The constant angle problem for mean curvature flow inside rotational tori

TL;DR

This paper studies the long-term behavior of mean curvature flow of hypersurfaces inside rotational tori, proving convergence to flat cross-sections under specific boundary and initial conditions.

Contribution

It establishes the global existence and convergence of mean curvature flow inside rotational tori with Neumann boundary conditions, extending understanding of geometric flows in complex boundary settings.

Findings

Flow exists for all time under given conditions

Flow converges to flat cross-sections as time approaches infinity

Provides conditions for initial hypersurface compatibility

Abstract

We flow a hypersurface in Euclidean space by mean curvature flow with a Neumann boundary condition, where the boundary manifold is any torus of revolution. If we impose the conditions that the initial manifold is compatible and does not contain the rotational vector field in its tangent space, then mean curvature flow exists for all time and converges to a flat cross-section as $t \rightarrow \infty$.