Optimal asymptotic volume ratio for noncompact 3-manifolds with asymptotically nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound

TL;DR

This paper establishes an upper bound on the asymptotic volume ratio for certain noncompact 3-manifolds with specific curvature conditions, revealing linear volume growth and optimal volume ratio limits.

Contribution

It provides the first sharp upper bound on the asymptotic volume ratio for 3-manifolds with asymptotically nonnegative Ricci curvature and positive scalar curvature.

Findings

Manifolds have at most linear volume growth.

The asymptotic volume ratio is bounded above by 4kπ under certain decay conditions.

Results apply to manifolds with nonnegative Ricci curvature and positive scalar curvature.

Abstract

In this paper, we study 3-dimensional complete non-compact Riemannian manifolds with asymptotically nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound. Our main result is that, if this manifold has $k$ ends and finite first Betti number, then it has at most linear volume growth, and furthermore, if the negative part of Ricci curvature decays sufficiently fast at infinity, then we have an optimal asymptotic volume ratio $\limsup_{r\rightarrow\infty}\frac{\mathrm{Vol}(B(p, r))}{r}\leq4k\pi$. In particular, our results apply to 3-dimensional complete non-compact Riemannian manifolds with nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound.