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

In this paper, we show the following main results. Let (Mⁿ,g(t)), t ∈ 0,T), be a solution of the unnormalized H^k ? flow on a closed manifold, and λ_1,p(t) be the first eigenvalue of the p‐Laplace operator. If there exists a nonnegative constant ε such that in M × 0,T) and in M × 0,T),then λ_1,p(t) is increasing and the differentiable almost everywhere along the unnormalized H^k ? flow on 0,T). At last, we discuss some useful monotonic quantities. Copyright © 2013 John Wiley & Sons, Ltd.