Dhruv Kohli, Jeffrey M. Rabin
In an earlier paper we showed that the radial expansion of a hyperbolic convex set in the Poincaré disk about any point inside it results in a hyperbolic convex set. In this work, we generalize this result by showing that the asymmetric expansion of a hyperbolic convex set about any point inside it also results in a hyperbolic convex set.
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On a flat plane, convexity of a set is preserved by both radial expansion and contraction of the set about any point inside it. Using the Poincaré disk model of hyperbolic geometry, we prove that radial expansion of a hyperbolic convex set about a point inside it always preserves hyperbolic convexity. Using stereographic projection of a sphere, we prove that radial contraction of a spherical convex set about a point inside it, such that the initial set is contained in the closed hemisphere centred at that point, always preserves spherical convexity.
The Hessian of the renormalized volume of geometrically finite hyperbolic $3$-manifolds without rank-$1$ cusps, computed at the hyperbolic metric $g$ with totally geodesic boundary of the convex core, is shown to be a strictly positive bilinear form on the tangent space to Teichmüller space. The metric $g$ is known from results of Bonahon and Storm to be an absolute minimum for the volume of the convex core. We deduce the strict convexity of the functional volume of the convex core at its minimum point.
We prove that the static convexity is preserved along two kinds of locally constrained curvature flows in hyperbolic space. Using the static convexity of the flow hypersurfaces, we prove new family of geometric inequalities for such hypersurfaces in hyperbolic space.
Almgren frequency and Agmon convexity on rotationally symmetric Cartan-Hadamard manifolds
本文在旋转对称 Cartan-Hadamard 流形上引入全局频率函数，并在实双曲空间中证明相关频率函数的单调性，进而得到对数凸性定理和双曲三球定理。
Power convexity of the torsion function on horo-convex domains in hyperbolic space
该研究针对双曲空间中严格 horo-凸区域的扭转问题，证明当区域直径足够小时，扭转函数的负平方根严格凸，且指数 $1/2$ 最优，核心方法含常秩定理等，贡献了幂凸性结论。
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…
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.
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.
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…
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。
We prove: "If $M$ is a compact hypersurface of the hyperbolic space, convex by horospheres and evolving by the volume preserving mean curvature flow, then it flows for all time, convexity by horospheres is preserved and the flow converges, exponentially, to a geodesic sphere". In addition, we show that the same conclusions about long time existence and convergence hold if $M$ is not convex by horospheres but it is close enough to a geodesic sphere.
Generalized convexity and quantitative estimates for constant mean curvature spacelike hypersurfaces in Anti-de Sitter space
本文推广凸包概念，给出反德西特空间中常平均曲率超曲面主曲率的上界，并由此界定截面曲率及双曲平面上θ-滑坡的拟共形伸缩商。
Gauss curvature type flow and Alexandrov-Fenchel inequalities in the hyperbolic space
本文研究双曲空间中一致凸超曲面的高斯曲率型流，证明其长时间存在并指数收敛于测地球面，进而给出相关 Alexandrov-Fenchel 不等式的新证明。
The quermassintegral inequalities for horo-convex domains in the sphere
本文研究了球面上horo-凸域的quermass积分不等式，通过证明Guan/Li流的光滑收敛性，建立了该类超曲面的完整不等式。
Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space
本文研究双曲空间中凸超曲面的保体积高斯曲率流，证明解长期存在且指数收敛到测地球，是首个仅要求初始凸性的非局部型保体积曲率流结果。
This survey is an introduction to the geometry of co-Minkowksi space, the space of unoriented spacelike hyperplanes of the Minkowski space. Affine deformations of cocompact lattices of hyperbolic isometries act on it, in a way similar to the way that quasi-Fuchsian groups act on hyperbolic space. In particular, there is a convex core. There is also a unique "mean" hypersurface, i.e. with traceless second fundamental form. The mean distance between the mean hypersurface and the lower boundary of the convex core endows the space of affine deformations of a given lattice with an asymmetric norm. …
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。
$β$-Uniform Convexity and Divisible Domains
本文证明严格凸可除域上的自然芬斯勒度量具有$\eta$-一致凸性，常数$\eta$与边界正则性相关，并借助AI辅助证明希尔伯特度量在适当假设下也满足该性质。
Locally constrained curvature flows and geometric inequalities in hyperbolic space
本文研究双曲空间中两类局部约束曲率流，证明h-凸性保持及光滑收敛性，并由此建立多组新的尖锐几何不等式，肯定回答了Ge-Wang-Wu猜想，同时给出Andrews-Chen等人不等式的新证明及其高阶推广。
Geometric inequalities and their stabilities for modified quermassintegrals in hyperbolic space
本文研究双曲空间中Hu-Li-Wei流的曲线情形，利用Heintze-Karcher型不等式证明严格h-凸闭曲线全局存在并指数收敛为测地圆，进而给出修正quermassintegral相关几何不等式的新证明及稳定性讨论。
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。
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…
A compact Fuchsian manifold with boundary is a hyperbolic 3-manifold homeomorphic to $S_g \times [0; 1]$ such that the boundary component $S_g \times \{ 0\}$ is geodesic. We prove that a compact Fuchsian manifold with convex boundary is uniquely determined by the induced path metric on $S_g \times \{1\}$. We do not put further restrictions on the boundary except convexity.
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