}-\xi_{b})$ -set contains only $(\xi_{a},-\xi_{b})$ .

Proof.

Suppose there exist two distinct pairs $(\xi_{a},-\xi_{b})$ and $(\xi_{c},-\xi_{d})$ in an $\eta$ -set. By Definition 2.2, we have

a<b,~~~c<d,~~~\xi_{c}-\xi_{d}=\xi_{a}-\xi_{b},

which yields that $\xi_{a}+\xi_{d}=\xi_{c}+\xi_{b}$ . Without loss of generality, assume $a<c$ . Then, from Lemma 5.2 we obtain a contradiction that $a<c<d=a+1$ . ∎

Considering $(\xi_{a}+\xi_{b})$ -set and $(\xi_{a}-\xi_{b})$ -set for $(a,b)\not=(2k-1,2k)$ , we get from (5) and (6) that the block of $AA^{t}$ (see (3))

A_{ab}=O,

(25)

which obviously satisfies the linear system (8) and (9) for $r=a,s=b$ . Similarly, considering $(\xi_{2k-1}-\xi_{2k})$ -set, we derive that

A_{ij}^{11}+A_{ij}^{22}=A_{ij}^{12}-A_{ij}^{21}=0,\quad i=2k-1,~j=2k,~1\leq k\leq 6.

(26)

Combining (4), (25) and (26), we obtain that $AA^{t}$ has the following block structure:

AA^{t}=\begin{pmatrix}B_{1}&&&\\ &B_{2}&&\\ &&\ddots&\\ &&&B_{6}\end{pmatrix},\quad B_{i}=\begin{pmatrix}a_{2i-1}I_{2}&A_{2i-1\,2i}\\ A_{2i-1\,2i}^{t}&a_{2i}I_{2}\end{pmatrix},\quad A_{2i-1\,2i}=\begin{pmatrix}\alpha_{i}&\beta_{i}\\ \beta_{i}&-\alpha_{i}\end{pmatrix}.

We now turn our attention to solving equations (8) and (9) for the $\eta_{0}$ -set specified above, noting that these equations incorporate (5) and (6) as special cases. A direct verification shows that the solutions are given by

\left\{\begin{aligned} \alpha_{1}&=-R_{1}\left(1+\frac{p^{2}-q^{2}}{r^{2}}(\cos^{2}\phi_{1}-\cos^{2}\psi_{1})\right),&&\alpha_{3}=R_{1}\cos^{2}\phi_{1},&&\alpha_{5}=R_{1}\sin^{2}\psi_{1},\\ \alpha_{2}&=-R_{1}\left(1+\frac{p^{2}-q^{2}}{r^{2}}(\cos^{2}\psi_{1}-\cos^{2}\phi_{1})\right),&&\alpha_{4}=R_{1}\cos^{2}\psi_{1},&&\alpha_{6}=R_{1}\sin^{2}\phi_{1},\\ \beta_{1}&=-R_{2}\left(1+\frac{p^{2}-q^{2}}{r^{2}}(\cos^{2}\phi_{2}-\cos^{2}\psi_{2})\right),&&\beta_{3}=R_{2}\cos^{2}\phi_{2},&&\beta_{5}=R_{2}\sin^{2}\psi_{2},\\ \beta_{2}&=-R_{2}\left(1+\frac{p^{2}-q^{2}}{r^{2}}(\cos^{2}\psi_{2}-\cos^{2}\phi_{2})\right),&&\beta_{4}=R_{2}\cos^{2}\psi_{2},&&\beta_{6}=R_{2}\sin^{2}\phi_{2},\end{aligned}\right.

(27)

in which $R_{1},R_{2},\phi_{1},\phi_{2},\psi_{1}$ and $\psi_{2}$ are constant parameters.

By a long but straightforward computation, the equation (7) can be transformed to the following linear system,

\left\{\begin{aligned} a_{5}+a_{6}+a_{11}+a_{12}&=a_{7}+a_{8}+a_{9}+a_{10},\\ a_{1}+\dfrac{p(p+r)}{2r^{2}}(a_{5}+a_{9})+\dfrac{p(p-r)}{2r^{2}}(a_{6}+a_{10})&+\dfrac{q(q+r)}{2r^{2}}(a_{8}+a_{11})+\dfrac{q(q-r)}{2r^{2}}(a_{7}+a_{12})=\dfrac{1}{4},\\ a_{3}+\dfrac{q(q+r)}{2r^{2}}(a_{5}+a_{10})+\dfrac{q(q-r)}{2r^{2}}(a_{6}+a_{9})&+\dfrac{p(p+r)}{2r^{2}}(a_{7}+a_{11})+\dfrac{p(p-r)}{2r^{2}}(a_{8}+a_{12})=\dfrac{1}{4},\\ a_{2}+\dfrac{p(p-r)}{2r^{2}}(a_{5}+a_{9})+\dfrac{p(p+r)}{2r^{2}}(a_{6}+a_{10})&+\dfrac{q(q-r)}{2r^{2}}(a_{8}+a_{11})+\dfrac{q(q+r)}{2r^{2}}(a_{7}+a_{12})=\dfrac{1}{4},\\ a_{4}+\dfrac{q(q-r)}{2r^{2}}(a_{5}+a_{10})+\dfrac{q(q+r)}{2r^{2}}(a_{6}+a_{9})&+\dfrac{p(p-r)}{2r^{2}}(a_{7}+a_{11})+\dfrac{p(p+r)}{2r^{2}}(a_{8}+a_{12})=\dfrac{1}{4}.\end{aligned}\right.

(28)

Clearly, $a_{1},\cdots,a_{4}$ can be expressed in terms of the remaining eight parameters. Moreover, the first equation shows that the solution space of (28) is 7-dimensional.

Finally, to obtain a minimal isometric immersion, we need only further impose that $AA^{t}$ is positive semidefinite, or equivalently, each block $B_{i}$ must be positive semidefinite. This is equivalent to requiring that $\alpha_{i}^{2}+\beta_{i}^{2}$ , the eigenvalue of $A_{2i-1\,2i}^{t}A_{2i-1\,2i}$ , does not exceed $a_{2i-1}a_{2i}$ , a condition that can be satisfied by choosing $R_{1}$ and $R_{2}$ sufficiently small.

In summary, once the entries $a_{i},\alpha_{j},\beta_{j}$ of $AA^{t}$ have been determined, we take the square root of $AA^{t}$ to obtain the required matrix $A$ in (2), which then yields a minimal isometric immersion of $\mathbb{T}^{3}_{r}$ into $\mathbb{S}^{23}$ . As long as one of $\{\alpha_{i},\beta_{i}\mid 1\leq i\leq 6\}$ is nonzero, the corresponding minimal immersion is non-homogeneous. Since each parameter $\alpha_{i}$ and $\beta_{i}$ can be deformed continuously to $0$ , every such minimal isometric immersion can be deformed continuously into a homogeneous example, which itself forms a $7$ -parameter family. Moreover, any homogeneous case can be further deformed into $a_{2i-1}=a_{2i}=0$ for all $i>2$ , resulting in a homogeneous minimal immersion in $\mathbb{S}^{7}$ . Thus we arrive at the following proposition.

Proposition 5.4.

For each $r\in\mathfrak{P}$ , the flat $3$ -torus $\mathbb{T}^{3}_{r}$ admits a $13$ -parameter family of non-congruent minimal isometric immersions into spheres. Apart from a $7$ -dimensional subset, every minimal immersion of $\mathbb{T}^{3}_{r}$ in this family is non-homogeneous.

Next, we show the matrix data associated with such minimal immersions and use it to analyze their embeddedness. Define

Q=\frac{1}{3}\left(\begin{array}[]{ccc}4&-2&-2\\ -2&\frac{r^{2}+2}{r^{2}}&1\\ -2&1&\frac{r^{2}+2}{r^{2}}\\ \end{array}\right),

Y=\left(\begin{array}[]{cccccccccccc}\frac{r+1}{2}&\frac{1-r}{2}&\frac{r+1}{2}&\frac{1-r}{2}&\frac{1+p+q}{2}&\frac{1-p-q}{2}&\frac{1+p-q}{2}&\frac{1-p+q}{2}&\frac{1+p-q}{2}&\frac{1-p+q}{2}&\frac{1+p+q}{2}&\frac{1-p-q}{2}\\ r&-r&0&0&p&-p&-q&q&p&-p&q&-q\\ 0&0&r&-r&q&-q&p&-p&-q&q&p&-p\\ \end{array}\right).

(29)

Note that, by definition, $r$ and $p$ are odd while $q$ is even. Hence the columns of $Y$ are all integral vectors. Moreover, one can verify directly that $\xi_{j}$ is obtained as the linear combination of $\eta_{1},\eta_{2},\eta_{3}$ (given in (24)) with coefficients given by the $j$ -th column vector $Y_{j}$ . Consequently, for a minimal immersion constructed above that uses all vectors $\{\xi_{j}\}_{j=1}^{12}$ , the pair $\{Q,Y\}$ can be regarded as its matrix data.

Proposition 5.5.

For each $r\in\mathfrak{P}$ , every full minimal isometric immersion of the flat $3$ -torus $\mathbb{T}^{3}_{r}$ into $\mathbb{S}^{23}$ is embedded.

Proof.

By the fullness assumption, the pair $\{Q,Y\}$ provides the matrix data of such an immersion. One can see that in (12), if $\mathbf{u}Y$ is a nonzero integer vector with its coordinates satisfy $0\leq u_{j}<1$ , then from $\mathbf{u}(Y_{1}+Y_{2})\in\mathbb{Z}$ we have $\mathrm{u}_{1}=0$ . By

\mathbf{u}Y_{1}\in\mathbb{Z},~~~\mathbf{u}(Y_{5}+Y_{9})\in\mathbb{Z},~~~\mathbf{u}(Y_{7}+Y_{12})\in\mathbb{Z},

we derive

r\mathrm{u}_{2}\in\mathbb{Z},~~~2p\mathrm{u}_{2}\in\mathbb{Z},~~~2q\mathrm{u}_{2}\in\mathbb{Z},

which yields that $\mathrm{u}_{2}=0$ , since $(p,q,r)=1$ and $r$ is odd. Similarly, there holds $\mathrm{u}_{3}=0$ . Hence $\mathbf{u}=\mathbf{0}$ . By Remark 2.6, every homogeneous minimal isometric immersion of $\mathbb{T}^{3}_{r}$ associated to the matrix data $\{Q,Y\}$ is an embedding. In fact, the analysis in Remark 2.6 extends to non-homogeneous full immersions as well, since in this setting the matrix $A$ in (2) has full rank. This completes the proof of this proposition. ∎

Remark 5.6.

Because the minimal isometric immersion does not require the use of all eigenvectors, we can construct examples of the type described above even when $r$ is the hypotenuse of multiple Pythagorean triples. Consequently, for any integer that can be expressed as a sum of two distinct nonzero squares, one can associate a flat $3$ -torus together with non-homogeneous minimal isometric immersions (embeddings) into spheres.

Remark 5.7.

By Corollary 2.8, for flat $2$ -tori, the image of every minimal isometric immersion into a sphere is embedded. For higher-dimensional flat tori, Proposition 2.7 demonstrates that the same conclusion remains valid for any homogeneous minimal isometric immersions. Although Proposition 5.5 establishes the existence of abundant non-homogeneous minimal isometric embeddings of flat $3$ -tori into spheres, we shall present an example in the following subsection to illustrate that, even in dimension three, the image of a non-homogeneous minimal isometric immersion is not always embedded.

5.2 A minimal flat $3$ -torus in $\mathbb{S}^{7}$ whose image exhibits self-intersections

We continue to adopt the notation introduced in the previous subsection.

Example 5.8.

In (27) and (28), choose the following solution,

\alpha_{1}=\alpha_{2}=-\alpha_{3}=-\alpha_{4}=\frac{1}{8},~~~\alpha_{5}=\alpha_{6}=0,~~~\beta_{1}=\cdots=\beta_{6}=0,

a_{1}=\cdots=a_{8}=\frac{1}{8},~~~a_{9}=\cdots=a_{12}=0.

Then we can obtain a matrix

A=\frac{1}{\sqrt{8}}\begin{pmatrix}A_{1}&&&&\\ &\ddots&&&\\ &&A_{4}&&\\ &&&O&\\ &&&&O\end{pmatrix},\quad A_{1}=A_{2}=\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 1&0&0&0\\ 0&-1&0&0\end{pmatrix},\quad A_{3}=A_{4}=\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ -1&0&0&0\\ 0&1&0&0\end{pmatrix},

which yields a non-homogeneous minimal isometric immersion of $\mathrm{x}_{r}:\mathbb{T}^{3}_{r}\rightarrow\mathbb{S}^{7}\subset\mathbb{S}^{15}$ by (2).

We first show that this immersion is not an embedding. Set

\widetilde{Q}=\left(\begin{array}[]{ccc}1&0&0\\ 0&m_{1}&m_{2}\\ 0&-m_{2}&m_{1}\\ \end{array}\right)Q\left(\begin{array}[]{ccc}1&0&0\\ 0&m_{1}&-m_{2}\\ 0&m_{2}&m_{1}\\ \end{array}\right),

\widetilde{Y}=\left(\begin{array}[]{cccccccccccc}\frac{r+1}{2}&\frac{1-r}{2}&\frac{r+1}{2}&\frac{1-r}{2}&\frac{1+p+q}{2}&\frac{1-p-q}{2}&\frac{1+p-q}{2}&\frac{1-p+q}{2}\\ m_{1}&-m_{1}&m_{2}&-m_{2}&m_{1}&-m_{1}&-m_{2}&m_{2}\\ -m_{2}&m_{2}&m_{1}&-m_{1}&m_{2}&-m_{2}&m_{1}&-m_{1}\\ \end{array}\right),

(30)

where $\{m_{1},m_{2}\}$ is the unique ordered pair of coprime positive integers satisfying $r=m_{1}^{2}+m_{2}^{2}$ . Let $\widetilde{\Lambda}_{3,r}^{*}$ be the lattice generated by

\widetilde{\eta}_{1}\triangleq\eta_{1},~~~\widetilde{\eta}_{2}\triangleq m_{1}\eta_{2}+m_{2}\eta_{3},~~~\widetilde{\eta}_{3}\triangleq m_{1}\eta_{3}-m_{2}\eta_{2}.

(31)

It takes $\widetilde{Q}$ as the Gram matrix. We denote by $\widetilde{\Lambda}_{3,r}$ its dual lattice, and $\widetilde{\mathbb{T}}^{3}_{r}=\mathbb{R}^{3}/\widetilde{\Lambda}_{3,r}$ the corresponding flat $3$ -torus. Note that $\mathbb{T}^{3}_{r}$ forms a $r$ -fold cover of $\widetilde{\mathbb{T}}^{3}_{r}$ . For $1\leq j\leq 8$ , one can verify directly that $\xi_{j}$ is exactly the linear combination of $\widetilde{\eta}_{1},\widetilde{\eta}_{2},\widetilde{\eta}_{3}$ with coefficients given by the $j$ -th column vector $\widetilde{Y}_{j}$ . Consequently, the minimal isometric immersion $\mathrm{x}_{r}:\mathbb{T}^{3}_{r}\rightarrow\mathbb{S}^{7}$ factors through a minimal isometric immersion $\widetilde{\mathrm{x}}_{r}:\widetilde{\mathbb{T}}^{3}_{r}\rightarrow\mathbb{S}^{7}$ , whose matrix data is $\{\widetilde{Q},\widetilde{Y}\}$ . By (2), (12) and (30), the immersion $\widetilde{\mathrm{x}}_{r}$ admits the following explicit expression:

\begin{split}\widetilde{\mathrm{x}}_{r}(\mathrm{u}_{1},\mathrm{u}_{2},\mathrm{u}_{3})=\frac{1}{\sqrt{2}}\Bigl(&\cos(\mathrm{u}_{1}\pi)\,e^{2\pi\mathbf{i}\bigl(\frac{r}{2}\mathrm{u}_{1}+m_{1}\mathrm{u}_{2}-m_{2}\mathrm{u}_{3}\bigr)},\ \cos(\mathrm{u}_{1}\pi)\,e^{2\pi\mathbf{i}\bigl(\frac{r}{2}\mathrm{u}_{1}+m_{2}\mathrm{u}_{2}+m_{1}\mathrm{u}_{3}\bigr)},\\ &\sin(\mathrm{u}_{1}\pi)\,e^{2\pi\mathbf{i}\bigl(\frac{p+q}{2}\mathrm{u}_{1}+m_{1}\mathrm{u}_{2}+m_{2}\mathrm{u}_{3}\bigr)},\ \sin(\mathrm{u}_{1}\pi)\,e^{2\pi\mathbf{i}\bigl(\frac{p-q}{2}\mathrm{u}_{1}-m_{2}\mathrm{u}_{2}+m_{1}\mathrm{u}_{3}\bigr)}\Bigr),\end{split}

(32)

with $0\leq\mathrm{u}_{1},\mathrm{u}_{2},\mathrm{u}_{3}<1$ .

Next, we show that even $\widetilde{\mathrm{x}}_{r}:\widetilde{\mathbb{T}}^{3}_{r}\rightarrow\mathbb{S}^{7}$ fails to be an embedding, and its image possesses self-intersections. Note that for any $\mathrm{u}_{1},\mathrm{u}_{1}^{\prime}\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$ , $\widetilde{\mathrm{x}}_{r}(\mathrm{u}_{1},\mathrm{u}_{2},\mathrm{u}_{3})=\widetilde{\mathrm{x}}_{r}(\mathrm{u}_{1}^{\prime},\mathrm{u}_{2}^{\prime},\mathrm{u}_{3}^{\prime})$ implies $\mathrm{u}_{1}=\mathrm{u}_{1}^{\prime}$ or $\mathrm{u}_{1}=1-\mathrm{u}_{1}^{\prime}$ . Set $\delta_{2}\triangleq\mathrm{u}_{2}-\mathrm{u}_{2}^{\prime}$ , $\delta_{3}\triangleq\mathrm{u}_{3}-\mathrm{u}_{3}^{\prime}$ . Then in the former case, we get

m_{1}\delta_{2}-m_{2}\delta_{3}\in\mathbb{Z},~~m_{2}\delta_{2}+m_{1}\delta_{3}\in\mathbb{Z},~~m_{1}\delta_{2}+m_{2}\delta_{3}\in\mathbb{Z},~~-m_{2}\delta_{2}+m_{1}\delta_{3}\in\mathbb{Z}.

Then we have

	$\displaystyle r\delta_{2}$	$\displaystyle=m_{1}(m_{1}\delta_{2}-m_{2}\delta_{3})+m_{2}(m_{2}\delta_{2}+m_{1}\delta_{3})\in\mathbb{Z},$		(33)
	$\displaystyle 2p\delta_{2}$	$\displaystyle=(m_{1}-m_{2})[(m_{1}\delta_{2}-m_{2}\delta_{3})+(m_{2}\delta_{2}+m_{1}\delta_{3})+(m_{1}\delta_{2}+m_{2}\delta_{3})+(m_{2}\delta_{2}-m_{1}\delta_{3})]\in\mathbb{Z}.$		(33)

Since $(r,p)=1$ , this forces $\delta_{2}\in\mathbb{Z}$ , and consequently $\delta_{2}=0$ . It follows that $m_{1}\delta_{3}\in\mathbb{Z}$ and $m_{2}\delta_{3}\in\mathbb{Z}$ , which further implies $\delta_{3}=0$ since $(m_{1},m_{2})=1$ . In the latter case, we obtain

\begin{gathered}\frac{1}{2}+\frac{r}{2}(2\mathrm{u}_{1}-1)+m_{1}\delta_{2}-m_{2}\delta_{3}\in\mathbb{Z},~~\frac{1}{2}+\frac{r}{2}(2\mathrm{u}_{1}-1)+m_{2}\delta_{2}+m_{1}\delta_{3}\in\mathbb{Z},\\ \frac{p+q}{2}(2\mathrm{u}_{1}-1)+m_{1}\delta_{2}+m_{2}\delta_{3}\in\mathbb{Z},~~\frac{p-q}{2}(2\mathrm{u}_{1}-1)-m_{2}\delta_{2}+m_{1}\delta_{3}\in\mathbb{Z}.\end{gathered}

which is equivalent to

	$\displaystyle m_{1}(\delta_{2}+(m_{1}+m_{2})\mathrm{u}_{1})-m_{2}(\delta_{3}+(m_{1}-m_{2})\mathrm{u}_{1})\in\mathbb{Z},$		(34)
	$\displaystyle m_{2}(\delta_{2}+(m_{1}+m_{2})\mathrm{u}_{1})+m_{1}(\delta_{3}+(m_{1}-m_{2})\mathrm{u}_{1})\in\mathbb{Z},$		(35)
	$\displaystyle\frac{1}{2}+m_{1}(\delta_{2}+(m_{1}+m_{2})\mathrm{u}_{1})+m_{2}(\delta_{3}+(m_{1}-m_{2})\mathrm{u}_{1})\in\mathbb{Z},$		(36)
	$\displaystyle-\frac{1}{2}+m_{2}(\delta_{2}+(m_{1}+m_{2})\mathrm{u}_{1})-m_{1}(\delta_{3}+(m_{1}-m_{2})\mathrm{u}_{1})\in\mathbb{Z}.$		(37)

The linear combination of (34) and (35) with coefficients $m_{1}$ and $m_{2}$ yields that $r\big(\delta_{2}+(m_{1}+m_{2})\mathrm{u}_{1}\big)\in\mathbb{Z}$ . Summing equations (34) through (37) and multiplying $m_{1}-m_{2}$ gives $2p\bigl(\delta_{2}+(m_{1}+m_{2})\mathrm{u}_{1}\bigr)\in\mathbb{Z}$ . Hence by $(r,p)=1$ , we derive that $\delta_{2}+(m_{1}+m_{2})\mathrm{u}_{1}\in\mathbb{Z}$ and therefore,

m_{2}(\delta_{3}+(m_{1}-m_{2})\mathrm{u}_{1})\in\mathbb{Z},\quad\frac{1}{2}+m_{2}(\delta_{3}+(m_{1}-m_{2})\mathrm{u}_{1})\in\mathbb{Z}

which can never occur.

This means the immersion $\widetilde{\mathrm{x}}_{r}(\mathrm{u}_{1},\mathrm{u}_{2},\mathrm{u}_{3})$ is embedded at

\{(\mathrm{u}_{1},\mathrm{u}_{2},\mathrm{u}_{3})\mid\mathrm{u}_{1}\in(0,\frac{1}{2})\cup(\frac{1}{2},1),0\leq\mathrm{u}_{2},\mathrm{u}_{3}<1\}.

However, it is easy to see that

\widetilde{\mathrm{x}}_{r}(0,\mathrm{u}_{2},\mathrm{u}_{3})=x(0,\mathrm{u}_{2}+\frac{m_{1}}{r},\mathrm{u}_{3}-\frac{m_{2}}{r})=x(0,\mathrm{u}_{2}+\frac{m_{2}}{r},\mathrm{u}_{3}+\frac{m_{1}}{r}),

\widetilde{\mathrm{x}}_{r}(\frac{1}{2},\mathrm{u}_{2},\mathrm{u}_{3})=x(\frac{1}{2},\mathrm{u}_{2}+\frac{m_{1}}{r},\mathrm{u}_{3}+\frac{m_{2}}{r})=x(\frac{1}{2},\mathrm{u}_{2}-\frac{m_{2}}{r},\mathrm{u}_{3}+\frac{m_{1}}{r}).

So the image of the minimal immersion $\widetilde{\mathrm{x}}_{r}:\widetilde{\mathbb{T}}^{3}_{r}\rightarrow\mathbb{S}^{7}$ has self-intersections.

6 Isometric deformation and irrationality degree of minimal flat $n$ -tori

In this section, we first address the minimal target dimension problem for minimally immersed flat $n$ -tori through isometric deformation analysis. Then, we establish an upper bound for the irrationality degrees of minimal flat $n$ -tori.

6.1 The isometric deformation of minimal flat $n$ -tori in spheres

Theorem 6.1.

Proof.

We denote by $\{Q,Y\}$ the matrix data of $x$ , where $Q$ is the Gram matrix of $\Lambda_{n}$ with respect to a chosen generator, and $Y=\{Y_{1},\cdots,Y_{N}\}$ is the integer vectors appearing in the minimal immersion $x$ . First, the immersion $x$ can deform into a homogeneous immersion into $\mathbb{S}^{2N-1}$ (may not be linearly full). Then from Theorem 2.4, $\frac{Q^{-1}}{n}$ lies in $C_{Y}$ as the maximal point of the determinant function restricted on $C_{Y}$ , where $C_{Y}$ is a convex polytope decomposed (not necessarily uniquely) as a union of some simplices of dimension at most $\frac{n(n+1)}{2}{-1}$ . Therefore, $\frac{Q^{-1}}{n}$ must lie in the interior of some simplex of dimension at most $\frac{n(n+1)}{2}{-1}$ , which means there exists a subset $\tilde{Y}\subset Y$ whose cardinality at most $\frac{n(n+1)}{2}$ such that $\frac{Q^{-1}}{n}\in\overset{\circ}{C_{\tilde{Y}}}$ . As a maximum point of $\det$ on $C_{Y}$ , it automatically maximizes $\det$ on $C_{\tilde{Y}}$ . It follows from Theorem 2.4 that the matrix data $\{Q,\tilde{Y}\}$ provides a homogeneous minimal and isometric immersion of $T^{n}$ in $\mathbb{S}^{p}$ . Such immersion is obviously a special solution of (11) and hence this completes the proof. ∎

Combining Theorem 6.1 with Proposition 3.5, we obtain the following corollary.

Corollary 6.2.

For every rational flat $n$ -torus $T^{n}$ , there exist infinitely many positive integers $k$ such that $T^{n}$ admits a minimal immersion into $\mathbb{S}^{n^{2}+n-1}$ realized by the $k$ -th eigenfunctions.

Remark 6.3.

1.

Combining Theorem 3.4 with Theorem 6.1 we obtain that up to a dilation, every flat rational $n$ -torus admits a minimal and isometric immersion in $\mathbb{S}^{n^{2}+n-1}$ .
2.

In particular, by Theorem 6.1, every minimal flat $2$ -torus admits an isometric and minimal immersion in $\mathbb{S}^{5}$ . See below for an illustration by some special examples. Moreover, for any such flat $2$ -torus, there exist infinitely many integers $k$ so that the minimal immersions into $\mathbb{S}^{5}$ can be realized by the $k$ -th eigenfunctions.

Example 6.4.

By Bryant’s theorem, a flat torus $T_{a,b}=\mathbb{R}^{2}/2\pi(\mathbb{Z}\oplus(a+ib)\mathbb{Z})$ can be immersed into $S^{n}$ by flat minimal isometric immersions if and only if both $a$ and $b^{2}$ are rational numbers. It is well-known that the eigenfunctions and eigenvalues of $T_{a,b}$ are

\cos(pu+\frac{q-pa}{b}v),~~\sin(pu+\frac{q-pa}{b}v),~~\lambda_{p,q}=p^{2}+\frac{(q-pa)^{2}}{b^{2}}.

Consider a special family of $2$ -tori with

a=\frac{m}{n}<\frac{1}{2},~~b=\sqrt{1-a^{2}}=\frac{\sqrt{n^{2}-m^{2}}}{n}.

For such $2$ -tori we have

\lambda_{p,q}=\frac{(p^{2}+q^{2})n^{2}-2pqmn}{n^{2}-m^{2}}.

One can check that when $\{p,q\}=\{n,0\}$ and $\{p,q\}=\{n,2m\}$ , we always have

\lambda_{p,q}=\frac{n^{4}}{n^{2}-m^{2}}

with corresponding eigenfunctions

e^{\pm i\frac{n^{2}v}{\sqrt{n^{2}-m^{2}}}},e^{\pm i\left(nu+\frac{mnv}{\sqrt{n^{2}-m^{2}}}\right)},e^{\pm i\left(nu-\frac{mnv}{\sqrt{n^{2}-m^{2}}}\right)},e^{\pm i\left(2mu+\frac{(n^{2}-2m^{2})v}{\sqrt{n^{2}-m^{2}}}\right)}.

Then we see that the immersion $f:T_{a,b}\rightarrow S^{7}\subset\mathbb{R}^{8}=\mathbb{C}^{4}$

f:=\left(r_{1}e^{i\frac{n^{2}v}{\sqrt{n^{2}-m^{2}}}},r_{2}e^{i\left(nu+\frac{mnv}{\sqrt{n^{2}-m^{2}}}\right)},r_{3}e^{i\left(nu-\frac{mnv}{\sqrt{n^{2}-m^{2}}}\right)},r_{4}e^{i\left(2mu+\frac{(n^{2}-2m^{2})v}{\sqrt{n^{2}-m^{2}}}\right)}\right)

is a minimal immersion if and only if

\left\{\begin{aligned} &r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r_{4}^{2}=1,\\ &n^{4}r_{1}^{2}-n^{2}(n^{2}-2m^{2})(r_{2}^{2}+r_{3}^{2})+(n^{4}-8m^{2}n^{2}+8m^{4})r_{4}^{2}=0,\\ &n^{2}(r_{2}^{2}-r_{3}^{2})+2(n^{2}-2m^{2})r_{4}^{2}=0.\\ \end{aligned}\right.

The second and third equation can be re-written as

\left\{\begin{aligned} &r_{1}^{2}-(b^{2}-a^{2})(r_{2}^{2}+r_{3}^{2})+(1-8a^{2}+8a^{4})r_{4}^{2}=0,\\ &r_{3}^{2}-r_{2}^{2}=2(b^{2}-a^{2})r_{4}^{2}.\\ \end{aligned}\right.

Setting $r_{4}=\rho\geq 0$ , we obtain the solutions to the above equations are

\left\{\begin{aligned} &r_{1}^{2}=\frac{b^{2}-a^{2}}{2b^{2}}-(b^{2}-3a^{2})\rho^{2},\\ &r_{2}^{2}=\frac{1}{4b^{2}}-\rho^{2},\\ &r_{3}^{2}=\frac{1}{4b^{2}}+(b^{2}-3a^{2})\rho^{2},\\ \end{aligned}\right.\hbox{ with }0\leq\rho\leq\frac{1}{2b}.

This provides a family of minimal flat $2$ -tori in $S^{7}$ , which is full in $S^{7}$ when $0<\rho<\frac{1}{2b}$ , and is contained in some $S^{5}\subset S^{7}$ when $\rho=0$ or $\frac{1}{2b}$ .

Note that when $n=2k$ , one can choose $\{p,q\}=\{k,0\}$ and $\{p,q\}=\{k,m\}$ with $\lambda_{p,q}=\frac{k^{4}}{4k^{2}-m^{2}}$ , to construct a family of flat minimal tori in $\mathbb{S}^{7}$ . We leave it to interested readers.

Remark 6.5.

For a minimal flat $n$ -torus $x:T^{n}\rightarrow\mathbb{S}^{m}$ , if $m>n(n+1)-1$ , then the minimal immersion of $T^{n}$ in $\mathbb{S}^{n(n+1)-1}$ , as described in Theorem 6.1, is situated on the boundary of the moduli space $\mathcal{M}(T^{n})$ of minimal isometric immersions of $T^{n}$ . If $\dim{\mathcal{M}(T^{n})}>0$ , then from the convexity of $\mathcal{M}(T^{n})$ , any two minimal isometric immersions of $T^{n}$ can be deformed to each other by a smooth homotopy of minimal isometric immersions. We note that during the deformation of two minimal isometric immersions with equal codimension, the codimension can first increase and later decrease.

6.2 The algebraic irrationality of minimal flat $n$ -tori

Employing methods analogous to those developed in Subsection 4.1, we derive an upper bound estimate for the algebraic irrationality of minimal flat $n$ -tori. The following algebraic lemma will be used.

Lemma 6.6.

Let $P_{1},P_{2},\cdots,P_{m}\in\mathbb{Q}[x_{1},x_{2},\ldots,x_{m}]$ be polynomials with respective degrees $d_{1},d_{2},\cdots,d_{m}$ , and $(\alpha_{1},\alpha_{2},\cdots,\alpha_{m})$ be an isolated common zero point of these polynomials. Then the field extension degree $[\mathbb{Q}(\alpha_{1},\alpha_{2},\cdots,\alpha_{m}):\mathbb{Q}]$ satisfies

[\mathbb{Q}(\alpha_{1},\alpha_{2},\cdots,\alpha_{m}):\mathbb{Q}]\leq d_{1}d_{2}\cdots d_{m}.

Proof.

For each $1\leq i\leq m$ , let $\widetilde{P}_{i}\in\mathbb{Q}[x_{1},x_{2},\cdots,x_{m},x_{m+1}]$ be the homogenization of the polynomial $P_{i}$ . We denote by $V$ the projective scheme defined by $\widetilde{P}_{1},\widetilde{P}_{2},\cdots,\widetilde{P}_{m}$ in $\mathbb{P}^{m}_{\mathbb{Q}}$ .

Let $p$ be the corresponding closed point of $(\alpha_{1},\alpha_{2},\cdots,\alpha_{m})$ in $V$ . It follows from the generalized Bézout theorem over the non-algebraically closed field [14, Proposition 8.4, Page 145] (see also [23]) that

l(\mathcal{O}_{p}(V))[\mathbb{Q}(\alpha_{1},\alpha_{2},\cdots,\alpha_{m}):\mathbb{Q}]\leq d_{1}d_{2}\cdots d_{m},

where $l(\mathcal{O}_{p}(V))$ denotes the length of the local ring $\mathcal{O}_{p}(V)$ of $V$ at $p$ . Since $l(\mathcal{O}_{p}(V))\geq 1$ , the conclusion immediately follows. ∎

Theorem 6.7.

Suppose $T^{n}=\mathbb{R}^{n}/\Lambda_{n}$ is a flat $n$ -torus that admits a minimal isometric immersion into some sphere, equipped with the corresponding matrix data $\{Q,Y\}$ . Let $K/\mathbb{Q}$ be the minimal field extension containing all entries of $Q$ . Then the extension degree satisfies

[K:\mathbb{Q}]\leq\min\{(n-1)^{k-1},(n-1)^{s}\},

where $k=\mathrm{rank}\{Y_{j}Y_{j}^{t}\,|\,Y_{j}\in Y\},~s=\frac{n(n+1)}{2}-k$ .

Proof.

Without loss of generality, we assume that

\mathrm{rank}\{Y_{1}Y_{1}^{t},Y_{2}Y_{2}^{t},\cdots,Y_{k}Y_{k}^{t}\}=k.

On the one hand, from the proof of Theorem 6.1, we may assume $\frac{Q^{-1}}{n}$ lies in the convex hull of $\{Y_{1}Y_{1}^{t},Y_{2}Y_{2}^{t},\cdots,Y_{k}Y_{k}^{t}\}$ , which may be locally parameterized as

Y_{k}Y_{k}^{t}+\lambda_{1}(Y_{1}Y_{1}^{t}-Y_{k}Y_{k}^{t})+\lambda_{2}(Y_{2}Y_{2}^{t}-Y_{k}Y_{k}^{t})+\cdots+\lambda_{k-1}(Y_{k-1}Y_{k-1}^{t}-Y_{k}Y_{k}^{t}).

Then it follows from Theorem 2.4 that, as the unique critical point of the determinant function restricted on this convex hull, the parameter of $\frac{Q^{-1}}{n}$ satisfies a system of $k-1$ polynomial equations in the variables $(\lambda_{1},\cdots,\lambda_{k-1})$ , with rational coefficients and degree $n-1$ . Moreover, it is an isolated common zero point. Therefore, by Lemma 6.6 we have

[K:\mathbb{Q}]\leq(n-1)^{k-1}.

On the other hand, similar to the approach introduced at the beginning of Section 4, we can parameterize $W_{Y}$ as follows,

W_{Y}=\{Q_{0}+t_{1}Q_{1}+\cdots+t_{s}Q_{s}|t_{i}\in\mathbb{R}\}\cap\Sigma_{+},

where $Q_{0}\in{\mathrm{Sym_{n}}}\cap GL(n,\mathbb{Q})$ is the unique matrix in $\mathrm{Span}\{Y_{j}Y_{j}^{t}\,|\,Y_{j}\in Y\}$ satisfying

\langle Q_{0},Y_{i}Y_{i}^{t}\rangle=1,~~~1\leq i\leq\sharp(Y),

and $\{Q_{1},\cdots,Q_{s}\}\subset{\mathrm{Sym_{n}}}\cap GL(n,\mathbb{Q})$ is a basis of the orthogonal complement of $\mathrm{Span}\{Y_{j}Y_{j}^{t}\,|\,Y_{j}\in Y\}$ . Consequently, as the critical point of the determinant function restricted on $W_{Y}$ , the parameter of $Q$ satisfies a system of $s$ polynomial equations in the variables $(t_{1},\cdots,t_{s})$ , with rational coefficients and degree $n-1$ . So by Lemma 6.6, we also have

[K:\mathbb{Q}]\leq(n-1)^{s}.

∎

Proof of Theorem 2.

Note that

\max_{n\leq k\leq\frac{n(n+1)}{2}}\min\{k-1,\frac{n(n+1)}{2}-k\}={[\frac{(n-1)(n+2)}{4}]}.

It follows that the extension degree in Theorem 6.7 satisfies

[K:\mathbb{Q}]\leq(n-1)^{\lfloor\frac{(n-1)(n+2)}{4}\rfloor}.

(38)

∎

Remark 6.8.

For the case of $3$ -tori, the upper bound given in (38) equals $4$ , and it is optimal by Theorem 4.2 and examples in Section 4. For the case of $n$ -tori with $n\geq 4$ , it is unknown whether the corresponding upper bound is sharp.

Acknowledgement: The first author and the third author is supported by NSFC No. 12171473. The second author is supported by NSFC No. 12371052. The third author is also partially supported by the Fundamental Research Funds for Central Universities.

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Minimal Isometric Immersions of Flat nn-tori into Spheres

Abstract

1 Introduction

Theorem 1.

Theorem 2.

Theorem 3.

2 Preliminaries

2.1 Flat tori and lattices

Definition 2.1.

2.2 Algebraic characterizations of minimal isometric immersions of flat nn-tori

Definition 2.2.

Proposition 2.3.

2.3 A variational characterization

Theorem 2.4.

Remark 2.5.

Remark 2.6.

Proposition 2.7.

Proof.

Corollary 2.8.

3 Minimal isometric immersions of rational flat nn-tori into spheres

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Lemma 3.3.

Theorem 3.4.

Proof.

Proposition 3.5.

Remark 3.6.

4 Minimal isometric immersions of flat 33-tori

4.1 Irrational minimal flat 33-tori

Example 4.1.

Theorem 4.2.

Proof.

Remark 4.3.

Example 4.4.

Example 4.5.

Example 4.6.

4.2 The uniqueness of minimal immersion for irrational flat 33-torus

Theorem 4.7.

Proof.

Remark 4.8.

4.3 Homogeneity of irrational minimal flat 33-torus

Theorem 4.9.

Proof.

5 Examples of non-homogeneous minimal flat 33-tori in spheres

5.1 A family of non-homogeneous minimal isometric immersions of 𝕋r3\mathbb{T}^{3}_{r} into spheres

Lemma 5.1.

Lemma 5.2.

Proof.

Lemma 5.3.

Proof.

Proposition 5.4.

Proposition 5.5.

Proof.

Remark 5.6.

Remark 5.7.

5.2 A minimal flat 33-torus in 𝕊7\mathbb{S}^{7} whose image exhibits self-intersections

Example 5.8.

6 Isometric deformation and irrationality degree of minimal flat nn-tori

6.1 The isometric deformation of minimal flat nn-tori in spheres

Theorem 6.1.

Proof.

Corollary 6.2.

Remark 6.3.

Example 6.4.

Remark 6.5.

6.2 The algebraic irrationality of minimal flat nn-tori

Lemma 6.6.

Proof.

Theorem 6.7.

Proof.

Proof of Theorem 2.

Remark 6.8.

References

Minimal Isometric Immersions of Flat $n$ -tori into Spheres

2.2 Algebraic characterizations of minimal isometric immersions of flat $n$ -tori

3 Minimal isometric immersions of rational flat $n$ -tori into spheres

4 Minimal isometric immersions of flat $3$ -tori

4.1 Irrational minimal flat $3$ -tori

4.2 The uniqueness of minimal immersion for irrational flat $3$ -torus

4.3 Homogeneity of irrational minimal flat $3$ -torus

5 Examples of non-homogeneous minimal flat $3$ -tori in spheres

5.1 A family of non-homogeneous minimal isometric immersions of $\mathbb{T}^{3}_{r}$ into spheres

5.2 A minimal flat $3$ -torus in $\mathbb{S}^{7}$ whose image exhibits self-intersections

6 Isometric deformation and irrationality degree of minimal flat $n$ -tori

6.1 The isometric deformation of minimal flat $n$ -tori in spheres

6.2 The algebraic irrationality of minimal flat $n$ -tori