Flow by mean curvature inside a moving ambient space

TL;DR

This paper investigates the behavior of submanifolds evolving by mean curvature flow within a dynamically changing ambient space, exploring generalizations of key monotonicity formulas and differential inequalities.

Contribution

It introduces potential extensions of Huisken's monotonicity formula and examines their relation to Li--Yau--Hamilton inequalities in a moving Riemannian setting.

Findings

Connections between mean curvature flow and Ricci flow in a dynamic ambient space

Potential generalizations of Huisken's monotonicity formula

Insights into differential Harnack inequalities in evolving manifolds

Abstract

We show some computations related to the motion by mean curvature flow of a submanifold inside an ambient Riemannian manifold evolving by Ricci or backward Ricci flow. Special emphasis is given to the possible generalization of Huisken's monotonicity formula and its connection with the validity of some Li--Yau--Hamilton differential Harnack--type inequalities in a moving Riemannian manifold.