Volume growth, entropy and stability for translating solitons

TL;DR

This paper investigates the geometric and spectral properties of translating solitons in mean curvature flow, establishing volume growth rates, entropy calculations, curvature bounds, and stability criteria.

Contribution

It provides new results on volume growth, entropy, curvature estimates, and stability spectra for translating solitons, including rigidity results for stable translators.

Findings

Complete translators have at least linear volume growth.

Entropy of grim reaper and bowl solitons computed.

Curvature estimates for translators with small entropy.

Abstract

We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. We also give a curvature estimate for translators in $\mathbf{R}^3$ with small entropy. Finally, we estimate the spectrum of the stability operator $L$ for translators and give a rigidity result of $L$-stable translators.