Remarks on topology of stable translating solitons

TL;DR

This paper proves that complete f-stable translating solitons cannot contain certain cycles and that two-dimensional stable solitons must have genus zero, revealing topological restrictions on their structure.

Contribution

It establishes topological constraints on complete f-stable translating solitons, showing they cannot contain non-disconnecting cycles and must have genus zero in two dimensions.

Findings

No codimension one cycle without disconnecting the manifold

Two-dimensional complete f-stable translating solitons have genus zero

Provides topological restrictions on stable translating solitons

Abstract

We show that any complete $f$-stable translating soliton $M$ admits no codimension one cycle which does not disconnect $M$. As a corollary, it follows that any two dimensional complete $f$-stable translating soliton has genus zero.