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An Obata type result for the first eigenvalue of the sub-Laplacian on a CR manifold with a divergence-free torsion

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Abstract

We prove a CR version of the Obata’s result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian manifold which satisfies a Lichnerowicz type condition and has a divergence free pseudohermitian torsion. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standard Sasakian unit sphere. We also give a version of this theorem using the existence of a function with traceless horizontal Hessian on a complete, with respect to Webster’s metric, pseudohermitian manifold.

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Authors and Affiliations

  1. Faculty of Mathematics and Informatics, University of Sofia, blvd. James Bourchier 5, 1164, Sofia, Bulgaria

    Stefan Ivanov

  2. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico, 87131-0001, USA

    Dimiter Vassilev

Authors
  1. Stefan Ivanov
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  2. Dimiter Vassilev
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Correspondence to Stefan Ivanov.

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Ivanov, S., Vassilev, D. An Obata type result for the first eigenvalue of the sub-Laplacian on a CR manifold with a divergence-free torsion. J. Geom. 103, 475–504 (2012). https://doi.org/10.1007/s00022-013-0145-7

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  • DOI: https://doi.org/10.1007/s00022-013-0145-7

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