Notes on scalar curvature lower bounds of steady gradient Ricci solitons

TL;DR

The paper introduces new decay estimates for scalar curvatures in steady gradient Ricci solitons and bounds on manifold diameter under Bakry--Emery Ricci tensor constraints, using Gromov's $$-bubbles.

Contribution

It provides novel decay estimates and diameter bounds for specific Riemannian manifolds, employing Gromov's $$-bubbles technique.

Findings

New decay estimate for scalar curvature of steady gradient Ricci solitons.

Upper bound on the diameter of manifolds with bounded $$-Bakry--Emery Ricci tensor.

Abstract

We provide new type of decay estimate for scalar curvatures of steady gradient Ricci solitons. We also give certain upper bound for the diameter of a Riemannian manifold whose $\infty$-Bakry--Emery Ricci tensor is bounded by some positive constant from below. For the proofs, we use $\mu$-bubbles introduced by Gromov.