Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Beomjun Choi, Kyeongsu Choi, Soojung Kim

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 $\det D^2 u=(1+|Du|^2)^{2-\frac{1}{2α}}$ on $\mathbb{R}^2$ 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.

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 on in the range . The result also reveals that the moduli spaces of solutions are homeomorphic to either Euclidean spaces or cylinders.

Paper Structure

This paper contains 6 sections, 10 theorems, 56 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline
  4. Preliminary and Notation
  5. Change of coordinates and entropy
  6. ODE formulation

Key Result

Theorem A

Let $\Sigma\subset \mathbb{R}^3$ be a translator with $\alpha \in (0,1/4)$ in a geometric Alexandrov sense. Then, its profile $u: \mathbb{R}^2\to\mathbb{R}$ is an entire smooth strictly convex function such that holds for some uniform constant $C>1$ depending only on $\alpha$.

Figures (1)

  • Figure 1: possible blow-downs in different ranges of $\alpha$

Theorems & Definitions (21)

  • Theorem A: Growth rate CCK_regularity
  • Theorem 1.1: Uniqueness of blow-downs
  • Remark 1.2
  • Theorem B: Existence CCK_existence
  • Remark 1.3
  • Theorem 1.4: Classification
  • Theorem 1.5: Topology of moduli space
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 11 more