Showing 1–13 of 13 results for author: Shen, D

Vacuum initial data with minimal decay and borderline decay

Authors: Dawei Shen, Jingbo Wan

Abstract: In this note, we show that the conical solution-operator method of Mao-Tao in [Localized initial data for Einstein equations] applies to a simple construction of vacuum asymptotically flat initial data at minimal and borderline decay thresholds, corresponding to the global and exterior stability of Minkowski spacetime proved by the first named author in [Global stability of Minkowski spacetime wit… ▽ More In this note, we show that the conical solution-operator method of Mao-Tao in [Localized initial data for Einstein equations] applies to a simple construction of vacuum asymptotically flat initial data at minimal and borderline decay thresholds, corresponding to the global and exterior stability of Minkowski spacetime proved by the first named author in [Global stability of Minkowski spacetime with minimal decay] and [Exterior stability of Minkowski spacetime with borderline decay]. △ Less

Submitted 1 February, 2026; originally announced February 2026.

Comments: 17 pages. All comments are welcome

Cauchy Data for Formation of Multiple Black Holes with Prescribed ADM Parameters

Authors: Dawei Shen, Jingbo Wan

Abstract: We give a simple construction of smooth, asymptotically flat vacuum initial data modeling a relativistic collapsing $N$--body system, with independently prescribed ADM energy, linear momentum, and angular momentum for each component, subject to the timelike condition $\E>|¶|$. The initial data contain no trapped surfaces, and the future development contains multiple causally independent trapped re… ▽ More We give a simple construction of smooth, asymptotically flat vacuum initial data modeling a relativistic collapsing $N$--body system, with independently prescribed ADM energy, linear momentum, and angular momentum for each component, subject to the timelike condition $\E>|¶|$. The initial data contain no trapped surfaces, and the future development contains multiple causally independent trapped regions that dynamically form from localized subsets of the initial slice. In particular, the maximal development of data with well-separated collapsing components and relative motion is expected to yield spacetimes containing multiple black holes. △ Less

Submitted 4 January, 2026; originally announced January 2026.

Comments: 27 pages, 1 figure. All comments are welcome

Cauchy data for multiple collapsing boson stars

Authors: Elena Giorgi, Dawei Shen, Jingbo Wan

Abstract: We construct Cauchy initial data for the Einstein-Maxwell-Klein-Gordon (EMKG) system, which evolves in finite time into spacetimes containing multiple trapped surfaces. From a physical perspective, this corresponds to preparing multiple well-separated boson stars, each of which collapses to form a spacelike black hole region. In particular, this extends the result of the second and third named aut… ▽ More We construct Cauchy initial data for the Einstein-Maxwell-Klein-Gordon (EMKG) system, which evolves in finite time into spacetimes containing multiple trapped surfaces. From a physical perspective, this corresponds to preparing multiple well-separated boson stars, each of which collapses to form a spacelike black hole region. In particular, this extends the result of the second and third named authors on the formation of multiple trapped surfaces in vacuum to the EMKG system. △ Less

Submitted 1 December, 2025; originally announced December 2025.

Comments: 53 pages, 5 figures. All comments are welcome

Stability of Big Bang singularity for the Einstein-Maxwell-scalar field-Vlasov system in the full strong sub-critical regime

Authors: Xinliang An, Taoran He, Dawei Shen

Abstract: In $3+1$ dimensions, we study the stability of Kasner solutions for the Einstein-Maxwell-scalar field-Vlasov system. This system incorporates gravity, electromagnetic, weak and strong interactions for the initial stage of our universe. Due to the presence of the Vlasov field, various new challenges arise. By observing detailed mathematical structures and designing new delicate arguments, we identi… ▽ More In $3+1$ dimensions, we study the stability of Kasner solutions for the Einstein-Maxwell-scalar field-Vlasov system. This system incorporates gravity, electromagnetic, weak and strong interactions for the initial stage of our universe. Due to the presence of the Vlasov field, various new challenges arise. By observing detailed mathematical structures and designing new delicate arguments, we identify a new strong sub-critical regime and prove the nonlinear stability with Kasner exponents lying in this full regime. This extends the result of Fournodavlos-Rodnianski-Speck [8] from the Einstein-scalar field system to the physically more complex system with the Vlasov field. △ Less

Submitted 25 August, 2025; v1 submitted 24 July, 2025; originally announced July 2025.

Comments: 56 pages

Formation of multiple Black Holes from Cauchy data

Authors: Dawei Shen, Jingbo Wan

Abstract: We construct a family of asymptotically flat Cauchy initial data for the Einstein vacuum equations that contain no trapped surfaces, yet whose future development admits multiple causally independent trapped surfaces. Assuming the weak cosmic censorship conjecture, this implies the formation of multiple black holes in finite time. The initial data are obtained by surgically modifying a vacuum geome… ▽ More We construct a family of asymptotically flat Cauchy initial data for the Einstein vacuum equations that contain no trapped surfaces, yet whose future development admits multiple causally independent trapped surfaces. Assuming the weak cosmic censorship conjecture, this implies the formation of multiple black holes in finite time. The initial data are obtained by surgically modifying a vacuum geometrostatic manifold equipped with the Brill-Lindquist metric: the data agree exactly with the Brill-Lindquist metric outside a collection of balls centered at its multiple poles, and each ball is replaced by a constant-time slice of a well-prepared dynamical spacetime. The construction is based on Christodoulou's short-pulse framework, a stability analysis in the finite future of the short-pulse region, the geometry of geometrostatic manifolds with small mass and large separation, and the obstruction-free annular gluing method developed by Mao-Oh-Tao. The absence of trapped surfaces in the initial data is verified via a standard mean curvature comparison argument. △ Less

Submitted 19 June, 2025; originally announced June 2025.

Comments: 88 pages, 8 figures. Comments are welcome

Formation of trapped surfaces for the Einstein--Maxwell--charged scalar field system

Authors: Dawei Shen, Jingbo Wan

Abstract: In this paper, we prove a scale-critical trapped surface formation result for the Einstein--Maxwell--charged scalar field (EMCSF) system, without any symmetry assumptions. Specifically, we establish a scale-critical semi-global existence theorem from past null infinity and show that the focusing of gravitational waves, the concentration of electromagnetic fields, or the condensation of complex sca… ▽ More In this paper, we prove a scale-critical trapped surface formation result for the Einstein--Maxwell--charged scalar field (EMCSF) system, without any symmetry assumptions. Specifically, we establish a scale-critical semi-global existence theorem from past null infinity and show that the focusing of gravitational waves, the concentration of electromagnetic fields, or the condensation of complex scalar fields, each individually, can lead to the formation of a trapped surface. In addition, we capture a nontrivial charging process along past null infinity, which introduces new difficulties due to the abnormal behavior of the matter fields. Nevertheless, the semi-global existence result and the formation of a trapped surface remain valid. △ Less

Submitted 28 April, 2025; originally announced April 2025.

Comments: 99 pages, 3 figures. All comments are welcome

A canonical foliation on null infinity in perturbations of Kerr

Authors: Sergiu Klainerman, Dawei Shen, Jingbo Wan

Abstract: Kerr stability for small angular momentum has been proved in the series of works by Klainerman-Szeftel, Giorgi-Klainerman-Szeftel and Shen. Some of the most basic conclusions of the result, concerning various physical quantities on the future null infinity are derived in the work of Klainerman-Szeftel. Further important conclusions were later derived in An-He-Shen and Chen-Klainerman. In this pape… ▽ More Kerr stability for small angular momentum has been proved in the series of works by Klainerman-Szeftel, Giorgi-Klainerman-Szeftel and Shen. Some of the most basic conclusions of the result, concerning various physical quantities on the future null infinity are derived in the work of Klainerman-Szeftel. Further important conclusions were later derived in An-He-Shen and Chen-Klainerman. In this paper, based on the existence and uniqueness results for GCM spheres by Klainerman-Szeftel, we establish the existence of a canonical foliation on the future null infinity for which the null energy, linear momentum, center of mass and angular momentum are well defined and satisfy the expected physical laws of gravitational radiation. The rigid character of this foliation eliminates the usual ambiguities related to these quantities in the physics literature. We also show that under the initial assumption of Klainerman-Szeftel, the center of mass of the black hole has a large deformation (recoil) after the perturbation. △ Less

Submitted 28 December, 2024; originally announced December 2024.

Comments: 82 pages, 5 figures, 2 tables. Comments are welcome

Exterior stability of Minkowski spacetime with borderline decay

Authors: Dawei Shen

Abstract: In 1993, the global stability of Minkowski spacetime was proved in the celebrated work of Christodoulou and Klainerman. In 2003, Klainerman and Nicolò revisited Minkowski stability in the exterior of an outgoing null cone. In 2023, the author extended the results of Christodoulou-Klainerman to minimal decay assumptions. In this paper, we prove that the exterior stability of Minkowski holds with de… ▽ More In 1993, the global stability of Minkowski spacetime was proved in the celebrated work of Christodoulou and Klainerman. In 2003, Klainerman and Nicolò revisited Minkowski stability in the exterior of an outgoing null cone. In 2023, the author extended the results of Christodoulou-Klainerman to minimal decay assumptions. In this paper, we prove that the exterior stability of Minkowski holds with decay that is borderline compared to the minimal decay considered in 2023. △ Less

Submitted 3 February, 2026; v1 submitted 29 April, 2024; originally announced May 2024.

Comments: 39 pages, 2 figures. Accepted for publication in Annales scientifiques de l'École normale supérieure

Angular Momentum Memory Effect

Authors: Xinliang An, Taoran He, Dawei Shen

Abstract: Utilizing recent mathematical advances in proving stability of Minkowski spacetime with minimal decay rates and nonlinear stability of Kerr black holes with small angular momentum, we investigate the detailed asymptotic behaviors of gravitational waves generated in these spacetimes. Here we report and propose a new angular momentum memory effect along future null infinity. This accompanies Christo… ▽ More Utilizing recent mathematical advances in proving stability of Minkowski spacetime with minimal decay rates and nonlinear stability of Kerr black holes with small angular momentum, we investigate the detailed asymptotic behaviors of gravitational waves generated in these spacetimes. Here we report and propose a new angular momentum memory effect along future null infinity. This accompanies Christodoulou's nonlinear displacement memory effect and the spin memory effect. The connections and differences to these effects are also addressed. △ Less

Submitted 10 April, 2024; v1 submitted 17 March, 2024; originally announced March 2024.

Global stability of Minkowski spacetime with minimal decay

Authors: Dawei Shen

Abstract: The global stability of Minkowski spacetime, a milestone in the field, has been proven in the celebrated work of Christodoulou and Klainerman \cite{Ch-Kl} in 1993. In 2007, Bieri \cite{Bieri} has extended the result of \cite{Ch-Kl} under lower decay and regularity assumptions on the initial data. In this paper, we extend the result of \cite{Bieri} to minimal decay assumptions. Also, concerning the… ▽ More The global stability of Minkowski spacetime, a milestone in the field, has been proven in the celebrated work of Christodoulou and Klainerman \cite{Ch-Kl} in 1993. In 2007, Bieri \cite{Bieri} has extended the result of \cite{Ch-Kl} under lower decay and regularity assumptions on the initial data. In this paper, we extend the result of \cite{Bieri} to minimal decay assumptions. Also, concerning the treatment of curvature estimates, we replace the vectorfield method used in \cite{Ch-Kl,Bieri} by the $r^p$--weighted estimates of Dafermos and Rodnianski \cite{Da-Ro}. △ Less

Submitted 11 October, 2023; originally announced October 2023.

Comments: 94 pages, 3 figures

Kerr stability in external regions

Authors: Dawei Shen

Abstract: In 2003, Klainerman and Nicolò \cite{Kl-Ni} proved the stability of Minkowski in the case of the exterior of an outgoing null cone. Relying on the method used in \cite{Kl-Ni}, Caciotta and Nicolò \cite{Ca-Ni} proved the stability of Kerr spacetime in external regions, i.e. outside an outgoing null cone far away from the Kerr event horizon. In this paper, we give a new proof of \cite{Ca-Ni}. Compar… ▽ More In 2003, Klainerman and Nicolò \cite{Kl-Ni} proved the stability of Minkowski in the case of the exterior of an outgoing null cone. Relying on the method used in \cite{Kl-Ni}, Caciotta and Nicolò \cite{Ca-Ni} proved the stability of Kerr spacetime in external regions, i.e. outside an outgoing null cone far away from the Kerr event horizon. In this paper, we give a new proof of \cite{Ca-Ni}. Compared to \cite{Ca-Ni}, we reduce the number of derivatives needed in the proof, simplify the treatment of the last slice, and provide a unified treatment of the decay of initial data which contains in particular the initial data considered by Klainerman and Szeftel in \cite{KS:main}. Also, concerning the treatment of curvature estimates, similar to \cite{ShenMink}, we replace the vectorfield method used in \cite{Kl-Ni,Ca-Ni} by $r^p$-weighted estimates introduced by Dafermos and Rodnianski in \cite{Da-Ro}. △ Less

Submitted 15 April, 2024; v1 submitted 22 March, 2023; originally announced March 2023.

Comments: 119 pages, 7 figures, accepted in Annals of PDE

Stability of Minkowski spacetime in exterior regions

Authors: Dawei Shen

Abstract: In 1993, the global stability of Minkowski spacetime has been proven in the celebrated work of Christodoulou and Klainerman \cite{Ch-Kl} in a maximal foliation. In 2003, Klainerman and Nicòlo \cite{Kl-Ni} gave a second proof of the stability of Minkowski in the case of the exterior of an outgoing null cone. In this paper, we give a new proof of \cite{Kl-Ni}. Compared to \cite{Kl-Ni}, we reduce the… ▽ More In 1993, the global stability of Minkowski spacetime has been proven in the celebrated work of Christodoulou and Klainerman \cite{Ch-Kl} in a maximal foliation. In 2003, Klainerman and Nicòlo \cite{Kl-Ni} gave a second proof of the stability of Minkowski in the case of the exterior of an outgoing null cone. In this paper, we give a new proof of \cite{Kl-Ni}. Compared to \cite{Kl-Ni}, we reduce the number of derivatives needed in the proof, simplify the treatment of the last slice, and provide a unified treatment of the decay of initial data. Also, concerning the treatment of curvature estimates, we replace the vectorfield method used in \cite{Kl-Ni} by the $r^p$--weighted estimates of Dafermos and Rodnianski \cite{Da-Ro}. △ Less

Submitted 26 August, 2023; v1 submitted 28 November, 2022; originally announced November 2022.

Comments: 82 pages, 3 figures, minor improvements, accepted in Pure and Applied Mathematics Quarterly

Construction of GCM hypersurfaces in perturbations of Kerr

Authors: Dawei Shen

Abstract: This is a follow-up of \cite{KS:Kerr1} on the general covariant modulated (GCM) procedure in perturbations of Kerr. In this paper, we construct GCM hypersurfaces, which play a central role in extending GCM admissible spacetimes in \cite{KS:main} where decay estimates are derived in the context of nonlinear stability of Kerr family for $|a|\ll m$. As in \cite{KS}, the central idea of the constructi… ▽ More This is a follow-up of \cite{KS:Kerr1} on the general covariant modulated (GCM) procedure in perturbations of Kerr. In this paper, we construct GCM hypersurfaces, which play a central role in extending GCM admissible spacetimes in \cite{KS:main} where decay estimates are derived in the context of nonlinear stability of Kerr family for $|a|\ll m$. As in \cite{KS}, the central idea of the construction of GCM hypersurfaces is to concatenate a $1$--parameter family of GCM spheres of \cite{KS:Kerr1} by solving an ODE system. The goal of this paper is to get rid of the symmetry restrictions in the GCM procedure introduced in \cite{KS} and thus remove an essential obstruction in extending the results to a full stability proof of the Kerr family. △ Less

Submitted 7 May, 2023; v1 submitted 20 May, 2022; originally announced May 2022.

Comments: 113 pages, 2 figures, minor improvements, accepted in Annals of PDE. arXiv admin note: substantial text overlap with arXiv:1911.00697 by other authors