Translating surfaces under flows by sub-affine-critical powers of Gauss curvature

2021/04/27 by Beomjun Choi, Kyeongsu Choi, Choi, Beomjun +3 · 1 citation
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research

paper · pdf · doi:10.48550/arxiv.2104.13186

Abstract

We classify the surfaces translating under the flows by sub-affine-critical powers of the Gauss curvature. This, in particular, lists all translating solitons possibly model Type II singularities for convex closed solutions in all positive powers. The surfaces are entire graphs, and therefore our result corresponds to the Liouville theorem for the degenerate Monge--Ampère equations det D² u=(1+|Du|²)^2-(1)/(2α) on ℝ² in the range 0<α<1/4. The result also reveals that the moduli spaces of solutions are homeomorphic to either Euclidean spaces or cylinders.

Citations

• Boundary regularity of maps with convex potentials
• Ancient solutions to the Ricci flow in dimension 3

Cited by

• Ancient Gauss Curvature Flows of Bounded Width

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