Scott A. Wolpert
New results on the convexity of geodesic-length functions on Teichmüller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-length functions to Weil-Petersson distance is described. Applications to the behavior of Weil-Petersson geodesics are discussed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The usual notion of set-convexity, valid in the classical Euclidean context, metamorphoses into several distinct convexity types in the more general Riemannian setting. By studying this phenomenon in reverse, we characterize complete manifolds for which certain convexity types are assumed a priori to coincide.
Some nilpotent Lie groups possess a transformation group analogous to the similarity group acting on the Euclidean space. We call such a pair a nilpotent similarity structure. It is notably the case for all Carnot groups and their dilatations. We generalize a theorem of Fried: closed manifolds with a nilpotent similarity structure are either complete or radiant and, in the latter case, complete for the structure of the space deprived of a point. The proof relies on a generalization of convexity arguments in a setting where, in the coordinates given by the Lie algebra, we study geodesic segment…
Convex optimization is a vibrant and successful area due to the existence of a variety of efficient algorithms that leverage the rich structure provided by convexity. Convexity of a smooth set or a function in a Euclidean space is defined by how it interacts with the standard differential structure in this space -- the Hessian of a convex function has to be positive semi-definite everywhere. However, in recent years, there is a growing demand to understand non-convexity and develop computational methods to optimize non-convex functions. Intriguingly, there is a type of non-convexity that disap…
Let $π:\mc{X}\to \mc{T}$ be Teichmüller curve over Teichmüller space $\mc{T}$, such that the fiber $\mc{X}_z=π^{-1}(z)$ is exactly the Riemann surface given by the complex structure $z\in \mc{T}$. For a fixed Riemannian manifold $M$ and a continuous map $u_0: M\to \mc{X}_{z_0}$, let $E(z)$ denote the energy function of the harmonic map $u(z):M\to \mc{X}_z$ homotopic to $u_0$, $z\in \mathcal T$. We obtain the first and the second variations of the energy function $E(z)$, and show that $\log E(z)$ is strictly plurisubharmonic on Teichmüller space, from which we give a new proof on the Steinness …
We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature Kähler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays, and the uniform convexity properties of the space of Kähler metrics.
Given a Riemannian manifold M and a hypersurface H in M, it is well known that infinitesimal convexity on a neighborhood of a point in H implies local convexity. We show in this note that the same result holds in a semi-Riemannian manifold. We make some remarks for the case when only timelike, null or spacelike geodesics are involved. The notion of geometric convexity is also reviewed and some applications to geodesic connectedness of an open subset of a Lorentzian manifold are given.
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。
There are two parts of this paper. First, we discovered an explicit formula for the complex Hessian of the weighted log-Bergman kernel on a parallelogram domain, and utilised this formula to give a new proof about the strict convexity of the Mabuchi functional along a smooth geodesic. Second, when a C^{1,1}-geodesic connects two non-degenerate energy minimizers, we also proved this strict convexity, by showing that such a geodesic must be non-degenerate and smooth.
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.
If $Ω$ is the interior of a convex polygon in $\mathbb{R}^{2}$ and $f,g$ two asymptotic geodesics, we show that the distance function $d\left(f\left(t\right),g\left(t\right)\right)$ is convex for $t$ sufficiently large. The same result is obtained in the case $\partial Ω$ is of class $C^{2}$ and the curvature of $\partial Ω$ at the point $f\left(\infty\right)=g\left(\infty\right) $ does not vanish. An example is provided for the necessity of the curvature assumption.
Geometric $C^1$-stability of geodesic spheres under volume-preserving mean curvature flow
该研究在单连通空间形式中，通过定量Alexandrov反射论证，证明保体积平均曲率流下测地球面的几何C¹稳定性，得到长时间存在性、最终凸性及光滑指数收敛结果。
A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function $d_S$ for a convex subset $S$ in the cases where the boundary of $S$ contains a geodesic segment, the boundary of $S$ is $C^2$ or the boundary of $S$ is not regular is discussed. Furthermore, a nonsmooth version of positive semi-definiteness of Hessian of convex functions on Riemannian manifolds is established.
In this paper we analyze the problem of the geodesic connectedness of subsets of Riemannian manifolds. By using variational methods, the geodesic connectedness of open domains (whose boundaries can be not differentiable and not convex) of a smooth Riemannian manifold is proved. In some cases also the convexity of the domain is obtained. Moreover we present examples of the applicability and of the independence of the assumptions. Finally we give an application to the existence of trajectories with fixed energy of dynamical systems.
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.
In this paper, we show a local energy convexity of $W^{1,2}$ maps into $CAT(K)$ spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting.
We are analysing the convexity and continuity properties of the Mabuchi functional along weak geodesics. The key technical point in our paper is the global approximation of weak geodesics obtained via a well-chosen family of Monge-Ampère equations.
We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry--Emery and Otto--Villani. Furthermore we show that Ricc…
Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a reconstruction formula. We also show that under an assumption on the existence of a global foliation by strictly convex hypersurfaces the geodesic X-ray transform is globally injective. In addition we prove stability estimates and propose a layer stripping type algorithm for reconstruction.
We prove a geodesic distance formula for quasi-psh functions with finite entropy, extending results by Chen and Darvas. We work with big and nef cohomology classes: a key result we establish is the convexity of the $K$-energy in this general setting. We then study Monge-Ampère measures on contact sets, generalizing a recent result by the first author and Trapani.
The Teichmüller space $\mathcal{T}_S(\mathbf{b})$ of hyperbolic metrics on a surface $S$ with fixed lengths at the boundary components is symplectic. We prove that any sum of infinitesimal earthquakes on $S$ that is tangent to $\mathcal{T}_S(\mathbf{b})$ is Hamiltonian, by providing a Hamiltonian $\mathbb{L}$. Such function extends the classical length map associated to a compactly supported measured geodesic lamination and shares with it some peculiar properties, such as properness and strict convexity along earthquakes paths under usual topological conditions. As an application, we prove tha…
This paper is concerned with the problem of determining whether a projective-equivalence class of sprays is the geodesic class of a Finsler function. We address both the local and the global aspects of this problem. We present our results entirely in terms of a multiplier, that is, a type (0,2) tensor field along the tangent bundle projection. In the course of the analysis we consider several related issues of interest including the positivity and strong convexity of positively-homogeneous functions, the relation to the so-called Rapcsák conditions, some peculiarities of the two-dimensional ca…
Displacement convexity of Boltzmann's entropy characterizes the strong energy condition from general relativity
本文将黎曼流形中里奇曲率下界与熵的测地凸性之间的对应关系推广到洛伦兹时空，证明 Hawking-Penrose 强能量条件等价于 Boltzmann-Shannon 熵在洛伦兹距离分数次幂诱导的概率测度空间中的测地凸性，为在非光滑时空中研究强能量条件奠定基础。
We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this functional over Teichmuller space (due to Wolpert (1987)). We then estimate this Hessian from below in terms of local quantities and distance along the geodesic. The formula extends to proper arcs on punctured hyperbolic surfaces, and the estimate to laminations. Wolpert's result…
Horospherically Convex Optimization on Hadamard Manifolds Part I: Analysis and Algorithms
本文针对Hadamard流形上的优化问题，提出水平凸性（h-凸性）概念，并设计曲率无关的次梯度下降与Nesterov加速算法，匹配欧氏空间复杂度，并扩展到函数和最小化场景。
Convexity properties of the condition number II
本文研究最大秩矩阵空间中的条件度量诱导的Lipschitz-Riemann结构，证明矩阵最小奇异值的倒数沿测地线是对数凸函数，并将类似结果推广到线性系统的解簇。
The main goal of this paper is to present results of existence and non-existence of convex functions on Riemannian manifolds and, in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove that the conservativity of the geodesic flow on a Rieman- nain manifold with infinite volume is an obstruction to the existence of convex functions. Next, we present a geometric condition that ensures the existence of (strictly) convex functions on a particular class of complete non-compact man- ifolds, and, we use this fact to construct a manifold whose se…
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.
In this note, we shall prove geodesic convexity of the space of Kähler potentials on an ALE Kähler manifold. This extends earlier results in the compact case proved in the fundamental work of X-X. Chen. We further prove the boundedness from below of the Mabuchi energy, and give an alternative proof for the uniqueness of scalar-flat metrics in the category of ALE Kähler manifold.
Consider a compact Riemannian manifold of dimension $\geq 3$ with strictly convex boundary, such that the manifold admits a strictly convex function. We show that the attenuated ray transform in the presence of an arbitrary connection and Higgs field is injective modulo the natural obstruction for functions and one-forms. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo gauge transformations. The proofs involve a reduction to a local result showing that the geodesic X-ray transform with a matrix weight can be inverted locally near a…
Fractional Integral Estimates of Hermite-Hadamard type in Global Nonpositive Curvature Spaces
该研究在整体非正曲率空间中引入测地$h$-凸函数以拓展函数凸性概念，通过Katugampola分数阶积分证明Hermite-Hadamard型估计，得到涉及平方距离映射的本质精确推论，丰富了弯曲几何空间的分析理论。
The sparseness of g-convex functions
本文研究流形上g-凸光滑函数，建立了黎曼度量或联络存在性判据，并据此得出紧流形上g-凸函数集无处稠密等三个稀疏性结果。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。