Evolution of the first eigenvalue along the inverse mean curvature flow in space forms

Abstract

In this paper, we show that the first nonzero eigenvalues λ1 of the Laplacian and the p-Laplacian are decreasing along the inverse mean curvature flow in the hyperbolic space and in the sphere. For a convex closed surface M⊂S3, we show that the geometric quantity λ1⋅Area(M) is monotonically decreasing along the inverse mean curvature flow in S3.