f|^2V^{\tau+\g-\a} d\O}{\int_M f^2V^\tau d\O}.
             \label{eigenvalue Neumann}
         \end{aligned}$$ The assertion follows immediately from Theorem \[Poincare\] above by using the curvature condition $Ric^D\ge (n-1)V^{\a-\g}\bar g$.

[*Proof of Corollary \[coro\].*]{} Let $\a=0, \g=1$. It is sufficient to observe the following fact: the Dirichlet boundary problem for PDEs $$\left\{
    \begin{array}[]{rlll}
        \di \bd^D\phi &=&-\lambda_1^{Dir}
        \phi&\mathrm{ in }\ \O,\\
        \phi&=& 0 &\mathrm{ on }\ \S,\\
    \end{array}
    \right.$$ and $$\left\{
    \begin{array}[]{rlll}
        \di V\bar \De f-\bar \De Vf&=&-\lambda_1^{Dir}
       f&\mathrm{ in }\ \O,\\
        f&=& 0 &\mathrm{ on }\ \S,\\
    \end{array}
    \right.$$ are equivalent under the correspondence $\phi=\frac{f}{V}$. The same phenomenon holds for Neumann boundary problems. Then Corollary \[coro\] follows from Theorem \[thm ineq\] (iii).

Appendix: Proof of Proposition \[thm Ricci\] {#appenA}
============================================

In this appendix we prove Proposition \[thm Ricci\]. We only need to prove it at any fixed point $P$ under normal coordinates system $\{\P_i\}_{i=1}^n$ such that $\bar g_{ij}(P)=\delta_{ij}$, $\P_k\bar g_{ij}(P)=0$, and $\nb_{\P_i}\P_j(P)=0$. Throughout the proof we will use these properties implicitly. We use $u_i:=du(\P_i)$ and let $u_{ij}$ be the Hessian of $u$ with respect to $\nb$.

By definition, we first observe that $$\begin{aligned}
        D_i\P_j=\nb_i\P_j+\a u_i\P_j+\a u_j\P_i+\g \d_{ij}\nb u,
        \label{thm Ricci equ1}
    \end{aligned}$$ and $$\begin{aligned}
        D_k(D_i\P_j)=&\nb_k(D_i\P_j)+\a u_k(D_i\P_j)+\a du(D_i\P_j)\P_k+\g \bar g(\P_k,D_i\P_j)\nb u.
        \label{thm Ricci equ2}
    \end{aligned}$$ Next, we compute 