【作者】 陈伟；

【导师】 李光汉；

【作者基本信息】 武汉大学 ， 基础数学， 2022， 博士

【摘要】 本文主要研究了空间形式上一类完全非线性逆曲率流的内外非塌缩性和球面上一类分段对数高斯曲率流的长时间存在性.本文分为五章去说明.第一章是引言,介绍了曲率流和逆曲率流以及Minkowski问题的研究背景、研究现状和本文的主要结果.在第二章,我们介绍了一些预备知识,包括逆凹曲率函数,粘性解和球曲率函数几何,球面Sⁿ⁺¹上的凸体几何,分段对数高斯曲率流,以及测地极坐标下基本公式和投影坐标.在第三章中我们考察了球曲率函数,也即一个两点函数所具有的一些特殊性质.通过这些性质,并且将球和双曲空间分别放在更高一维的欧氏空间和Minkowski空间中考虑,选取特殊的坐标系,利用两点函数的极值原理,我们证明了一个微分不等式.然后,我们去考虑沿着曲率函数是单调、对称、一次齐次以及逆凹的逆曲率流的外球估计.选取合适的辅助函数,以及利用这个微分不等式,用反证法得到了沿着逆曲率流初始超曲面具有外部非塌缩性.最后,我们通过对之前关于逆曲率流外部非塌缩性的文章中得到的微分不等式的各项进行更加精确的估计以及选取不同的辅助函数,得到了最优的内部非塌缩性估计.第四章主要研究一类分段对数高斯曲率流的长时间存在性.考虑球中心对切平面的投影,将半球面上的凸超曲面转化为欧氏平面上的凸超曲面,得到了这类高斯曲率流的先验估计.于是,我们证明了在给定参数条件下,这类分段对数高斯曲率流的长时间存在性.最后,我们给出了所研究的这类分段对数高斯曲率流的一个应用.利用给定泛函J的值做区别,在不同的连续变动超曲面集合中,给出了一类预定曲率问题在β>0至少有两个解.在第五章中,我们将本文的主要内容做了总结,进而提出了一些可以继续研究的问题.更多还原

【Abstract】 In this thesis,we research exterior non-collapsing estimates and interior noncollapsing estimates for a fully nonlinear inverse curvature flow for inverse-concave speed functions in space forms,and longtime existence of a class of piece-wise logarithmic Gauss curvature flow.This thesis is divided into five chapters to illustrate.Chapter 1 is introduction,which introduces the background and the developments of curvature flows,inverse curvature flows and Minkowski problem,and the main results of this thesis.In Chapter 2,we introduce the preliminaries of this thesis.It contains the properties of inverse-concave curvature function,viscosity solution and geometry of ball curvature,convex geometry in the sphere,the piece-wise logarithmic Gauss curvature flow,the fundamental formulas and the projection coordinate.In Chapter 3,we study the special properties of ball curvature,which is a twopoints function.We consider the sphere and the hyperbolic space as the embedded submanifold of Euclidean space and Minkowski space,respectively.By a suitable choice of the coordinates system,we obtain a differential inequality by maximum principle of a two-point function.Then,we consider exterior ball curvature estimates for inverse curvature flow,for which the speed is a monotone increasing,symmetric,homogeneous of degree one and inverse-concave function.Selecting a suitable auxiliary function and using the differential inequality,we derive exterior non-collapsing estimates for a fully nonlinear inverse curvature flow.Finally,we investigate terms of the differential inequality in previous article,and obtain optimal inscribed estimates by choosing a different auxiliary function.In Chapter 4,we mainly research longtime existence of a class of piece-wise logarithmic Gauss curvature flow.By the projection of hemisphere onto the hyperplane of Euclidean space,we transform convex hypersurface in the sphere to convex hypersurface in Euclidean hyperplane,and get the a priori estimates of these piece-wise logarithmic Gauss curvature flows.Then,we obtain longtime existence of a class of piece-wise logarithmic Gauss curvature flow in giving index condition.Finally,we give an application of these piece-wise logarithmic Gauss curvature flows.Make use of different value of the monotone functional J,we give that prescribed curvature problem has at least two solutions when β > 0.In Chapter 5,we summarize the main contents of this thesis.Moreover,we put forward some questions that will be further studied in the future.更多还原