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-1n-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) (w w) ^n-3{n-1}π. 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+.