Triviality results for quasi $k$-Yamabe solitons

TL;DR

This paper proves that compact quasi k-Yamabe gradient solitons have constant sigma_k-curvature and establishes conditions for gradient and rigidity results for both compact and noncompact cases.

Contribution

It demonstrates that all compact quasi k-Yamabe gradient solitons must have constant sigma_k-curvature and provides new conditions for gradient and rigidity in these solitons.

Findings

Compact quasi k-Yamabe gradient solitons have constant sigma_k-curvature.

Conditions are established for a compact quasi k-Yamabe soliton to be gradient.

Rigidity results for noncompact solitons under decay conditions on the soliton field.

Abstract

In this paper, we show that any compact quasi $k$-Yamabe gradient solitons must have constant $\sigma_{k}$-curvature. Moreover, we provide a certain condition for a compact quasi $k$-Yamabe soliton to be gradient, and for noncompact solitons, we present a geometric rigidity under a decaiment condition on the norm of the soliton field.