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Long time regularity of the $p$-Gauss curvature flow with flat side

G. Huang, X. -J. Wang, Y. Zhou

Abstract

In this paper, we prove the long time regularity of the interface in the $p$-Gauss curvature flow with flat side in all dimensions for $p>\frac1n$. Here the interface is the boundary of the flat part in the flow. In dimension $2$, this problem was solved in \cite{DL2004} for $p=1$ and in \cite{KimLeeRhee2013} for $p\in(1/2,1)$. We utilize the duality method to transform the Gauss curvature flow to a singular parabolic Monge-Ampère equation, and prove the regularity of the interface by studying the asymptotic cone of the parabolic Monge-Ampère equation in the polar coordinates.

Long time regularity of the $p$-Gauss curvature flow with flat side

Abstract

In this paper, we prove the long time regularity of the interface in the -Gauss curvature flow with flat side in all dimensions for . Here the interface is the boundary of the flat part in the flow. In dimension , this problem was solved in \cite{DL2004} for and in \cite{KimLeeRhee2013} for . We utilize the duality method to transform the Gauss curvature flow to a singular parabolic Monge-Ampère equation, and prove the regularity of the interface by studying the asymptotic cone of the parabolic Monge-Ampère equation in the polar coordinates.
Paper Structure (8 sections, 35 theorems, 250 equations)

This paper contains 8 sections, 35 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Estimates for the speed of the interface
  3. Growth estimates at the singular point
  4. Bernstein theorem for a singular parabolic Monge-Ampère equation
  5. Estimates for the modulus of continuity of $\zeta_t$ and $D^2\zeta$
  6. Higher regularity for $\zeta$
  7. Regularity for linear parabolic equations
  8. Higher regularity for $\zeta$

Key Result

Theorem 1.1

Assume conditions (I1)-(I3). Then if $p>\frac{1}{n}$, the interface $\Gamma_t$ is smooth and uniformly convex $\forall\ t\in(0,T^*)$, where $T^*>0$ is the time when the flat region disappears.

Theorems & Definitions (58)

  • Theorem 1.1
  • Proposition 2.1: Theorem 9.1,DH1999
  • Remark 2.1
  • Proposition 2.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.1
  • proof
  • ...and 48 more