Mean curvature flow in a Ricci flow background

TL;DR

This paper investigates the behavior of mean curvature flow within a Ricci flow background, deriving evolution equations and monotonicity formulas, extending classical results to more general geometric flows.

Contribution

It introduces a new weighted functional and derives its properties under Ricci flow, including evolution equations and monotonicity for mean curvature flow in this setting.

Findings

Derived evolution equations for second fundamental form and mean curvature.

Established a boundary term involving Hamilton's Harnack expression.

Discussed mean curvature solitons and Huisken monotonicity in gradient Ricci soliton backgrounds.

Abstract

Following work of Ecker, we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-with-boundary. We compute its variational properties and its time derivative under Perelman's modified Ricci flow. The answer has a boundary term which involves an extension of Hamilton's Harnack expression for the mean curvature flow in Euclidean space. We also derive the evolution equations for the second fundamental form and the mean curvature, under a mean curvature flow in a Ricci flow background. In the case of a gradient Ricci soliton background, we discuss mean curvature solitons and Huisken monotonicity.