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Half-space theorems for translating solitons of the r-mean curvature flow

Hilário Alencar, G. Pacelli Bessa, Gregório Silva Neto

TL;DR

The paper establishes half-space-type nonexistence results for translating solitons of the $r$-mean curvature flow in $\mathbb{R}^{n+1}$. By employing the $(r-1)$-th Newton transformation $P_{r-1}$ and the operator $L_{r-1}$ together with Omori–Yau maximum principles, it derives contradictions under growth constraints on $\sigma_{r-1}$ and on the second fundamental form, ruling out translators in the complement of a cone, in a closed half-space opposite the translation, and in the intersection of two transversal vertical half-spaces. The results generalize classical half-space theorems to the $r$-mean curvature flow and clarify the geometric constraints on translators, with the special case $r=1$ recovering and refining known corollaries. The work also addresses gaps in prior Omori–Yau-based proofs and situates these nonexistence results within a broader framework of curvature-flow solitons.

Abstract

In this paper, we establish nonexistence results for complete translating solitons of the r-mean curvature flow under suitable growth conditions on the (r-1)-mean curvature and on the norm of the second fundamental form. We first show that such solitons cannot be entirely contained in the complement of a right rotational cone whose axis of symmetry is aligned with the translation direction. We then relax the growth condition on the (r-1)-mean curvature and prove that properly immersed translating solitons cannot be confined to certain half-spaces opposite to the translation direction. We conclude the paper by showing that complete, properly immersed translating solitons satisfying appropriate growth conditions on the (r-1)-mean curvature cannot lie completely within the intersection of two transversal vertical half-spaces.

Half-space theorems for translating solitons of the r-mean curvature flow

TL;DR

The paper establishes half-space-type nonexistence results for translating solitons of the -mean curvature flow in . By employing the -th Newton transformation and the operator together with Omori–Yau maximum principles, it derives contradictions under growth constraints on and on the second fundamental form, ruling out translators in the complement of a cone, in a closed half-space opposite the translation, and in the intersection of two transversal vertical half-spaces. The results generalize classical half-space theorems to the -mean curvature flow and clarify the geometric constraints on translators, with the special case recovering and refining known corollaries. The work also addresses gaps in prior Omori–Yau-based proofs and situates these nonexistence results within a broader framework of curvature-flow solitons.

Abstract

In this paper, we establish nonexistence results for complete translating solitons of the r-mean curvature flow under suitable growth conditions on the (r-1)-mean curvature and on the norm of the second fundamental form. We first show that such solitons cannot be entirely contained in the complement of a right rotational cone whose axis of symmetry is aligned with the translation direction. We then relax the growth condition on the (r-1)-mean curvature and prove that properly immersed translating solitons cannot be confined to certain half-spaces opposite to the translation direction. We conclude the paper by showing that complete, properly immersed translating solitons satisfying appropriate growth conditions on the (r-1)-mean curvature cannot lie completely within the intersection of two transversal vertical half-spaces.
Paper Structure (5 sections, 8 theorems, 127 equations, 2 figures)

This paper contains 5 sections, 8 theorems, 127 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Omori-Yau type maximum principles
  3. proof of Theorem \ref{['thm-Cone']}
  4. Proof of Theorem \ref{['thm-HS-1']}
  5. Proof of Theorem \ref{['thm-HS-2']}

Key Result

Theorem 1.1

There are no complete, $n$-dimensional, translating solitons of the $r$-mean curvature flow $\Sigma^n\subset\mathbb{R}^{n+1}$ with velocity $V$ with $P_{r-1}$ positive semidefinite, contained in the complement of the open cone $\mathcal{C}_{V,a}$, satisfying one of the following conditions:

Figures (2)

  • Figure 1: The Grim Reaper cylinder and the sets $\mathcal{H}_{V}$ and $\left( \mathcal{C}_{V,a}\right)^{c}.$
  • Figure 2: Regions in the intersection of vertical halfspaces

Theorems & Definitions (25)

  • Remark 1.1
  • Theorem 1.1
  • Corollary 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2
  • Remark 1.4
  • Corollary 1.2
  • Remark 1.5
  • Example 1.1
  • ...and 15 more