Volume growth and Bernstein theorems for translating solitons

Abstract

In this paper, we study properties of the potential function of a translating soliton M in R n + 1 and the volume growth of the intersection of Euclidean balls with M . We show that if M is a convex translating soliton, then the infimum of the mean curvature function on M is zero, i.e., inf M ⁡ H = 0 . We give a condition to obtain the Bernstein theorem for the translating solitons. We also give an outline of a simple proof of the Bernstein theorem due to Bombieri–De Giorgi–Miranda.