Geometry & Topology - Joel Spruck - Convexity of translating solitons for the mean curvature flow
Speaker: Joel Spruck (Johns Hopkins University)
Title: Convexity of translating solitons for the mean curvature flow
Abstract: Deciding when solutions of geometric problems (and more generally elliptic pde’s) are convex is a complicated subject with a long history. In this talk we will sketch some new methods to study this question for translating solitons of the mean curvature flow. With Ling Xiao, we showed that any complete two dimensional mean convex translator in R3 is actually convex and this leads to a complete classification. The same statement in higher dimensions is probably false. In new work with Liming Sun, we prove that complete uniformly 2-convex translat- ing solitons in Rn+1 are convex. Our theorem implies that a uniformly 2-convex, entire graphical translating solitons must be the bowl soliton.
We also prove the convexity of graphical translating solitons defined over convex domains with constant boundary data. This theorem im- plies the existence of (n-2) dimensional families of locally strictly convex translators in Rn+1. The same method also establishes an interesting convexity theorem for the constant mean curvature graph equation.