The first eigenvalue of the p -Laplace operator under powers of mean curvature flow

Abstract

In this paper, we derive the evolution equation for the first eigenvalue of Laplace operator along pow- ers of mean curvature flow. Considering a compact, strictly convex n-dimensional surface M without boundary, which is smoothly immersed in Rn+1, we prove that if the initial 2-dimensional surface M is totally umbilical, then the first eigenvalue is nondecreasing along the unnormalized H k -flow. Moreover, as applications of the evolution equation, we construct some monotonic quantities along this kind of flow.