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

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 $\det D^{2}u=(1+|Du{|}^2{)}^{2-\frac{1}{2\alpha }}$ on ${\mathbb{R}}^2$ in the range $0<\alpha <1/4$. The result also reveals that the moduli spaces of solutions are homeomorphic to either Euclidean spaces or cylinders.

arXiv commentIn this revision, we (i) fixed a gap by proving a rigidity result in Section 3.5, and (ii) improved the result by determining the topology of the moduli space of translators in Section 5

References in corpus (5)

• Uniqueness of closed self-similar solutions to the Gauss curvature flow
• Unique Asymptotics of Compact Ancient Solutions to three-dimensional Ricci flow
• A Liouville theorem for solutions of degenerate Monge-Ampère equations
• Ancient finite entropy flows by powers of curvature in ${\mathbb{R}}^2$
• Classification of ancient flows by sub-affine-critical powers of curvature in ${\mathbb{R}}^2$