Parabolic Frequency on Manifolds

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