Showing 1–13 of 13 results for author: Chow, T A

Normal Curvature and the Projective Systole

Authors: Tsz-Kiu Aaron Chow, Jingbo Wan

Abstract: For a smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright \overline{\mathbb{B}}^{N}(1)$, we observe that the sharp systolic inequality forces a sharp lower bound on its maximal normal curvature $κ(F)$. In dimensions $m=2,3$, the sharp inequalities of Pu and Bray--Brendle--Eichmair--Neves therefore give $κ(F)^2\ge \frac{2m}{m+1}$. Equality holds precisely for the Veronese embedding. This reco… ▽ More For a smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright \overline{\mathbb{B}}^{N}(1)$, we observe that the sharp systolic inequality forces a sharp lower bound on its maximal normal curvature $κ(F)$. In dimensions $m=2,3$, the sharp inequalities of Pu and Bray--Brendle--Eichmair--Neves therefore give $κ(F)^2\ge \frac{2m}{m+1}$. Equality holds precisely for the Veronese embedding. This recovers Petrunin's theorem for $\mathbb{R}\mathbb{P}^2$ and, for $\mathbb{R}\mathbb{P}^3$, confirms the first open case of his question for real projective spaces. △ Less

Submitted 18 August, 2026; originally announced August 2026.

Maximal Normal Curvature and Veronese Rigidity

Authors: Tsz-Kiu Aaron Chow, Jingbo Wan

Abstract: We prove a sharp Veronese rigidity theorem for closed immersed submanifolds of the Euclidean unit ball under intrinsic harmonic-structure assumptions. For an isometric immersion $F:(Σ,g)\looparrowright\overline B(1)$, define the maximal normal curvature by \[ κ(F):= \sup_{x\inΣ} \sup_{\substack{v\in T_xΣ\\ |v|_g=1}} |A_x(v,v)|. \] If $Σ^{2n}$ is almost Hermitian with harmonic fundamental t… ▽ More We prove a sharp Veronese rigidity theorem for closed immersed submanifolds of the Euclidean unit ball under intrinsic harmonic-structure assumptions. For an isometric immersion $F:(Σ,g)\looparrowright\overline B(1)$, define the maximal normal curvature by \[ κ(F):= \sup_{x\inΣ} \sup_{\substack{v\in T_xΣ\\ |v|_g=1}} |A_x(v,v)|. \] If $Σ^{2n}$ is almost Hermitian with harmonic fundamental two-form, or $Σ^{4n}$ is almost quaternion-Hermitian with harmonic fundamental four-form, $n\ge2$, then \[ κ(F)\ge \sqrt{\frac{2n}{n+1}} . \] In the equality case the harmonic form is parallel and the immersion is, up to a totally geodesic inclusion, the standard complex or quaternionic Veronese embedding of projective spaces. The key input is a Bochner--Gauss mechanism that turns the Bochner curvature term of the harmonic form into a sharp algebraic estimate for the shape operators. △ Less

Submitted 1 July, 2026; originally announced July 2026.

Comments: Comments are welcome!

Sharp focal radius estimate and rigidity of hypersurfaces in manifolds with positive curvature

Authors: Tsz-Kiu Aaron Chow, Jingbo Wan

Abstract: We prove a sharp Clifford-threshold focal-radius estimate and rigidity for immersed hypersurfaces. Under a $p$-form curvature condition, formulated by the Weitzenböck curvature term together with $\mathrm{Ric}_p\ge p$, any closed two-sided immersion $F:Σ^m\to M^{m+1}$ with $b_p(Σ;\mathbb R)\neq0$ and $1\le p\le m/2$ satisfies \[ r_f(F,M)\le\fracπ{4}. \] The equality case is rigid: if the ambient… ▽ More We prove a sharp Clifford-threshold focal-radius estimate and rigidity for immersed hypersurfaces. Under a $p$-form curvature condition, formulated by the Weitzenböck curvature term together with $\mathrm{Ric}_p\ge p$, any closed two-sided immersion $F:Σ^m\to M^{m+1}$ with $b_p(Σ;\mathbb R)\neq0$ and $1\le p\le m/2$ satisfies \[ r_f(F,M)\le\fracπ{4}. \] The equality case is rigid: if the ambient manifold is complete, equality forces the hypersurface to be locally the Clifford hypersurface $S^p(1/\sqrt2)\times S^{m-p}(1/\sqrt2)\subset S^{m+1}(1)$; if the ambient manifold is compact and connected, it is a spherical space form. The curvature condition follows from $\sec\ge1$ for $p=1$, from normalized $\mathrm{PIC1}\ge1$ for $p=2$, and from curvature operator bounded below by one in all degrees. By quotient lifting and the Hopf fibrations, we also obtain focal-radius estimates in $\mathbb{CP}^n$ and $\mathbb{HP}^n$, with projective Clifford rigidity, without any Betti-number assumption. △ Less

Submitted 1 June, 2026; originally announced June 2026.

A Spinorial Perelman's Functional: Critical Points and Gradient Flow

Authors: Tsz-Kiu Aaron Chow, Frederick Tsz-Ho Fong

Abstract: In this article, we introduce an energy functional on closed Riemannian spin manifolds which unifies Perelman's W- and F-functionals, Baldauf-Ouzch's E-functional, and Dirchlet energy for spinors. We compute its first variation formula, and show that its critical points under natural constraints are twisted Ricci solitons and eigen-spinsors of the weighted Dirac operator. We introduce a negative L… ▽ More In this article, we introduce an energy functional on closed Riemannian spin manifolds which unifies Perelman's W- and F-functionals, Baldauf-Ouzch's E-functional, and Dirchlet energy for spinors. We compute its first variation formula, and show that its critical points under natural constraints are twisted Ricci solitons and eigen-spinsors of the weighted Dirac operator. We introduce a negative L^2-gradient flow of this functional, and establish its short-time existence and uniqueness via contraction mapping methods. △ Less

Submitted 5 January, 2026; originally announced January 2026.

Comments: 23 pages. Comments are welcome!

MSC Class: 53E20; 53C27

Scalar curvature rigidity for products of spheres and tori

Authors: Tsz-Kiu Aaron Chow

Abstract: We prove Llarull-type rigidity for $S^{n-m}\times\mathbb{T}^m$ ($3\le n\le 7$, $1\le m\le n-2$). If a closed spin $(M^n,g)$ admits a degree-nonzero map to $S^{n-m}\times\mathbb{T}^m$ whose spherical projection is area non-increasing, and there exists $ψ\in C^\infty(M)$ with $-Δ_Mψ-\frac{1}{2}|D_Mψ|^2+\frac{1}{2}\big(R_M-(n-m)(n-m-1)\big)\ge0$, then $(M,g)$ is isometrically covered by… ▽ More We prove Llarull-type rigidity for $S^{n-m}\times\mathbb{T}^m$ ($3\le n\le 7$, $1\le m\le n-2$). If a closed spin $(M^n,g)$ admits a degree-nonzero map to $S^{n-m}\times\mathbb{T}^m$ whose spherical projection is area non-increasing, and there exists $ψ\in C^\infty(M)$ with $-Δ_Mψ-\frac{1}{2}|D_Mψ|^2+\frac{1}{2}\big(R_M-(n-m)(n-m-1)\big)\ge0$, then $(M,g)$ is isometrically covered by $S^{n-m}\times\mathbb{R}^m$. For bands, we extend Gromov's torical inequality and obtain sharp width bounds: $\text{dist}(\partial_-M,\partial_+M)\le 2π\sqrt{n/((n+1)σ)}$ when $R_M\ge (n-m)(n-m-1)+σ$. The method combines stable weighted slicing with a spectral Dirac operator argument. △ Less

Submitted 6 November, 2025; originally announced November 2025.

Comments: 24 pages, comments are welcome!

MSC Class: 53C21; 53C24; 53C27

Bandwidth and focal radius with positive isotropic curvature

Authors: Tsz-Kiu Aaron Chow, Jingze Zhu

Abstract: This paper investigates quantitative metric inequalities for manifolds with positive isotropic curvature (PIC). Our results include upper bounds on the bandwidth and focal radius of hypersurfaces in PIC manifolds, contingent on boundary convexities and Betti numbers. The proof is based on exploiting the spectral properties of a twisted de Rham-Hodge operator on manifolds with boundary. This paper investigates quantitative metric inequalities for manifolds with positive isotropic curvature (PIC). Our results include upper bounds on the bandwidth and focal radius of hypersurfaces in PIC manifolds, contingent on boundary convexities and Betti numbers. The proof is based on exploiting the spectral properties of a twisted de Rham-Hodge operator on manifolds with boundary. △ Less

Submitted 30 May, 2024; originally announced May 2024.

Comments: 44 pages, comments are welcome!

Rigidity results for initial data sets satisfying the dominant energy condition

Authors: Christian Baer, Simon Brendle, Tsz-Kiu Aaron Chow, Bernhard Hanke

Abstract: Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is based on the solution of a boundary value problem for Dirac operators. For convex polytopes we use approximations by manifolds with smooth boundary. Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is based on the solution of a boundary value problem for Dirac operators. For convex polytopes we use approximations by manifolds with smooth boundary. △ Less

Submitted 10 November, 2025; v1 submitted 8 April, 2023; originally announced April 2023.

Comments: Published version (up to layout)

MSC Class: 53C20; 53C23; 53C24

Journal ref: J. reine angew. Math.2025

Preserving positive intermediate curvature

Authors: Tsz-Kiu Aaron Chow, Florian Johne, Jingbo Wan

Abstract: Consider a compact manifold $N$ (with or without boundary) of dimension $n$. Positive $m$-intermediate curvature interpolates between positive Ricci curvature ($m = 1$) and positive scalar curvature ($m = n-1$), and it is obstructed on partial tori $N^n = M^{n-m} \times \mathbb{T}^m$. Given Riemannian metrics $g, \bar{g}$ on $(N, \partial N)$ with positive $m$-intermediate curvature and $m$-positi… ▽ More Consider a compact manifold $N$ (with or without boundary) of dimension $n$. Positive $m$-intermediate curvature interpolates between positive Ricci curvature ($m = 1$) and positive scalar curvature ($m = n-1$), and it is obstructed on partial tori $N^n = M^{n-m} \times \mathbb{T}^m$. Given Riemannian metrics $g, \bar{g}$ on $(N, \partial N)$ with positive $m$-intermediate curvature and $m$-positive difference $h_g - h_{\bar{g}}$ of second fundamental forms we show that there exists a smooth family of Riemannian metrics with positive $m$-intermediate curvature interpolating between $g$ and $\bar{g}$. Moreover, we apply this result to prove a non-existence result for partial torical bands with positive $m$-intermediate curvature and strictly $m$-convex boundaries. △ Less

Submitted 2 November, 2023; v1 submitted 18 January, 2023; originally announced January 2023.

Comments: final version; the article is published in the Journal of Geometric Analysis

Journal ref: J Geom Anal 33, 366 (2023)

Equivariant $3$-manifolds with positive scalar curvature

Authors: Tsz-Kiu Aaron Chow, Yangyang Li

Abstract: In this paper, for any compact Lie group $G$, we show that the space of $G$-invariant Riemannian metrics with positive scalar curvature (PSC) on any closed three-manifold is either empty or contractible. In particular, we prove the generalized Smale conjecture for spherical three-orbifolds. Moreover, for connected $G$, we make a classification of all PSC $G$-invariant three-manifolds. In this paper, for any compact Lie group $G$, we show that the space of $G$-invariant Riemannian metrics with positive scalar curvature (PSC) on any closed three-manifold is either empty or contractible. In particular, we prove the generalized Smale conjecture for spherical three-orbifolds. Moreover, for connected $G$, we make a classification of all PSC $G$-invariant three-manifolds. △ Less

Submitted 20 April, 2022; v1 submitted 26 September, 2021; originally announced September 2021.

Comments: 22 pages, improved exposition

Area and boundary length of surfaces diffeomorphic to annuli

Authors: Tsz-Kiu Aaron Chow

Abstract: In this paper, we give a proof to a statement in Perelman's paper for finite extinction time of Ricci flow. Our proof draws on different techniques from the one given in Morgan-Tian's exposition and is extrinsic in nature, which relies on the co-area formula instead of the Gauss-Bonnet theorem, and is potentially generalizable to higher dimensions. In this paper, we give a proof to a statement in Perelman's paper for finite extinction time of Ricci flow. Our proof draws on different techniques from the one given in Morgan-Tian's exposition and is extrinsic in nature, which relies on the co-area formula instead of the Gauss-Bonnet theorem, and is potentially generalizable to higher dimensions. △ Less

Submitted 1 July, 2021; originally announced July 2021.

Comments: 16 pages

Ricci Flow on Manifolds with Boundary with Arbitrary Initial Metric

Authors: Tsz-Kiu Aaron Chow

Abstract: In this paper, we study the Ricci flow on manifolds with boundary. In the first part of the paper, we prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously umbilic for positive time. In the second part of the paper, we prove that the flow we constructed in the first part preserves natural boundary conditions. More specifically, if the initial metr… ▽ More In this paper, we study the Ricci flow on manifolds with boundary. In the first part of the paper, we prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously umbilic for positive time. In the second part of the paper, we prove that the flow we constructed in the first part preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition. △ Less

Submitted 7 August, 2021; v1 submitted 8 December, 2020; originally announced December 2020.

Comments: 55 pages, improved exposition

Positivity of Curvature on Manifolds with Boundary

Authors: Tsz-Kiu Aaron Chow

Abstract: Consider a compact manifold $M$ with smooth boundary $\partial M$. Suppose that $g$ and $\tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ which agrees with $g$ outside a neighborhood of $\partial M$ and agrees with $\tilde{g}$ in a neighborhood of $\partial M$. We prove that the family of metrics preserves various natural curvature conditions under suitable ass… ▽ More Consider a compact manifold $M$ with smooth boundary $\partial M$. Suppose that $g$ and $\tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ which agrees with $g$ outside a neighborhood of $\partial M$ and agrees with $\tilde{g}$ in a neighborhood of $\partial M$. We prove that the family of metrics preserves various natural curvature conditions under suitable assumptions on the boundary data. Moreover, under suitable assumptions on the boundary data, we can deform a metric to one with totally geodesic boundary while preserving various natural curvature conditions. △ Less

Submitted 10 March, 2021; v1 submitted 30 November, 2020; originally announced December 2020.

Comments: 19 pages

Self-Expanders to Inverse Curvature Flows by Homogeneous Functions

Authors: Tsz-Kiu Aaron Chow, Ka-Wing Chow, Frederick Tsz-Ho Fong

Abstract: In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-expanders to any of these flows are round spheres. Secondly, we show that comp… ▽ More In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-expanders to any of these flows are round spheres. Secondly, we show that complete non-compact self-expanders to any of these flows with asymptotically cylindrical ends must be rotationally symmetric. Thirdly, we show that when such a flow is uniformly parabolic, there exist complete rotationally symmetric self-expanders which are asymptotic to two round cylinders with different radii. These extend some earlier results of inverse mean curvature flow to a wider class of flows. △ Less

Submitted 16 June, 2018; v1 submitted 15 January, 2017; originally announced January 2017.

Comments: to appear in Comm. Anal. Geom

MSC Class: 53C44