Boundary entropy of a hyperbolic group

Lipschitz Continuous Hypersurfaces with Prescribed Curvature and Asymptotic Boundary in Hyperbolic Space

We prove the existence of a complete locally Lipschitz continuous hypersurface in weak sense with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under certain assumptions.

On the diastatic entropy and C^1-rigidity of complex hyperbolic manifolds

Let f:(Y,g)->(X,g_0) be a non zero degree continuous map between compact Kähler manifolds of dimension greater or equal to 2, where g_0 has constant negative holomorphic sectional curvature. Adapting the Besson-Courtois-Gallot barycentre map techniques to the Kähler setting, we prove a gap theorem in terms of the degree of f and the diastatic entropies of (Y, g) and (X,g_0), which extends the rigidity result proved by the author in [13].

Minimal entropy for uniform lattices in PSL$_2(\mathbb{R})^n$

We prove that, among metrics on a compact quotient of $\left(\mathbb{H}^2\right)^n$ (product of hyperbolic planes) of prescribed total volume, the product of hyperbolic metrics has minimal volume entropy.

Colding Minicozzi Entropy in Hyperbolic Space

This note introduces a notion of entropy for submanifolds of hyperbolic space analogous to the one introduced by Colding and Minicozzi for submanifolds of Euclidean space. Several properties are proved for this quantity including monotonicity along mean curvature flow in low dimensions and a connection with the conformal volume.

Diastatic entropy and rigidity of hyperbolic manifolds

Let $f: Y \rightarrow X$ be a continuous map between a compact real analytic Kähler manifold $(Y,g)$ and a compact complex {hyperbolic manifold} $(X,g_0)$. In this paper we give a lower bound of the diastatic entropy of $(Y,g)$ in terms of the diastatic entropy of $(X,g_0)$ and the degree of $f$. When the lower bound is attained we get geometric rigidity theorems for the diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and S. Gallot [2] for the volume entropy. As a corollary, when $X=Y$, we show that the minimal diastatic entropy is achieved if and only if $g$ is holo…

双曲3-流形的数量曲率与体积熵

Scalar curvature and volume entropy of hyperbolic 3-manifolds

本文证明任意闭双曲3-流形上存在数量曲率不小于-6但体积熵严格大于2的黎曼度量，从而构造出Agol–Storm–Thurston猜想的反例。

无共轭点粗双曲流形上测地流最大熵测度的唯一性

Uniqueness of the measure of maximal entropy for geodesic flows on coarse hyperbolic manifolds without conjugate points

本文在无共轭点且基本群为Gromov双曲、剩余有限的闭黎曼流形上，通过对扩张集施加适当条件，证明了测地流最大熵测度的唯一性，从而推广了此前仅在负截面曲率背景度量下成立的相关结果。

极小曲面熵与 Ricci 流在有限体积双曲 3-流形上的应用

Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds

本文研究有限体积双曲 3-流形上的极小曲面熵，利用曲率条件及 Ricci 流收敛速率，证明双曲度量分别对应熵的最小值或唯一最大值。

Asymptotically hyperbolic manifold with a horospherical boundary

We discuss asymptotically hyperbolic manifold with a noncompact boundary which is close to a horosphere in a certain sense. The model case is a horoball or the complement of a horoball in standard hyperbolic space. We show some geometric formulas.

双曲度量附近Ricci流的拓扑熵单调性

Monotonicity of topological entropy along the Ricci flow near a hyperbolic metric

本文解决了Manning提出的问题，证明了带双曲度量的闭黎曼流形的测地流拓扑熵，在与双曲度量足够接近的变负截面曲率度量出发的归一化Ricci流中严格递减。

Incidence axioms for the boundary at infinity of complex hyperbolic spaces

We characterize the boundary at infinity of a complex hyperbolic space as a compact Ptolemy space that satisfies four incidence axioms.

Local rigidity of hyperbolic manifolds with geodesic boundary

Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n>3 then the holonomy representation of pi_1 (W) into the isometry group of hyperbolic n-space is infinitesimally rigid.

测地电流在紧致双曲面上的熵与自交数

Entropy and self-intersection number of geodesic currents on compact hyperbolic surfaces

研究测地电流在双曲面上的熵与自交数关系，建立熵与自交数及系统最小长度的定量上界，证明小自交数导致小熵。

The mass of an asymptotically hyperbolic manifold with a noncompact boundary

We define a mass-type invariant for asymptotically hyperbolic manifolds with a noncompact boundary which are modelled at infinity on the hyperbolic half-space and prove a sharp positive mass inequality in the spin case under suitable dominant energy conditions. As an application we show that any such manifold which is Einstein and either has a totally geodesic boundary or is conformally compact and has a mean convex boundary is isometric to the hyperbolic half-space.

Inverse mean curvature flow with a free boundary in hyperbolic space

We study inverse mean curvature flow with free boundary supported on geodesic spheres in hyperbolic space. Starting from any convex hypersurface inside a geodesic ball with a free boundary, the flow converges to a totally geodesic disk in finite time. Using the convergence result, we show a Willmore type inequality.

Hyperbolic metrics on surfaces with boundary

We discuss an alternative approach to the uniformisation problem on surfaces with boundary by representing conformal structures on surfaces $M$ of general type by hyperbolic metrics with boundary curves of constant positive geodesic curvature. In contrast to existing approaches to this problem, the boundary curves of our surfaces $(M,g)$ cannot collapse as the conformal structure degenerates which is important in applications in which $(M,g)$ serves as domain of a PDE with boundary conditions.

Positive mass theorems for asymptotically hyperbolic Riemannian manifolds with boundary

We prove positive mass theorems for asymptotically hyperbolic and asymptotically locally hyperbolic Riemannian manifolds with black-hole-type boundaries.

双曲曲面的熵谱

The entropy spectrum of hyperbolic surfaces

本文研究双曲曲面的熵谱，证明其为具有限重数的逆良序多重集，在数值1附近存在可量化间隙，该间隙源于带边界曲面上曲线数量的几何估计。

Local marked boundary rigidity under hyperbolic trapping assumptions

Under the assumption that the X-ray transform over symmetric solenoidal 2-tensors is injective, we prove that smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set are locally marked boundary rigid.

Boundary rigidity of negatively-curved asymptotically hyperbolic surfaces

In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.

带类时边界的整体双曲流形上Friedrichs系统的柯西问题研究

On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary

本文研究带类时边界的整体双曲流形上Friedrichs系统的柯西问题，通过施加容许边界条件证明强解的存在唯一性，且双曲型Friedrichs系统的柯西问题在Hadamard意义下适定，并给出相关实例。

X-ray Transform and Boundary Rigidity for Asymptotically Hyperbolic Manifolds

We consider the boundary rigidity problem for asymptotically hyperbolic manifolds. We show injectivity of the X-ray transform in several cases and consider the non-linear inverse problem which consists of recovering a metric from boundary measurements for the geodesic flow.

The well-posedness of the Cauchy problem for the Dirac operator on globally hyperbolic manifolds with timelike boundary

We consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to MIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniqueness of weak solutions and then using local methods developed by Lax, Phillips and Rauch, Massey to show smoothness of the solutions. Our proof actually works for a slightly more general class of local boundary conditions.