14 jul 2016 -- 14:30
Aula Dal Passo, Dip.Matematica, Università "Tor Vergata", Roma
Abstract.
We will consider translating solutions of the mean curvature flow (or translators for short); that is, solutions of the mean curvature flow which evolve purely by translation. Strictly convex translators are of particular interest, as they arise as `blow-up limits' of singularities of mean convex mean curvature flow. It was a longstanding conjecture that strictly convex translators should be rotationally symmetric. This was famously proved, by X. J. Wang, to be true in two space dimensions, but false in all higher dimensions; however, the question remained open for translators which arise as blow-ups of singularities. It was recently proved by Haslhofer that two-convex, non-collapsing translating solutions are necessarily rotationally symmetric. It then follows from a result of Andrews that any translator which arises as a blow-up limit of a two-convex embedded mean curvature flow is necessarily rotationally symmetric. We will show how to remove the non-collapsing assumption in spatial dimensions $n\geq 3$ by instead making use of estimates of Huisken and Sinestrari. As a consequence, we deduce that any translator which arises as a blow-up limit of a two-convex immersed mean curvature flow is necessarily rotationally symmetric.