【作者】 郭健；

【导师】 朱高生；

【作者基本信息】 哈尔滨工业大学 ， 基础数学， 2025， 硕士

【摘要】 Laplace比较定理是黎曼几何中的一个重要结果,其基本想法是利用Ricci曲率的下界研究Laplace-Beltrami算子作用在距离函数上的性质.在完备黎曼流形的几何研究中,这一结果具有许多深刻应用,例如:Myers有限直径定理、Bishop-Gromov体积比较定理、Cheeger-Gromoll分裂定理等.目前,Laplace算子已有多种推广形式,如加权Laplace算子、散度型算子LA等,与之相对应的比较定理的研究需结合额外的几何结构,这导致证明过程的复杂性增加.研究Laplace比较定理的方法主要有以下几种:Jacobi场比较、Bochner公式比较、曲率维数不等式比较等.其中基于Bochner公式比较的方法是借助Bochner公式和曲率条件导出Δr的微分不等式,这一方法适用于光滑流形或者加权流形.本文借助Bochner公式,研究在Ricci曲率有下界的完备黎曼流形上,距离函数的Laplace算子的上界估计,主要分为两部分内容:第一,对于带有(1,1)型Codazzi自伴张量场A的完备无边流形M,证明散度型算子LA在距离函数满足π/(4(?))≤r≤π/(2(?))时的上界估计,这是经典Laplace比较定理的推广形式;在此基础上,得到该流形直径的上界.第二,对于完备带边黎曼流形,当Ricci曲率有负下界时,得到Δr的与平均曲率有关的上界估计,其中r表示流形上一点到边界的距离函数.更多还原

【Abstract】 Laplacian comparison theorem is an important result in Riemannian geometry,and its basic idea is to use the lower bound of Ricci curvature to study the properties of the Laplace-Beltrami operator acting on distance functions.In the geometric analysis of complete Riemannian manifolds,this result has many profound applications,such as,Myers’ theorem,Bishop-Gromov’s volume comparison theorem,Cheeger-Grooll splitting theorem,etc.At present,the Laplace operator has various forms of extension,such as weighted Laplace operator,divergence type operator LA,etc.The study of the corresponding comparison theorem requires additional geometric structures,which increases the complexity of the proof process.There are several methods for studying Laplacian comparison theorem,including:Jacobi field,Bochner formula,curvature dimension inequality,etc.The method based on Bochner formula is to obtain the differential inequality of Ar by using Bochner formula and curvature condition,which is applicable to smooth manifolds or weighted manifolds.This thesis uses the Bochner formula to study the upper bound estimation of the Laplace operator of the distance function on a complete Riemannian manifold with a lower bound on Ricci curvature.This thesis is mainly divided into two parts.Firstly,for a complete manifold with a(1,1)-type Codazzi self adjoint tensor field A,we prove an upper bound estimate of the elliptic operator LA when the distance function is satisfied byπ/(4(?))≤r≤π/(2(?)).This is a generalized form of the classical Laplace comparison theorem.Based on this,the upper bound of the diameter of the manifold is obtained.Secondly,for complete Riemannian manifolds with boundary,when the Ricci curvature has a negative lower bound,we prove that Ar has an upper bound,and this upper bound is related to the mean curvature,where r represents the distance from a point on the manifold to the boundary.更多还原