Communications in Analysis and Geometry

Volume 27 (2019)

Number 1

Volume growth, entropy and stability for translating solitons

Pages: 47 – 72



Qiang Guang (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.; and Department of Mathematics, University of California at Santa Barbara)


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.

Received 28 May 2015

Published 7 May 2019