A note on Huisken monotonicity-type formula for the mean curvature flow in a gradient shrinking extended Ricci soliton background

TL;DR

This paper extends Huisken's monotonicity formula to mean curvature flow within a manifold evolving under a shrinking Ricci soliton, providing new insights and results especially in noncompact settings.

Contribution

It adapts Huisken's formula to a Ricci soliton background and addresses noncompact cases under natural geometric assumptions.

Findings

Established a Huisken-type monotonicity formula in Ricci soliton backgrounds.

Proved results for noncompact manifolds under certain geometric conditions.

Extended previous techniques to new geometric contexts.

Abstract

We give an application of a Huisken monotonicity-type formula for the mean curvature flow in a compact smooth manifold with a Riemannian metric that evolves by a shrinking self-similar solution of the extended Ricci flow. Our investigation builds on previous articles by Huisken and the third author as we apply their techniques to establish new results in this geometric setting. Moreover, under some natural geometric assumptions, the noncompact case is also solved