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

TL;DR

This work studies mean curvature flow of hypersurfaces in a boundary-manifold $M$ whose ambient metric evolves under the coupled Ricci flow and harmonic map heat flow, the $(RH)_\alpha$ flow. It develops a variational framework via the weighted functional $\mathcal{F}_\infty^\alpha$, derives its first variation under weight-preserving variations, and proves a boundary extension of Hamilton's differential Harnack expression together with a Huisken-type monotonicity formula for MCF in this background. The authors analyze the modified RH$_\alpha$ flow and boundary MCF interactions, establish boundary evolution equations, extend Harnack-type identities to the boundary, and characterize and construct mean curvature solitons in this setting, including explicit Euclidean radial constructions. Overall, the paper unifies soliton theory, monotonicity, and boundary geometry for MCF in a Ricci-harmonic background, offering a cohesive variational framework with potential applications to geometric flows on manifolds with boundary.

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.

Theorems & Definitions (13)

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• proof : Proof of Theorem \ref{['principal_theorem']}
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• proof : Proof of Theorem \ref{['Huisken_monotonicity']}