On the uniqueness of the AdS space-time in higher dimensions

渐近平坦局部双曲静态真空黑洞的几何性质与唯一性

On the geometry and uniqueness of asymptotically locally hyperbolic static vacuum black holes

该研究针对带视界边界的三维渐近平坦局部双曲静态空间，建立几何刻画与刚性结果，推导相关不等式、单调量及正则性定理，证明特定静态黑洞的唯一性，支持相关猜想。

三维域中毛细盘的唯一性

Uniqueness of capillary disks in three-dimensional domains

论文证明了在三维域中，基于椭圆PDE的毛细盘的唯一性，扩展了Nitsche的结果，并受Gálvez和Mira对常均曲率球的Hopf定理的启发。

On uniqueness for the Harmonic Map Heat Flow in supercritical dimensions

We examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension. It is shown that, generically, singular data can give rise to two distinct solutions which are both stable, and satisfy the local energy inequality. We also discuss how uniqueness can be retrieved.

Maximal hypersurfaces in asymptotically Anti-de Sitter spacetime

We prove uniqueness, existence, and regularity results for maximal hypersurfaces in spacetimes with a conformal completion at timelike infinity and asymptotically constant scalar curvature, as relevant for asymptotically AdS spacetimes. This work is dedicated to Yvonne Choquet-Bruhat on the occasion of her upcoming 99th birthday.

Uniqueness of Weak Solutions for the Normalised Ricci Flow in Two Dimensions

We show uniqueness of classical solutions of the normalised two-dimensional Hamilton-Ricci flow on closed, smooth manifolds for smooth data among solutions satisfying (essentially) only a uniform bound for the Liouville energy and a natural space-time $L^2$-bound for the time derivative of the solution. The result is surprising when compared with results for the harmonic map heat flow, where non-uniqueness through reverse bubbling may occur.

On the uniqueness of complete biconservative surfaces in $3$-dimensional space forms

Biconservative surfaces are surfaces with divergence-free stress-bienergy tensor. Simply connected, complete, non-$CMC$ biconservative surfaces in $3$-dimensional space forms were constructed working in extrinsic and intrinsic ways. Then, one raises the question of the uniqueness of such surfaces. In this paper we give a positive answer to this question.

A Remark on a Uniqueness Property of High Multiplicity Tangent Flows in Dimension Three

In this note, we combine the work of Ilmanen and of Colding-Ilmanen-Minicozzi to observe a uniqueness property for tangent flows at the first singular time of a smooth mean curvature flow of a closed surface in 3-dimensional Euclidean space. Specifically, if, at a fixed singular point, one tangent flow is a positive integer multiple of a shrinking plane, cylinder or sphere, then, modulo rotations, all tangent flows at the point are the same.

Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

In this paper, we study the classification of $κ$-noncollapsed ancient solutions to n-dimensional Ricci flow on $S^n$, extending the result in [13] to higher dimensions. We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman.

Uniqueness of tangent cones for $2$-dimensional almost minimizing currents

We consider $2$-dimensional integer rectifiable currents which are almost area minimizing and show that their tangent cones are everywhere unique. Our argument unifies a few uniqueness theorems of the same flavor, which are all obtained by a suitable modification of White's original theorem for area minimizing currents in the euclidean space. This note is also the first step in a regularity program for semicalibrated $2$-dimensional currents and spherical cross sections of $3$-dimensional area minimizing cones.

A Uniqueness Theorem for Gluing Special Lagrangian Submanifolds

Butscher, D. Lee, Y. Lee, and Joyce constructed a special Lagrangian submanifold by gluing a Lawlor neck into a transverse intersection point of two special Lagrangian submanifolds. We prove a uniqueness theorem for the gluing of flat special Lagrangian tori of real dimension 3 in a flat complex torus of complex dimension 3.

Uniqueness and stability of singular Ricci flows in higher dimensions

In this short note, we observe that the Bamler-Kleiner proof of uniqueness and stability for 3-dimensional Ricci flow through singularities generalizes to singular Ricci flows in higher dimensions that satisfy an analogous canonical neighborhood property. In particular, this gives a canonical evolution through singularities for manifolds with positive isotropic curvature. The new ingredients we use are the recent classification of higher dimensional $κ$-solutions by Brendle, Daskalopoulos, Naff and Sesum, and the maximum principle for the linearized Ricci-DeTurck flow on locally conformally fl…

Photon sphere uniqueness in higher-dimensional electrovacuum spacetimes

We show a uniqueness result for the n-dimensional spatial Reissner-Nordström manifold: a static, electrovacuum, asymptotically flat system which is asymptotically Reissner-Nordström is a subextremal Reissner-Nordström manifold with positive mass, provided that its inner boundary is a (possibly disconnected) photon sphere that fulfils a suitably defined quasilocal subextremality condition. Our result implies a number of earlier uniqueness results for the Schwarzschild and the Reissner-Nordström manifolds in the static, (electro-)vacuum, asymptotically flat context, both for photon sphere and bl…