A Lichnerowicz estimate for the first eigenvalue of convex domains in K\"ahler manifolds

In this article, we prove a Lichnerowicz estimate for a compact convex domain of a K"ahler manifold whose Ricci curvature satisfies $\text{Ric}\ge k$ for some constant $k>0$. When equality is achieved, the boundary of the domain is totally geodesic and there exists a nontrivial holomorphic vector field. We show that a ball of sufficiently large radius in complex projective space provides an example of a strongly pseudoconvex domain which is not convex, and for which the Lichnerowicz estimate fails.