Optimal volume bound and volume growth for Ricci-nonnegative manifolds with positive bi-Ricci curvature

Abstract In this paper, we prove the optimal volume growth for complete Riemannian manifolds (M n, g) (M^n, g) with nonnegative Ricci curvature everywhere and bi-Ricci curvature bounded from below by n − 2 n-2 outside a compact set when the dimension is less than eight. This answers a question proposed by Antonelli–Xu in dimensions six and seven. As a by-product, we also prove an analogy of Gromov’s volume bound conjecture under the condition of positive bi-Ricci curvature.