Second fundamental form and gradient of Neumann semigroups

摘要

The second fundamental form for the boundary of a compact Riemannian manifold is described by a short time behavior for the gradient of Neumann semigroups. As an application, we prove that the manifold is convex if and only if the Neumann heat semigroup P_t satisfies the gradient estimate |i÷P_t, f|iU e～k~t P_t |i÷f| for some constant K iE R and all all t iY 0, f iE C～l.