On the existence and classification of $k$-Yamabe gradient solitons

TL;DR

This paper classifies rotationally symmetric conformally flat $k$-Yamabe gradient solitons, establishing existence and asymptotic behavior for certain dimensions, and proving non-existence in others, using dynamical systems analysis.

Contribution

It provides a complete classification of $k$-Yamabe gradient solitons under symmetry assumptions, including existence, asymptotics, and non-existence results.

Findings

Existence of complete expanding, steady, and shrinking solitons for $n \\geq 2k$.

Asymptotic descriptions of solitons at infinity.

Non-existence of steady and expanding solitons for $n < 2k$.

Abstract

In this paper we classify rotationally symmetric conformally flat admissible solitons to the $k$-Yamabe flow, a fully non-linear version of the Yamabe flow. For $n\geq 2k$ we prove existence of complete expanding, steady and shrinking solitons and describe their asymptotic behavior at infinity. For $n<2k$ we prove that steady and expanding solitons are not admissible. The proof is based on the careful analysis of an associated dynamical system.