论文标题
riemannian带和开放不完整的歧管的刚性比较几何形状
Rigid comparison geometry for Riemannian bands and open incomplete manifolds
论文作者
论文摘要
比较定理是我们对各种曲率约束所隐含的几何特征的理解的基础。本文认为在标量，2-RICCI或RICCI曲率上具有正下限的歧管，并包含多种定理，这些定理在{\ em {width}}的界限和概念之间提供牢固的关系。一些不平等的几何量（例如边界平均曲率），而另一些不平等量则涉及以链接要求或同源约束形式的拓扑条件。在其中一些结果中，研究了开放和不完整的歧管，其中之一部分解决了在这种情况下Gromov的猜想。大多数结果都伴随着刚度的刚度陈述，这些声明隔离了各种模型几何（完整和不完整），包括对圆形镜头空间的新表征以及其他在其他地方没有出现的模型。作为副产品，我们还提供了几种经典比较陈述（例如邦纳河床和弗兰克尔定理）的新的定量证明，以及Llarull定理的版本，以及关于渐近平面流形的显着事实。我们提出的结果在特征方面有很大差异，但是存在一个共同的主题，因为\ emph {spacetime谐波函数}在每个证明中的主要作用}，这是针对最初旨在研究数学一般相关性质量的某些椭圆方程的解决方案。
Comparison theorems are foundational to our understanding of the geometric features implied by various curvature constraints. This paper considers manifolds with a positive lower bound on either scalar, 2-Ricci, or Ricci curvature, and contains a variety of theorems which provide sharp relationships between this bound and notions of {\em{width}}. Some inequalities leverage geometric quantities such as boundary mean curvature, while others involve topological conditions in the form of linking requirements or homological constraints. In several of these results open and incomplete manifolds are studied, one of which partially addresses a conjecture of Gromov in this setting. The majority of results are accompanied by rigidity statements which isolate various model geometries -- both complete and incomplete -- including a new characterization of round lens spaces, and other models that have not appeared elsewhere. As a byproduct, we additionally give new and quantitative proofs of several classical comparison statements such as Bonnet-Myers' and Frankel's Theorem, as well as a version of Llarull's Theorem and a notable fact concerning asymptotically flat manifolds. The results that we present vary significantly in character, however a common theme is present in that the lead role in each proof is played by \emph{spacetime harmonic functions}, which are solutions to a certain elliptic equation originally designed to study mass in mathematical general relativity.
