Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow

TL;DR

This paper investigates the evolution of mean curvature flow within a Riemannian manifold whose metric evolves via Ricci flow coupled with harmonic map heat flow, deriving new monotonicity formulas and characterizations of solitons.

Contribution

It introduces a novel analysis of mean curvature flow in a dynamic ambient space evolving by coupled Ricci and harmonic map flows, extending Hamilton's Harnack expression and establishing a Huisken-type monotonicity formula.

Findings

Derived a variation formula for a functional along flow-preserving measures

Extended Hamilton's differential Harnack expression to boundary evolving by mean curvature flow

Constructed and characterized families of mean curvature solitons

Abstract

The main objective of this article is to study the mean curvature flow into an ambient compact smooth manifold M with boundary and with a Riemannian metric that evolves by a self-similar solution of the Ricci flow coupled with the harmonic map heat flow of a map from M to a Riemannian manifold N. In this context, we address a functional associated with this flow and calculate its variation along parameters that preserve the weighted volume measure. An extension of Hamilton's differential Harnack expression appears by considering the boundary of M evolving by mean curvature flow, which must vanish on the gradient steady soliton case. Next, we obtain a Huisken monotonicity-type formula for the mean curvature flow in the proposed background. We also show how to construct a family of mean curvature solitons and establish a characterization of such a family.