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

TL;DR

This paper establishes the optimal volume growth bounds for certain Ricci-nonnegative manifolds with positive bi-Ricci curvature, answering open questions in dimensions six and seven and extending Gromov's volume conjecture.

Contribution

It proves the optimal volume growth for manifolds with nonnegative Ricci curvature and bi-Ricci curvature bounds in dimensions less than eight, addressing key open problems.

Findings

Confirmed optimal volume growth bounds in specified manifolds.

Extended Gromov's volume bound conjecture under positive bi-Ricci curvature.

Answered open questions in dimensions six and seven.

Abstract

In this paper, we prove the optimal volume growth for complete Riemannian manifolds $(M^n,g)$ 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.