Local monotonicity and mean value formulas for evolving Riemannian manifolds

2006/08/18 by Klaus Ecker, Ecker, Klaus, Dan Knopf +5
Mathematics · #35K55 #53C44 #58J35 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.math/0608470

openalex publication_date 2006/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive identities for general flows of Riemannian metrics that may be regarded as local mean-value, monotonicity, or Lyapunov formulae. These generalize previous work of the first author for mean curvature flow and other nonlinear diffusions. Our results apply in particular to Ricci flow, where they yield a local monotone quantity directly analogous to Perelman's reduced volume V and a local identity related to Perelman's average energy F.

Citations

• On the parabolic kernel of the Schrödinger operator
• A Theory of Subtemperatures in Several Variables
• Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds

Related

• Mean value theorems on manifolds
• A Class of Monotonic Quantities along the Ricci Flow
• Some remarks on Huisken's monotonicity formula for mean curvature flow
• A note on Huisken monotonicity-type formula for the mean curvature flow in a gradient shrinking extended Ricci soliton background
• Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula