# Prescription de courbures sur l'espace hyperbolique

Abstract : The Ph.D is composed of two parts.

First part :
theme of the conformal scalar curvature on the hyperbolic
space. We bring here a study of the fine asymptotic behavior in any
dimension.
We always deal with general semi-linear equations, before applying our
results to the particular case of the geometric equation.

Second part :
theme of the Ricci curvature on the hyperbolic
space. We obtain the following result.
On the unit ball of $\Reel^n$, one considers the standard hyperbolic
metric $H_0$ whose Ricci curvature equals $R_0$ and Riemann-Christoffel
curvature is ${\cal R}_0$. We prove that in dimension $n\geq10$,
for any symetric
tensor $R$ near $R_0$, there exists a unique metric $H$ near $H_0$
whose Ricci curvature is $R$.
We deduce in the $C^\infty$ case that the image of the Riemann-Christoffel
operator is a submanifold in a neighborhood of ${\cal R}_0$.
We treat also in this part the contravariant Ricci curvature in all
dimensions, the Dirichlet problem at the infinity in dimension 2,
and some obstructions.
Keywords :
Document type :
Theses
Domain :

Cited literature [27 references]

https://tel.archives-ouvertes.fr/tel-00011944
Contributor : Erwann Delay <>
Submitted on : Tuesday, March 14, 2006 - 5:11:58 PM
Last modification on : Tuesday, January 14, 2020 - 10:38:02 AM
Long-term archiving on: : Saturday, April 3, 2010 - 8:57:53 PM

### Identifiers

• HAL Id : tel-00011944, version 1

### Citation

Erwann Delay. Prescription de courbures sur l'espace hyperbolique. Mathématiques [math]. Université Nice Sophia Antipolis, 1998. Français. ⟨tel-00011944⟩

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