A sharp lower bound for the scalar curvature of certain steady gradient Ricci solitons

TL;DR

This paper establishes a lower bound for the scalar curvature of specific steady gradient Ricci solitons, linking geometric properties with the hyperbolic secant function based on the distance from a point.

Contribution

It provides a new sharp lower bound for scalar curvature in steady gradient Ricci solitons under a specific ratio condition, advancing understanding of their geometric structure.

Findings

Scalar curvature is bounded below by the hyperbolic secant of half the distance from a fixed point.

The bound applies to solitons satisfying a ratio condition between Ricci tensor norm and scalar curvature.

The result enhances the geometric analysis of steady gradient Ricci solitons.

Abstract

We show that the scalar curvature of a steady gradient Ricci soliton satisfying that the ratio between the square norm of the Ricci tensor and the square of the scalar curvature is bounded by one half, is boundend from below by the hyperbolic secant of one half the distance function from a fixed point.