Evolution and monotonicity of eigenvalues under the Ricci flow  

作　　者：FANG ShouWen XU HaiFeng ZHU Peng 

机构地区：[1]School of Mathematical Science, Yangzhou University [2]School of Mathematics and Physics, Jiangsu University of Technology

出　　处：《Science China Mathematics》2015年第8期1737-1744,共8页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11401514,11371310,11101352 and 11471145);Natural Science Foundation of the Jiangsu Higher Education Institutions of China(Grant Nos.13KJB110029 and 14KJB110027);Foundation of Yangzhou University(Grant Nos.2013CXJ001 and 2013CXJ006);Fund of Jiangsu University of Technology(Grant No.KYY13005);Qing Lan Project

摘　　要：Let （M,g（t）） be a compact Riemannian manifold and the metric g（t） evolve by the Ricci flow. We derive the evolution equation for the eigenvalues of geometric operator --△φ＋ cR under the Ricci flow and the normalized Ricci flow, where A, is the Witten-Laplacian operator, φ∈C∞（M）, and R is the scalar curvature with respect to the metric g（t）. As an application, we prove that the eigenvalues of the geometric operator are nondecreasing along the Ricci flow coupled to a heat equation for manifold M with some Ricci curvature 1 condition when c 〉1/4.Let(M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. We derive the evolution equation for the eigenvalues of geometric operator-△φ+ c R under the Ricci flow and the normalized Ricci flow, where △φis the Witten-Laplacian operator, φ∈ C∞(M), and R is the scalar curvature with respect to the metric g(t). As an application, we prove that the eigenvalues of the geometric operator are nondecreasing along the Ricci flow coupled to a heat equation for manifold M with some Ricci curvature condition when c >14.

关 键 词：EIGENVALUE Witten-Laplacian Ricci flow 

分 类 号：O186.12[理学—数学] O174[理学—基础数学]