A Bernstein Type Theorem For Self-similar Shrinkers

TL;DR

This paper proves that smooth entire graph self-shrinkers in Euclidean space must be hyperplanes, removing previous growth restrictions and extending the classification of such geometric objects.

Contribution

It establishes that no growth condition at infinity is necessary for entire graph self-shrinkers to be hyperplanes, generalizing earlier results.

Findings

Entire graph self-shrinkers are hyperplanes

No growth assumptions are needed at infinity

Extends previous classification results

Abstract

In this note, we prove that smooth self-shrinkers in $\Real^{n+1}$, that are entire graphs, are hyperplanes. Previously Ecker and Huisken showed that smooth self-shrinkers, that are entire graphs and have at most polynomial growth, are hyperplanes. The point of this note is that no growth assumption at infinity is needed.