Asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends

TL;DR

This paper establishes sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian on conical ends, providing new insights into self-shrinkers and self-expanders in mean curvature flow.

Contribution

It introduces a weighted frequency function approach to derive asymptotic estimates and offers a new elliptic proof of the uniqueness of self-shrinkers asymptotic to a cone.

Findings

Sharp asymptotic estimates for eigenfunctions on conical ends

A new elliptic proof of the uniqueness of self-shrinkers

Unique continuation property for self-expanders from cones

Abstract

We use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness of self-shrinkers of the mean curvature flow asymptotic to a given cone. Another consequence is a unique continuation property for self-expanders of the mean curvature flow that flow from a cone.