Radial expansion preserves hyperbolic convexity and radial contraction preserves spherical convexity

Convexity on Complex Hyperbolic Space

In a Riemannian manifold a regular convex domain is said to be $λ$-convex if its normal curvature at each point is greater than or equal to $λ$. In a Hadamard manifold, the asymptotic behaviour of the quotient $\vol(Ω(t))/\vol(\partialΩ(t))$ for a family of $λ$-convex domains $Ω(t)$ expanding over the whole space has been studied and general bounds for this quotient are known. In this paper we improve this general result in the complex hyperbolic space $\CH$, a Hadamard manifold with constant holomorphic curvature equal to $-4k^2$. Furthermore, we give some specific properties of convex domain…

Local convexity of renormalized volume for rank-1 cusped manifolds

We study the critical points of the renormalized volume for acylindrical geometrically finite hyperbolic 3-manifolds that include rank-1 cusps, and show that the renormalized volume is locally convex around these critical points. We give a modified definition of the renormalized volume that is additive under gluing, and study some local properties.

Extension of the Weil-Petersson connection

Convexity properties of Weil-Petersson geodesics on the Teichmüller space of punctured Riemann surfaces are investigated. A normal form is presented for the Weil-Petersson Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considerations are combined to establish convexity along Weil-Petersson geodesics of the functions the distance between horocycles for a hyperbolic metric.

Non-contraction of heat flow on Minkowski spaces

We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in non-Riemannian Finsler manifolds. As an application, we show that the heat flow on Minkowski normed spaces other than inner product spaces is not contractive with respect to the quadratic Wasserstein distance.

截面曲率与矩阵位移凸性

Sectional curvature and matrix displacement convexity

该研究证明黎曼流形截面曲率非负等价于熵张量的矩阵位移凸性，还将此结论应用于强化HWI不等式等结果，且内在维数Wasserstein收缩不等式可刻画非负截面曲率

Spherical, hyperbolic and other projective geometries: convexity, duality, transitions

Since the end of the 19th century, and after the works of F. Klein and H. Poincaré, it is well known that models of elliptic geometry and hyperbolic geometry can be given using projective geometry, and that Euclidean geometry can be seen as a "limit" of both geometries. Then all the geometries that can be obtained in this way. Some of these geometries had a rich development, most remarkably hyperbolic geometry and the Lorentzian geometries of Minkowski, de Sitter and anti-de Sitter spaces, which in higher dimension have had large interest for a long time in mathematical physics and more precis…

Teichmüller测地线沿能量函数的拟凸性

Quasi-convexity of energy functions along Teichmüller geodesics

本文将双曲长度函数沿Teichmüller测地线的拟凸性推广到调和映射能量函数的两种情形，证明了相关拟凸性并推广了Masur的渐近增长结果，推导了能量函数的变分公式。

Contraction of surfaces in hyperbolic space and in sphere

In this paper, we consider the contracting curvature flow of smooth closed surfaces in $3$-dimensional hyperbolic space and in $3$-dimensional sphere. In the hyperbolic case, we show that if the initial surface $M_0$ has positive scalar curvature, then along the flow by a positive power $α$ of the mean curvature $H$, the evolving surface $M_t$ has positive scalar curvature for $t>0$. By assuming $α\in [1,4]$, we can further prove that $M_t$ contracts a point in finite time and become spherical as the final time is approached. We also show the same conclusion for the flows by powers of scalar c…

Schrödinger演化的几何不确定性原理在负曲率流形上

Geometric uncertainty principles for Schrödinger evolutions on negatively curved manifolds

本文研究了在特定Cartan-Hadamard流形上，配备渐近双曲度量的Schrödinger方程的不确定性原理，展示了在双曲几何中，尽管缺乏平移不变性和傅里叶表示，仍然存在类似的刚性现象，通过新的Carleman估计和对数凸性，揭示了曲率在散射方程定量唯一性性质中的作用。

柱面上的双曲 Monge-Ampère 方程：适定性与稳定性

Hyperbolic Monge-Ampère Equation on a Cylinder: Well-Posedness and Stability

研究利用部分凸性与 hodograph 变换将双曲 Monge-Ampère 方程化为阻尼波动方程，通过 parametrix-corrector 分解处理角点奇性，证明新弱解的适定性及曲率扰动下的定量稳定性。

Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and $F$ is a positive, strictly monotone and 1-homogeneous curvature function. In particular this class includes the mean curvature $F=H$. We prove that a certain initial pinching condition is preserved and the properly rescaled hypersurfaces converge smoothly to the unit sphere. We show that an example due to Andrews-McCoy-Zheng can be used to construct strictly convex initial hypersurfaces, for which the inverse mean curvature flow to the power…

双曲空间中的非齐次曲率流

Non-homogeneous curvature flows in hyperbolic space

本文研究双曲空间中非齐次曲率流，证明在权函数结构条件下流光滑存在、保持严格h-凸性并收缩至固定点，且归一化径向函数指数收敛。

球面与双曲正交环模式：可积性与变分原理

Spherical and hyperbolic orthogonal ring patterns: integrability and variational principles

本文在球面与双曲平面引入正交环模式，证明其半径由离散可积系统Q4描述，给出基于对数函数椭圆推广的变分原理，并证明了边值问题的存在唯一性及其与常均曲率离散表面的关系。