Vol. 6, No. 5, 2013

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 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
Milestones
Received: 25 January 2012
Revised: 10 June 2012
Accepted: 27 September 2012
Published: 3 November 2013
Authors
Vincent Guedj
Institut Universitaire de France and Institut Mathematiques de Toulouse
Universite Paul Sabatier
118 route de Narbonne
F-31062 Toulouse Cedex 9
France
Boris Kolev
Laboratoire d’Analyse, Topologie, Probabilités
CNRS & Aix-Marseille University
CMI
39 rue F. Joliot-Curie
13453 Marseille Cedex 13
France
Nader Yeganefar
Laboratoire d’Analyse, Topologie, Probabilités
Aix-Marseille University
CMI
39 rue F. Joliot-Curie
13453 Marseille Cedex 13
France