Variation of the first eigenvalue of p-Laplacian on evolving geometry and applications

Abstract

Let (M,g)be an n-dimensional compact Riemannian manifold whose metric g(t)evolves by the generalised abstractgeometric flow. This paper discusses the variation formulas, monotonicity and differentiability for the first eigenvalue of thep-Laplacian on (M,g(t)). It is shown that the first nonzero eigenvalue is monotonically nondecreasing along the flow undercertain geometric conditions and that it is differentiable almost everywhere. These results provide a unified approach to the study of eigenvalue variations and applications under many geometric flows