Research On The Eigenvalues ​​of Geometric Operators Along The Ricci Flow

In recent years,the study of eigenvalues of geometric operators has become a very powerful tool for studying geometry and topology on manifolds.In 2002,Perelman pioneered the definitions of two entropy functionals F and W for Ricci flow,which play a key role in the proof of the Poincare conjecture.The lower bounds of these two functionals are closely related to the eigenvalues of geometric operators.The research on them arouses many researchers’ interest in eigenvalues of geometric operators under geometric flow,especially the Ricci flow.In this paper,we mainly investigate the monotonicity of the geometric operators related to Perelman type F functional and W functional under the Ricci flow and Ricci-Bourguignon flow.The structure of this article is arranged as follows:In chapter one,there are some basic concepts and formulas of Riemannian ge-ometry need to be used in this paper,and some fundamental theories of Ricci flow and Ricci-Bourguignon flow.In chapter two,we consider the Ricci flow equation on a compact Riemannian manifold.Firstly,we define a geometric operator □ from Perelman’s W functional,where □f =-△φf+af In f+cRf,and derive the evolution equation of the geometric operator along the Ricci flow.Secondly,we consider the system of Ricci flow coupled to a heat equation,and derive the evolution equation of eigenvalues along the system,and obtain the monotonicity of eigenvalues.Finally,we also consider the normalized Ricci flow,and obtain the evolution equation and monotonicity of eigenvalues of the geometric operator □ under the normalized Ricci flow.In chapter three,we consider the Ricci-Bourguignon flow on a compact Rieman-nian manifold,which can be seen as an interpolation between the Ricci flow and the Yamabe flow.We study the evolution equation and monotonicity of eigenvalues of the geometric operator-△+cR along the Ricci-Bourguignon flow and the normalized Ricci-Bourguignon flow respectively.