Applications of Toponogov's comparison theorems for open triangles

洛伦兹长度空间的 Toponogov 全局化结果

A Toponogov globalisation result for Lorentzian length spaces

本文在合成洛伦兹长度空间中建立 Toponogov 全局化定理，利用猫摇篮构造及三角形细分和 Lebesgue 数引理，并应用于 Bonnet–Myers 定理、分裂定理及曲率界稳定性。

具有非负 Ricci 曲率和近最大局部重绕体积的开流形的有限拓扑类型定理

A finite Topological Type Theorem for open manifolds with Non-negative Ricci Curvature and Almost Maximal Local Rewinding Volume

针对非负 Ricci 曲率且局部重绕体积近最大的开流形，提出无需截面曲率、共轭半径或 Toponogov 型比较假设的有限拓扑类型定理，并扩展了既有方法的适用范围。

Topology of complete Finsler manifolds with radial flag curvature bounded below

We recently established a Toponogov type triangle comparison theorem for a certain class of Finsler manifolds whose radial flag curvatures are bounded below by that of a von Mangoldt surface of revolution (arXiv:1205.3913). In this article, as its applications, we prove the finiteness of topological type and a diffeomorphism theorem to Euclidean spaces.

Comparison theorems for Lorentzian length spaces with lower timelike curvature bounds

In this article we introduce a notion of normalized angle for Lorentzian pre-length spaces. This concept allows us to prove some equivalences to the definition of timelike curvature bounds from below for Lorentzian pre-length spaces. Specifically, we establish some comparison theorems known as the local Lorentzian version of the Toponogov theorem and the Alexandrov convexity property. Finally, as an application we obtain a first variation Formula for non-negatively curved globally hyperbolic Lorentzian length spaces.

Necessary and Sufficient Conditions for a Triangle Comparison Theorem

We prove a version of Topogonov's triangle comparison theorem with surfaces of revolution as model spaces. Given a model surface and a Riemannian manifold with a fixed base point, we give necessary and sufficient conditions under which every geodesic triangle in the manifold with a vertex at the base point has a corresponding Alexandrov triangle in the model. Under these conditions we also prove a version of the Maximal Radius Theorem and a Grove--Shiohama type Sphere Theorem.

比较几何讲义

Lecture Notes on Comparison Geometry

本比较几何讲义基于多伦多大学相关课程整理，涵盖多种经典比较定理及其应用、多位学者的核心定理与猜想证明，还介绍了Gromov-Hausdorff收敛等进阶内容。

Comparison geometry of holomorphic bisectional curvature for Kaehler manifolds and limit spaces

We give an analog of triangle comparison for Kaehler manifolds with a lower bound on the holomorphic bisectional curvature. We show that the condition passes to noncollapsed Gromov-Hausdorff limits. We discuss tangent cones and singular Kaehler spaces.

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. II

We prove, as our main theorem, the finiteness of topological type of a complete open Riemannian manifold $M$ with a base point $p \in M$ whose radial curvature at $p$ is bounded from below by that of a non-compact model surface of revolution $\tilde{M}$ which admits a finite total curvature and has no pair of cut points in a sector. Here a sector is, by definition, a domain cut off by two meridians emanating from the base point $\tilde{p} \in \tilde{M}$. Notice that our model $\tilde{M}$ does not always satisfy the diameter growth condition introduced by Abresch and Gromoll. In order to prove …

Geodesic period integrals of eigenfunctions on Riemannian surfaces and the Gauss-Bonnet Theorem

We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemann surfaces of negative curvature period integrals of eigenfunctions $e_λ$ over geodesics go to zero at the rate of $O((\logλ)^{-1/2})$ if $λ$ are their frequencies. As discussed in \cite{CSPer}, no such result is possible in the constant curvature case if the curvature is $\ge0$. Notwithstanding, we also show that these bounds for period integrals are valid provided that integrals of the curvature over all geodesic balls of radius $r\le 1$ are pinched from above by $-δr^N$ …

A rigidity theorem in Alexandrov spaces with lower curvature bound

Distance functions of metric spaces with lower curvature bound, by definition, enjoy various metric inequalities; triangle comparison, quadruple comparison and the inequality of Lang-Schroeder-Sturm. The purpose of this paper is to study the extremal cases of these inequalities and to prove rigidity results. The spaces which we shall deal with here are Alexandrov spaces which possibly have infinite dimension and are not supposed to be locally compact.

Shur type comparison theorems for affine curves with application to lattice point estimates

If $c, \overline c\colon [a,b]\to \mathbb R^2$ are two convex planar curve parameterized by affine arc length and $A\colon [a,b]\to [0,\infty)$ is the area bounded by the restriction $c\big|_{[a,s]}$ and the segment between $c(a)$ and $c(s)$ with $\overline A$ the corresponding function for $\overline c$, and the affine curvature of $c$ pointwise less than the affine curvature of $\overline c$. Then $A(s)\ge \overline A(s)$. Also for any point of a convex curve we define \emph{adapted affine coordinates} centered at the point and give sharp estimates on the coordinates of the curve in terms of…

Lorentz and semi-Riemannian spaces with Alexandrov curvature bounds

A semi-Riemannian manifold is said to satisfy $R\ge K$ (or $R\le K$) if spacelike sectional curvatures are $\ge K$ and timelike ones are $\le K$ (or the reverse). Such spaces are abundant, as warped product constructions show; they include, in particular, big bang Robertson-Walker spaces. By stability, there are many non-warped product examples. We prove the equivalence of this type of curvature bound with local triangle comparisons on the signed lengths of geodesics. Specifically, $R\ge K$ if and only if locally the signed length of the geodesic between two points on any geodesic triangle is …