本文選題：p-Laplacian算子 + Ricci流��； 參考：《河南師范大學(xué)》2017年碩士論文

【摘要】：這篇論文主要研究了三類問題:p-Laplacian第一特征值的上界估計(jì);Ricci流上幾何量的單調(diào)性;List流上特征值的單調(diào)性.在第一章中,考慮了n維完備黎曼流形(M,g)上有關(guān)方程△pu =-λ|u|p-2u,p1的正解的梯度估計(jì),得到了有關(guān)p-Laplacian第一特征值的上界估計(jì).在第二章中,在n維緊致無邊的黎曼流形(Mn,g(t))上考慮如下非線性方程-△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數(shù).g(t)沿著Ricci流和normalized Ricci流演化,得到了有關(guān)λab(t)的第一變分公式.特別地,這一章的結(jié)果推廣了[7]和[24]中的結(jié)論.在第三章中,首先研究n維緊致無邊的黎曼流形(Mn,g(t))上沿著Rescaled List's ex-tended Ricci 流(?)/(?)tgij=-2(Sij-r/ngij),φt = △φ特征值和能量泛函的單調(diào)性公式.得到Laplacian算子特征值的單調(diào)性公式,從而推廣了Li[29]和Cao-Hou-Ling[9]中的結(jié)論.此外,也考慮Fk泛函和Wk泛函的單調(diào)性公式,其中Fk被看作對(duì)于steady Ricci breathers的F泛函的推廣及Wk泛函被看作對(duì)于Shrinking Ricci breathers W泛函的推廣.
[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.
【學(xué)位授予單位】：河南師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O186.12

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相關(guān)期刊論文 前4條

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本文編號(hào)：1810078