first (nonzero) eigenvalue of the affine Laplacian $\bd^D$, i.e., there exists some non-trivial $\phi$ such that $\bd^D \phi=-\l_1\phi$ with Dirichlet boundary condition $\phi=0$ or Neumann boundary condition $\phi_\nu=0$.

-   Theorem \[thm ineq\] reduces to Heintze-Karcher, Minkowski, and Lichnerowicz inequalities for $\bar g$ in the case $\a=\g=0$ or $e^{2\a u} \bar g$ in the case $\a=-\g$. See Section 2.1 for an overview.

-   In the case $\a=0, \g=1$, the Heintze-Karcher type inequalities were first proved by Brendle [@Br], then by Wang-Wang [@WW] for non-homologous static manifolds without warped product structure by using the same method and by the authors for general cases using Reilly type formulas in [@LX]. The Minkowski type inequalities have been proved in our previous work [@LX].

-   Theorem \[thm ineq\] gives new geometric inequalities under the condition of nonnegative $1$-Bakry-Émery Ricci, which is the case $\a=\frac{1}{n-1}, \g=0$. To illustrate the idea, we only list the example of the Heintze-Karcher type inequality and the others hold true similarly.

    [**Corollary.**]{} [*Let $(M^n, \bar g, e^ud\O)$ be a smooth weighted Riemannian manifold and $\O$ be a bounded domain in $M$. If the $1$-Bakry-Émery Ricci is nonnegative, namely, $$\begin{aligned}
        &Ric-\bar\nabla^2 u+\frac{1}{n-1}du\otimes du\nonumber\ge 0,
            \label{Ric nonnegative tilde 1} 
        \end{aligned}$$ and the weighted 