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A Variational Analysis of Einstein–Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds

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Abstract

We establish new existence and non-existence results for positive solutions of the Einstein–scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein–scalar field system in general relativity. Our analysis introduces variational techniques, in the form of the mountain pass lemma, to the analysis of the Hamiltonian constraint equation, which has been previously studied by other methods.

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Authors and Affiliations

  1. Département de Mathématiques, Site de Saint-Martin, Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex, France

    Emmanuel Hebey

  2. Département de Mathématiques, Université Paris XII, 61 Avenue du Général de Gaulle, 94010, Créteil Cedex, France

    Frank Pacard

  3. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195-4350, USA

    Daniel Pollack

Authors
  1. Emmanuel Hebey
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  2. Frank Pacard
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  3. Daniel Pollack
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Correspondence to Daniel Pollack.

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Communicated by G.W. Gibbons

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Hebey, E., Pacard, F. & Pollack, D. A Variational Analysis of Einstein–Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds. Commun. Math. Phys. 278, 117–132 (2008). https://doi.org/10.1007/s00220-007-0377-1

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  • DOI: https://doi.org/10.1007/s00220-007-0377-1

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