Foliations by stable spheres with constant mean curvature for isolated systems without asymptotic symmetry

Hardt-Simon叶状结构附近的极小曲面

Minimal surfaces near Hardt-Simon foliations

本文借鉴Caffarelli-Hardt-Simon的方法求解球内Hardt-Simon曲面的极小曲面方程边值问题，并用粘合方法在Hardt-Simon曲面及二次锥附近构造极小曲面，丰富了带奇性极小曲面的研究结果。

Existence and uniqueness of constant mean curvature foliations of general asymptotically hyperbolic 3-manifolds

In 1996, Huisen-Yau proved that every three-dimensional, asymptotically Schwarzschilden manifold with positive mass is uniquely foliated by stable spheres of constant mean curvature and they defined the center of mass using this CMC-foliation. Rigger and Neves-Tian showed in 2004 and 2009/10 analogous existence and uniqueness theorems for three-dimensional, asymptotically Anti-de Sitter and asymptotically hyperbolic manifolds with positive mass aspect function, respectively. Last year, Cederbaum-Cortier-Sakovich proved that the CMC-foliation characterizes the center of mass in the hyperbolic s…

Existence and Uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds II

In a previous paper, the authors showed that metrics which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass admit a unique foliation by stable spheres with constant mean curvature. In this paper we extend that result to all asymptotically hyperbolic metrics for which the trace of the mass term is positive. We do this by combining the Kazdan-Warner obstructions with a theorem due to De Lellis and Müller.

Singular Riemannian Foliations and the prescribing scalar curvature problem

An orbit-like foliation is a singular foliation on a complete Riemannian manifold $M$ whose leaves are locally equidistant (i.e., a singular Riemannian foliation) and (transversely) infinitesimally homogenous. This class of singular foliation contains not only the classe of partion of the space into orbits of isometric actions, but also infinite many non homogenous examples and in particular the partition of $M$ into orbits of a proper groupoid. In this paper we prove a version of Kondrakov Embedding Theorem and an analogous Principle of Symmetric Criticality of Palais for basic funcions of or…

气泡片的典范叶状结构

Canonical foliation of bubblesheets

本文提出准平行平均曲率（QPMC）这一新曲率条件，并证明在接近标准度量的乘积流形上存在由QPMC球面构成的典范叶状结构，从而为几何流中的气泡片建立标准形式。

体积保持时空平均曲率流与初始数据集的叶状结构

Volume preserving spacetime mean curvature flow and foliations of initial data sets

本文研究具有正ADM能量的渐近欧几里得初始数据集中体积保持时空平均曲率流，证明其长期存在并收敛到常数曲率极限，为CSTMC叶状结构提供新构造，应用于广义相对论中孤立系统质心的定义。

Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. III. Two singularities

All $(-1)$-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus north and south poles have been classified in our earlier work as a four dimensional surface with boundary. In this paper, we establish near the no-swirl solution surface existence, non-existence and uniqueness results on $(-1)$-homogeneous axisymmetric solutions with nonzero swirl which are smooth on the unit sphere minus north and south poles.

球面上具有孤立奇点的常 Q-曲率完备度量空间中的紧性

Compactness within the space of complete, constant Q-curvature metrics on the sphere with isolated singularities

本文研究球面上带 k 个穿孔的常正 Q-曲率完备共形平坦度量模空间，证明当穿孔间距与渐近颈尺度均远离零时该集合序列紧，并引入径向 Pohozaev 不变量及改进共形因子先验界。

Minimal two-spheres in three-spheres

We prove that any manifold diffeomorphic to $S^3$ and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in three manifolds. We apply our methods to solve a problem posed by S.T. Yau in 1987 on whether the planar two-spheres are the only minimal spheres in ellipsoids centered about the origin in $\mathbb{R}^4$. Finally, considering the example…

Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. II. Classification of axisymmetric no-swirl solutions

We classify all (-1)-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles, parameterizing them as a four dimensional surface with boundary in appropriate function spaces. Then we establish smoothness properties of the solution surface in the four parameters. The smoothness properties will be used in a subsequent paper where we study the existence of (-1)-homogeneous axisymmetric solutions with non-zero swirl on $\mathbb{S}^2\setminus\{S,N\}$, emanating from the fou…

Surfaces of constant curvature in R^3 with isolated singularities

We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally convex curves in the 2-sphere with admissible cusp singularities, characterizing when the singularity is actually embedded. In the global setting, we describe the space of peaked spheres in R^3, i.e. compact convex surfaces of constant positive curvature with a finite number of singularities, …

Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. I. One singularity

We classify all $(-1)-$homogeneous axisymmetric no swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south pole, parameterize them as a two dimensional surface with boundary, and analyze their pressure profiles near the north pole. Then we prove that there is a curve of $(-1)-$homogeneous axisymmetric solutions with nonzero swirl, having the same smoothness property, emanating from every point of the interior and one part of the boundary of the solution surface. Moreover we prove that there is no such curve of …

等参超曲面导出的 Bernoulli 自由边界问题的奇点模型

Singularity models for the Bernoulli free boundary problem from isoparametric hypersurfaces

本文从等参叶状结构构造 Bernoulli 自由边界问题的齐次解，产生无穷多新奇点模型，建立几何刚性定理，并猜想最低密度锥体的形式。