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 $n\geq 3$, let $(M^n,g)$ be a simply connected compact smooth $n$-manifold with weakly convex boundary $\partial M$. If there exists a positive function $w\in C^{\infty}(M)$ that satisfies: \begin{equation*} \begin{cases} -\frac{n-1}{n-2}\Delta w+\Lambda_{\Ric} w\geq (n-1)w, \enspace in \enspace M, \frac{\partial w}{\partial \eta}=0, \enspace\enspace\enspace\enspace \enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace \enspace\enspace \enspace\enspace on \enspace\partial M, \end{cases} \end{equation*} where $\Lambda_{\Ric}$ denotes the smallest eigenvalue of the Ricci tensor, $\eta$ is the unit co-normal vector field of $\partial M$ in $M$, then the diameter of $M$ satisfies $\diam(M)\leq (\frac{\max w}{\min w})^{\frac{n-3}{n-1}}\pi$.\par If, in addition, $w$ attains its minimum on the boundary $\partial M$, we obtain a sharp upper bound for the volume of $M$: $\Vol(M)\leq \Vol(\bS^n_{+})$, with equality holding if and only if $M^n$ is isometric to the unit round hemisphere $\bS^{n}_{+}$.