Monotone Volume Formulas for Geometric Flows

TL;DR

This paper develops new monotone volume formulas for various geometric flows, extending Perelman's reduced volume concept to broader settings such as List's extended Ricci flow and Lorentzian mean curvature flow.

Contribution

The authors introduce a unified approach to construct monotone volume quantities for a class of geometric flows, generalizing known results like Perelman's reduced volume.

Findings

Established monotonicity formulas for extended Ricci flows.

Derived new volume monotonicity results for Lorentzian mean curvature flow.

Unified framework applicable to multiple geometric flow systems.

Abstract

We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter being non-decreasing along the flow. In the case where S=Ric is the Ricci curvature of M, the result corresponds to Perelman's well-known reduced volume monotonicity for the Ricci flow. Some other examples are given in the second section of this article, the main examples and motivation for this work being List's extended Ricci flow system, the Ricci flow coupled with harmonic map heat flow and the mean curvature flow in Lorentzian manifolds with nonnegative sectional curvatures. With our approach, we find new monotonicity formulas for these flows.