曲面上的预定高斯曲率问题

发布时间：2018-04-14 04:19

  本文选题：紧致无边曲面 + 高斯曲率　； 参考：《中国科学技术大学》2017年硕士论文

【摘要】：本论文考虑了2维紧致无边曲面上的预定高斯曲率问题.确切地说,预定高斯曲率问题是指给定一个带有黎曼度量g0的紧致无边曲面M以及定义在M上的一个光滑函数f,问是否可以找到一个与g0逐点共形的度量g(即存在某个光滑函数u可以将g表示为g = e2u.g0)使得在度量g下的高斯曲率Kg=f?在本论文中,我们将采用共形高斯曲率流的方法来研究这个预定高斯曲率问题.考虑一族依赖于时间参数t的度量g(t)满足如下的演化方程(?)=-2(K-λ(t)·f).g,其中λ(t)是只依赖于时间的函数.在对预先给定的函数f作适当假设后,我们将证明这个流的短时存在性,长时存在性以及收敛性.更进一步,当时间趋于无穷时得到的极限度量g∞即为预定高斯曲率问题的解。
[Abstract]:In this paper, we consider the problem of predetermined Gao Si curvature on two dimensional compact boundless surfaces.To be exact,The predetermined Gao Si curvature problem is to give a compact boundless surface M with Riemannian metric g 0 and a smooth function f defined on M, and ask if a metric g, which is conformal with g0 point by point, can be found (that is, there exists some smooth surface).The function u can express g as g = e2u.g0) such that the curvature of Gao Si under metric g is KgG f?In this thesis, the conformal Gao Si curvature flow is used to study this problem.Consider a family of metric gt dependent on time parameter t) and satisfy the following evolution equation, where 位 t) is a time-dependent function.After making appropriate assumptions for a given function f, we will prove the short time existence, long term existence and convergence of the flow.Furthermore, when the time tends to infinity, the limit metric g 鈭，

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