Mean curvature flow in submanifolds

TL;DR

This paper derives explicit solutions for the mean curvature flow within certain Euclidean submanifolds, including a hypersurface in a Lagrangian self-expander, demonstrating convergence to minimal surfaces.

Contribution

It provides explicit solutions for mean curvature flow in specific submanifolds and shows convergence to minimal surfaces, extending understanding of geometric flows.

Findings

Explicit solutions of mean curvature flow in Euclidean submanifolds.

Convergence of flow to minimal hypersurfaces.

Application to Lagrangian self-expanders.

Abstract

We obtain explicit solutions of the mean curvature flow in some submanifolds of the Euclidean space. We give particularly an explicit solution of the flow of a hypersurface in the Lagrangian self-expander $L$ which is constructed in the article of Joyce, Lee and Tsui and show that it converge to a minimal one.