The first eigenvalue of the $p$-Laplacian on time dependent Riemannian metrics

TL;DR

This paper investigates how the first eigenvalue of the $p$-Laplacian evolves on a compact Riemannian manifold with a time-dependent metric, establishing monotonicity and differentiability properties under geometric flows.

Contribution

It provides a unified framework for analyzing the evolution and monotonicity of the $p$-eigenvalue under various geometric flows, extending known results for the Laplace-Beltrami operator.

Findings

First eigenvalue is monotone nondecreasing under certain conditions.

The first eigenvalue is differentiable almost everywhere.

Results unify the study of $p$-eigenvalues across different geometric flows.

Abstract

Let $(M,g)$ be an $n$-dimensional compact Riemannian manifold ($n>1$) whose metric $g(t)$ evolves by the generalized abstract geometric flow. This paper discusses the evolution, monotonicity and differentiability for the first eigenvalue of the $p$-Laplacian on $(M,g(t))$ with respect to time evolution. We prove that the first nonzero $p$-eigenvalue is monotone nondecreasing along the flow under certain geometric condition and that the first eigenvalue is differentiable almost everywhere. When $p=2$, we recover the corresponding results for the usual Laplace-Beltrami operator. Our results provide a unified approach to the study of $p$-eigenvalue under various geometric flows