In this article, we prove a Lichnerowicz estimate for a compact convex domain of a Kähler manifold whose Ricci curvature satisfies Ric for some constant . 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.
"A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds." Anal. PDE 6 (5) 1001 - 1012, 2013. https://doi.org/10.2140/apde.2013.6.1001