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

TL;DR

This paper establishes a Lichnerowicz-type lower bound for the first eigenvalue of convex domains in K"ahler manifolds with Ricci curvature bounded below, revealing geometric conditions for equality and providing counterexamples.

Contribution

It extends the Lichnerowicz estimate to convex domains in K"ahler manifolds and characterizes the boundary and vector fields when equality holds.

Findings

Lichnerowicz estimate proven for convex domains with Ricci curvature ≥ k

Boundary is totally geodesic when equality is achieved

Large balls in complex projective space do not satisfy the estimate

Abstract

In this article, we prove a Lichnerowicz estimate for a compact convex domain of a K\"ahler manifold whose Ricci curvature satisfies $\Ric \ge k$ 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.