本文選題：緊致無(wú)邊曲面 + 高斯曲率��； 參考：《中國(guó)科學(xué)技術(shù)大學(xué)》2017年碩士論文

【摘要】：本論文考慮了2維緊致無(wú)邊曲面上的預(yù)定高斯曲率問(wèn)題.確切地說(shuō),預(yù)定高斯曲率問(wèn)題是指給定一個(gè)帶有黎曼度量g0的緊致無(wú)邊曲面M以及定義在M上的一個(gè)光滑函數(shù)f,問(wèn)是否可以找到一個(gè)與g0逐點(diǎn)共形的度量g(即存在某個(gè)光滑函數(shù)u可以將g表示為g = e2u.g0)使得在度量g下的高斯曲率Kg=f?在本論文中,我們將采用共形高斯曲率流的方法來(lái)研究這個(gè)預(yù)定高斯曲率問(wèn)題.考慮一族依賴于時(shí)間參數(shù)t的度量g(t)滿足如下的演化方程(?)=-2(K-λ(t)·f).g,其中λ(t)是只依賴于時(shí)間的函數(shù).在對(duì)預(yù)先給定的函數(shù)f作適當(dāng)假設(shè)后,我們將證明這個(gè)流的短時(shí)存在性,長(zhǎng)時(shí)存在性以及收斂性.更進(jìn)一步,當(dāng)時(shí)間趨于無(wú)窮時(shí)得到的極限度量g∞即為預(yù)定高斯曲率問(wèn)題的解。
[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，

本文編號(hào)：1747683