Vol. 6, No. 5, 2013

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

Vincent Guedj, Boris Kolev and Nader Yeganefar

Vol. 6 (2013), No. 5, 1001–1012
Abstract

In this article, we prove a Lichnerowicz estimate for a compact convex domain of a Kähler 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.

Keywords
Lichnerowicz estimate, first eigenvalue, convex domains in Kähler manifolds
Mathematical Subject Classification 2010
Primary: 35P15, 58C40