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Maximum Principles and Consequences for $γ$-translators in $\mathbb{R}^{n+1}$

José Torres Santaella

Abstract

In this paper we obtain several properties of translating solitons for a general class of extrinsic geometric curvature flows given by a homogeneous, symmetric, smooth non-negative function $γ$ defined in an open cone $Γ\subset\mathbb{R}^n$. The main results are tangential principles, nonexistence theorems for closed and entire solutions, and a uniqueness result that says that any strictly convex $γ$-translator defined on a ball with a single end $\mathcal{C}^2$-asymptotic to a cylinder is the ''bowl''-type solution found in the translator paper of S. Rengaswami.

Maximum Principles and Consequences for $γ$-translators in $\mathbb{R}^{n+1}$

Abstract

In this paper we obtain several properties of translating solitons for a general class of extrinsic geometric curvature flows given by a homogeneous, symmetric, smooth non-negative function defined in an open cone . The main results are tangential principles, nonexistence theorems for closed and entire solutions, and a uniqueness result that says that any strictly convex -translator defined on a ball with a single end -asymptotic to a cylinder is the ''bowl''-type solution found in the translator paper of S. Rengaswami.
Paper Structure (5 sections, 6 theorems, 46 equations, 4 figures)

This paper contains 5 sections, 6 theorems, 46 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof of Thoerem \ref{['T1']}
  3. Proof of Theorem \ref{['T2']}
  4. Proof of Theorem \ref{['T3']}
  5. Appendix: "Bowl"-type solutions asymptitics to round cylinders

Key Result

Theorem 1.2

Let $\Gamma=\left\{\lambda\in{\mathbb R}^n:\gamma(\lambda)>0\right\}$ and assume that $\gamma:\Gamma\to(0,\infty)$ satisfies properties a)-c). Then, there is no closed immersed $\gamma$-translator in ${\mathbb R}^{n+1}$ such that its principal curvatures belong to the cone $\Gamma$.

Figures (4)

  • Figure 1: Picture of the proof of Theorem \ref{['T1']}.
  • Figure 2: The moving plane method acting in a rotationally symmetric strictly convex surface $M$. Image courtesy of Francisco Martín.
  • Figure 3: Sequence $\nu(p_l)\to e_1$ as $l\to\infty$ where $p_l\in\Sigma_+(t_l)$. Image courtesy of Francisco Martín.
  • Figure 4: View from $x_{n+1}=h\gg1$ with $\Sigma_+^*(t_1-\varepsilon_1)$ is drawn in dashed lines. Image courtesy of Francisco Martín.

Theorems & Definitions (32)

  • Example 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Example 1.8
  • proof : Proof of Theorem \ref{['T1']}
  • proof : Proof of Thoerem \ref{['T2']}
  • ...and 22 more