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Mean curvature flow of graphs with asymptotic Dirichlet conditions in Cartan-Hadamard manifolds

Claudia Fernandes, Jorge de Lira, Matheus Soares

Abstract

A priori estimates for the mean curvature evolution of Killing graphs in Cartan-Hadamard manifolds with asymptotic Dirichlet conditions are established. As an application, the existence of the corresponding parabolic flow is proved, ensuring regularity of the obtained solutions through the construction of suitable barriers at points of the asymptotic boundary. Such a construction is made possible under an appropriate notion of convexity at infinity.

Mean curvature flow of graphs with asymptotic Dirichlet conditions in Cartan-Hadamard manifolds

Abstract

A priori estimates for the mean curvature evolution of Killing graphs in Cartan-Hadamard manifolds with asymptotic Dirichlet conditions are established. As an application, the existence of the corresponding parabolic flow is proved, ensuring regularity of the obtained solutions through the construction of suitable barriers at points of the asymptotic boundary. Such a construction is made possible under an appropriate notion of convexity at infinity.
Paper Structure (13 sections, 18 theorems, 299 equations)

This paper contains 13 sections, 18 theorems, 299 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric setting and main result
  4. Analytical inequalities
  5. Topological setting
  6. A priori estimates
  7. Height estimate
  8. Boundary gradient estimate
  9. Interior gradient estimate
  10. Curvature estimates
  11. Existence of the Flow
  12. Existence of the flow in compact case
  13. Existence of the flow in non-compact case

Key Result

Theorem 1

Let $P$ be a Cartan-Hadamard manifold and $\varrho\in C^\infty(P)$. Suppose that $P\times_\varrho \mathbb{R}$ satisfies the geometric conditions ricci-cond to cylinder-0. Given an entire smooth Killing graph $\Sigma_0$ with asymptotic boundary $\Gamma$, there exists a mean curvature flow of entire K

Theorems & Definitions (38)

  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 4
  • proof
  • ...and 28 more