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

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.
Comment: 21 pages. Some improvements on the main theorem: In (2) of Theorem 1.6, the assumption on the positive function f is weaken to $\int_{0}^{\infty}rf(r)dr<\infty$. Other minor changes in the exposition