Skip to Main content Skip to Navigation
Theses

CONSTRUCTION OF SOLUTIONS TO THE EINSTEIN CONSTRAINT EQUATIONS IN GENERAL RELATIVITY AND COMMENTS ON THE POSITIVE MASS THEOREM

Abstract : The aim of this thesis is the study of two topical issues arising from general relativity: finding initial data for the Cauchy problem with respect to the Einstein equations and the positive mass theorem. For the first issue, in the context of the conformal method introduced by Lichnerowicz [Lichnerowicz, 1944], Y. Choquet-Bruhat–J. York [Choquet-Bruhat et York, 1980] and Y. Choquet-Bruhat–J. Isenberg–D. Pollack [Choquet-Bruhat et al., 2007a], we consider the conformal constraint equations on compact Riemannian manifolds of dimension n > 3. In this thesis, we simplify the proof of [Dahl et al., 2012, Theorem 1.1], extend and sharpen the far-from CMC result proven by Holst– Nagy–Tsogtgerel [Holst et al., 2009], Maxwell [Maxwell, 2009] and give an unifying viewpoint of these results. Besides discussing the solvability of the conformal constraint equations, we will also show nonexistence and nonuniqueness results for solutions to the conformal constraint equations under certain assumptions. For the second one, we are interested in studying the positive mass theorem on asymptotically hyperbolic manifolds. More precisely, we prove that positivity of the mass of an asymptotically hyperbolic manifold is kept under a finite sequence of surgeries of codimension at least 3. As a consequence, we extend one of main results of Humbert–Hermann [Humbert et Herman, 2014, Theorem 8.5] to asymptotically hyperbolic manifolds, that is the positivity of the mass holds on all asymptotically hyperbolic manifolds of dimension n > 5, provided it is so on a single simply connected non-spin asymptotically hyperbolic manifold of the same dimension. This thesis is mainly self-contained except for minimum background in Geometric Analysis.
Complete list of metadatas

Cited literature [56 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-01245248
Contributor : The-Cang Nguyen <>
Submitted on : Wednesday, December 16, 2015 - 11:29:59 PM
Last modification on : Thursday, March 5, 2020 - 5:32:56 PM
Document(s) archivé(s) le : Thursday, March 17, 2016 - 5:20:18 PM

Identifiers

  • HAL Id : tel-01245248, version 1

Collections

Citation

The-Cang Nguyen. CONSTRUCTION OF SOLUTIONS TO THE EINSTEIN CONSTRAINT EQUATIONS IN GENERAL RELATIVITY AND COMMENTS ON THE POSITIVE MASS THEOREM. Differential Geometry [math.DG]. Laboratoire de Mathématiques et Physique Théorique, Université de Tours, 2015. English. ⟨tel-01245248⟩

Share

Metrics

Record views

335

Files downloads

202