Laplacian特征值的上界估计及在几何流上的单调性研究

发布时间：2018-04-27 08:55

本文选题：p-Laplacian算子 + Ricci流　；参考：《河南师范大学》2017年硕士论文

【摘要】：这篇论文主要研究了三类问题:p-Laplacian第一特征值的上界估计;Ricci流上几何量的单调性;List流上特征值的单调性.在第一章中,考虑了n维完备黎曼流形(M,g)上有关方程△pu =-λ|u|p-2u,p1的正解的梯度估计,得到了有关p-Laplacian第一特征值的上界估计.在第二章中,在n维紧致无边的黎曼流形(Mn,g(t))上考虑如下非线性方程-△u + aulogu + bRu =λabu,(?)u2 dv = 1的正解,其中λab(t)是使方程存在正解的最小常数.g(t)沿着Ricci流和normalized Ricci流演化,得到了有关λab(t)的第一变分公式.特别地,这一章的结果推广了[7]和[24]中的结论.在第三章中,首先研究n维紧致无边的黎曼流形(Mn,g(t))上沿着Rescaled List's ex-tended Ricci 流(?)/(?)tgij=-2(Sij-r/ngij),φt = △φ特征值和能量泛函的单调性公式.得到Laplacian算子特征值的单调性公式,从而推广了Li[29]和Cao-Hou-Ling[9]中的结论.此外,也考虑Fk泛函和Wk泛函的单调性公式,其中Fk被看作对于steady Ricci breathers的F泛函的推广及Wk泛函被看作对于Shrinking Ricci breathers W泛函的推广.
[Abstract]:In this paper, we study the monotonicity of geometric quantities on Ricci flow and the monotonicity of eigenvalues on list flows. In the first chapter, we consider the gradient estimates of positive solutions of the equation pu Pu p-2u p 1 on the n-dimensional complete Riemannian manifold, and obtain the upper bound estimate for the first eigenvalue of p-Laplacian. In chapter 2, on the n-dimensional compact Riemannian manifold, we consider the positive solution of the following nonlinear equation--u aulogu bRu = 位 u aulogu bRu 2dv = 1, where 位 abt) evolves along the Ricci and normalized Ricci flows with the minimum constant of positive solutions. The first variational formula about 位 abt) is obtained. In particular, the results of this chapter generalize the conclusions in [7] and [24]. In chapter 3, we first study the monotonicity formulas for the eigenvalues and energy functional of n dimensional compact Riemannian manifold, 蠁 t = 蠁, along the Rescaled List's ex-tended Ricci flow. The monotonicity formula of eigenvalues of Laplacian operator is obtained, which generalizes the conclusions in Li [29] and Cao-Hou-Ling [9]. In addition, the monotonicity formulas of Fk functional and Wk functional are also considered, where Fk is regarded as a generalization of steady Ricci breathers's F functional and Wk functional is regarded as a generalization of Shrinking Ricci breathers W functional.
【学位授予单位】：河南师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O186.12

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