Rigidity of complete self-shrinkers whose tangent planes omit a nonempty set

TL;DR

This paper establishes rigidity results for certain self-shrinkers and self-expanders in Euclidean space, showing that under specific geometric conditions, these are uniquely characterized as spheres, planes, or cylinders.

Contribution

It introduces a new rigidity condition based on tangent affine submanifolds omitting a set, leading to unique classification results for self-shrinkers and self-expanders.

Findings

Only spheres, planes, and cylinders satisfy the geometric assumption.

The assumption defines a new class of submanifolds beyond polynomial volume growth.

Analogous rigidity results hold for self-expanders.

Abstract

In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder as the only self-shrinkers satisfying a classic geometric assumption, namely the union of all tangent affine submanifolds of a complete self-shrinker omits a non-empty set of the Euclidean space. This assumption lead us to a new class of submanifolds, different from those with polynomial volume growth or the proper ones. We also prove an analogous result for self-expanders.