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 , let be a connected compact smooth ‐manifold with weakly convex boundary . If there exists a positive function that satisfies: where denotes the smallest eigenvalue of the Ricci tensor, is the unit co‐normal vector field of in , then the diameter of satisfies . If, in addition, attains its minimum on the boundary , we obtain a sharp upper bound for the volume of : , with equality holding if and only if is isometric to the unit round hemisphere .