On complete gradient Schouten solitons

TL;DR

This paper establishes key inequalities and curvature bounds for complete Schouten solitons, demonstrating linear growth of the potential function in shrinking cases and providing volume growth estimates without extra assumptions.

Contribution

It introduces an optimal inequality relating the potential function and gradient norm, and characterizes scalar curvature and potential function growth in Schouten solitons.

Findings

Proves an optimal inequality between potential function and gradient norm.

Shows scalar curvature of these metrics has a bounded, sign-defined nature.

Demonstrates linear growth of the potential function in shrinking solitons and estimates volume growth.

Abstract

In this paper, we prove an optimal inequality between the potential function of a complete Schouten soliton and the norm of its gradient. We also prove that these metrics have bounded scalar curvature of defined sign. As an application, we prove that the potential function of a Shrinking Schouten soliton grows linearly and provide optimal estimates for the growth of the volume of geodesic balls, without any further assumption.