A spectral volume comparison for manifolds with weakly convex boundary

We establish the Bonnet-Myers theorem and the Bishop-Gromov volume comparison theorem in the spectral sense for manifolds with weakly convex boundary. For n≥ 3, let (Mⁿ,g) be a simply connected compact smooth n-manifold with weakly convex boundary ∂ M. If there exists a positive function w∈ C∞(M) that satisfies: {equation*} {cases} -n-1/n-2Δw+Λ_{} w≥ (n-1)w, in M, ∂ w/∂ η=0, on ∂ M, {cases} {equation*} where Λ_ denotes the smallest eigenvalue of the Ricci tensor, $η$ is the unit co-normal vector field of ∂ M in M, then the diameter of M satisfies (M)≤ (max w/min w)n-3/n-1π. If, in addition, w attains its minimum on the boundary ∂ M, we obtain a sharp upper bound for the volume of M: (M)≤ (ⁿ₊), with equality holding if and only if Mⁿ is isometric to the unit round hemisphere ⁿ₊.