Curvature-free effects from volume growth and ends-counting and their applications

Abstract

In this paper, we investigate two curvature-free effects from volume growth and ends-counting, respectively. Motivated by generalizing classical results from Ricci curvature to other common curvatures, we establish two main theorems. First, any complete non-compact manifold with lower sublinear volume growth admits a smooth bounded mean-concave exhaustion. Second, any complete manifold with infinitely many ends contains escaping geodesic lines outside every compact subset. As applications, we provide new proofs of the Calabi--Yau minimal volume growth theorem and the Cai--Li--Tam finite-ends theorem for nonnegative Ricci curvature, without relying on the Bishop--Gromov volume comparison theorem or analytic tools specific to Ricci curvature. We further extend these results to Riemannian manifolds with nonnegative scalar curvature and K\"ahler manifolds with positive holomorphic sectional curvature.