arXiv subjects

Romain Gicquaud

Publications and source records attributed to Romain Gicquaud.

23 records · Page 2Linked to original sources

arXiv2011

A large class of non constant mean curvature solutions of the Einstein constraint equations on an asymptotically hyperbolic manifold

We construct solutions of the constraint equation with non constant mean curvature on an asymptotically hyperbolic manifold by the conformal method. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then in letting the exponent tend to its true value. We prove that the solutions of the sub-critical equations remain bounded which yields solutions of the constraint equation unless a certain limit equation admits a non-trivial solution. Finally, we give conditions which ensure that the limit equation admits no non-trivial solution.

gr-qc↗

arXiv2011

A limit equation associated to the solvability of the vacuum Einstein constraint equations using the conformal method

Let $(M,g)$ be a compact Riemannian manifold on which a trace-free and divergence-free $σ\in W^{1,p}$ and a positive function $τ\in W^{1,p}$, $p > n$, are fixed. In this paper, we study the vacuum Einstein constraint equations using the well known conformal method with data $σ$ and $τ$. We show that if no solution exists then there is a non-trivial solution of another non-linear limit equation on $1$-forms. This last equation can be shown to be without solutions no solution in many situations. As a corollary, we get existence of solutions of the vacuum Einstein constraint equation under explicit assumptions which in particular hold on a dense set of metrics $g$ for the $C^0$-topology.

gr-qc↗

arXiv2009

Linearization stability of the Einstein constraint equations on an asymptotically hyperbolic manifold

We study the linearization stability of the Einstein constraint equations on an asymptotically hyperbolic manifold. In particular we prove that these equations are linearization stable in the neighborhood of vacuum solutions for a non-positive cosmological constant and of Friedman--Lemaître--Robertson--Walker spaces in a certain range of decays. We also prove that this result is no longer true for faster decays. The construction of the counterexamples is based on a new construction of TT-tensors on the Euclidean space and on positive energy theorems.

gr-qc↗

arXiv2009

De l'équation de prescription de courbure scalaire aux équations de contrainte en relativité générale sur une variété asymptotiquement hyperbolique

Two problems concerning asymptotically hyperbolic manifolds with an inner boundary are studied. First, we study scalar curvature presciption with either Dirichlet or mean curvature prescription interior boundary condition. Then we apply those results to the Lichnerowicz equation with (future or past) apparent horizon interior boundary condition. In the last part we show how to construct TT-tensors. Thus we obtain Cauchy data with constant mean curvature for Einstein vacuum equations.

math.DG↗

arXiv2008

Conformal compactification of asymptotically locally hyperbolic metrics

In this paper we study the extent to which conformally compact asymptotically hyperbolic metrics may be characterized intrinsically. Building on the work of the first author, we prove that decay of sectional curvature to -1 and decay of covariant derivatives of curvature outside an appropriate compact set yield Hölder regularity for a conformal compactification of the metric. In the Einstein case, we prove that the estimate on the sectional curvature implies the control of all covariant derivatives of the Weyl tensor, permitting us to strengthen our result.

math.DG↗