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

In this paper, we prove the optimal volume growth for complete Riemannian manifolds (Mⁿ, g) with nonnegative Ricci curvature everywhere and bi-Ricci curvature bounded from below by n-2 outside a compact set when the dimension is less than eight. This answers a question AX24, Question 1 proposed by Antonelli-Xu in dimensions six and seven. As a by-product, we also prove an analogy of Gromov's volume bound conjecture Gro86, Open Question 2. A. (b) under the condition of positive bi-Ricci curvature.