Parabolic Frequency on Manifolds

Abstract

Abstract We prove monotonicity of a parabolic frequency on static and evolving manifolds without any curvature or other assumptions. These are parabolic analogs of Almgren’s frequency function. When the static manifold is Euclidean space and the drift operator is the Ornstein–Uhlenbeck operator, this can been seen to imply Poon’s frequency monotonicity for the ordinary heat equation. When the manifold is self-similarly evolving by the Ricci flow, we prove a parabolic frequency monotonicity for solutions of the heat equation. For the self-similarly evolving Gaussian soliton, this gives directly Poon’s monotonicity. Monotonicity of frequency is a parabolic analog of the 19th century Hadamard three-circle theorem about log convexity of holomorphic functions on C. From the monotonicity, we get parabolic unique continuation and backward uniqueness.