Complete Yamabe solitons with finite total scalar curvature

TL;DR

This paper proves that certain complete Yamabe solitons with finite total scalar curvature and specific curvature conditions are Ricci flat or have zero scalar curvature, advancing understanding of their geometric structure.

Contribution

It establishes new rigidity results for complete Yamabe solitons under finite total scalar curvature and curvature constraints, including Ricci flatness and zero scalar curvature.

Findings

Steady or shrinking Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat.

Under pinching conditions, such solitons with non-positive scalar curvature have zero scalar curvature.

Results contribute to the classification of Yamabe solitons with finite total scalar curvature.

Abstract

In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive scalar curvature have zero scalar curvature.