Compact translating solitons with non-empty planar boundary

Abstract

In this paper, we consider compact translating solitons with non-empty planar boundary. Each boundary component lies in a plane which is orthogonal to the translating direction. We firstly prove that when the planar boundary is either a circle or convex and the translating soliton meets the plane containing the boundary with a constant angle, then the compact translating soliton is part of an entire rotationally symmetric strictly convex graphical surface. Secondly, we show that a compact translating soliton spanning two horizontal planar Jordan curves inherits the symmetries of its boundary. We also show a balancing type formula for compact translating solitons with planar boundary.