A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds

Details A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds

2011-11-22

arxiv.org/abs/1111.5323v1

Mathematics

Differential Geometry

9 pages

Scientific paper

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.

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