EIGENVALUES VARIATION OF THE P-LAPLACIAN UNDER THE RICCI FLOW ON SM

Shahroud Azami

doi:10.22342/jims.22.2.215.157-177

Abstract

Let (M,F) be a compact Finsler manifold. Studying the eigenvalues and eigenfunctions for the linear and nonlinear geometric operators is a known problem. In this paper we will consider the eigenvalue problem for the p-laplace operator for Sasakian metric acting on the space of functions on SM. We find the first variation formula for the eigenvalues of p-Laplacian on SM evolving by the Ricci flow on M and give some examples.DOI : http://dx.doi.org/10.22342/jims.22.2.215.157-177

Highlights

For a compact Finsler manifold (M, F ), studying the eigenvalues of geometric operators plays a powerful role in geometric analysis
In the classical theory of the Laplace or p-Laplace equation several main parts of mathematics are joined in a fruitful way: Calculus of Variation, Partial Differential Equation, Potential Theory, 2000 Mathematics Subject Classification: Primary 58C40; Secondary 53C44
Ricci flow is a system of partial differential equations of parabolic type which was introduced by Hamilton on Riemannian manifolds for the first time in 1982 and Bao

Summary

Introduction

For a compact Finsler manifold (M, F ), studying the eigenvalues of geometric operators plays a powerful role in geometric analysis. Ricci flow is a system of partial differential equations of parabolic type which was introduced by Hamilton on Riemannian manifolds for the first time in 1982 and Bao (see [2, 17]). Studied Ricci flow equation in Finsler manifold. The Ricci flow has been proved to be a very useful tool to improve metrics in Finsler geometry, when M is compact. Under this normalized flow, the volume of the solution metrics remains constant in time. Short time exitance and uniqueness for solution to the Ricci flow on [0, T ) have been shown by Hamilton in [5] and by DeTurk in [7] for Riemannian manifolds and by the authors in [1] for important Berwald manifolds

Preliminaries

Examples