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

TL;DR

This paper proves nonexistence results for complete translating solitons of the r-mean curvature flow, showing they cannot be confined to certain geometric regions under specific growth conditions.

Contribution

It establishes new half-space theorems for translating solitons of the r-mean curvature flow, extending classical results to broader geometric and curvature conditions.

Findings

Translating solitons cannot be contained in the complement of a rotational cone.

Properly immersed solitons cannot be confined to certain half-spaces.

Complete solitons with growth conditions cannot lie within intersections of transversal half-spaces.

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.