The Frankel property for self-shrinkers from the viewpoint of elliptic PDE's

TL;DR

This paper proves that properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect, using a localized Reilly formula and f-harmonic functions, with implications for the geometry of self-shrinkers.

Contribution

It introduces a new method using localized Reilly formulas and f-harmonic functions to analyze the intersection properties of self-shrinkers, including a novel proof of the half-space property.

Findings

Properly embedded self-shrinkers separated at infinity must intersect.

A localized Reilly formula approach is effective for studying self-shrinkers.

A new proof of the generalized half-space property for immersed self-shrinkers.

Abstract

We show that two properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect at a finite point. The proof is based on a localized version of the Reilly formula applied to a suitable f-harmonic function with controlled gradient. In the immersed case, a new direct proof of the generalized half-space property is also presented.