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Romain Gicquaud

Publications and source records attributed to Romain Gicquaud.

At least 19 recordsLinked to original sources
arXiv2026

Mass-like invariants for asymptotically hyperbolic metrics. Part I: Classification

In this article, we classify the set of asymptotic mass-like invariants for asymptotically hyperbolic metrics. It turns out that the standard mass is just one example among the two families of invariants we find. These invariants correspond to finite-dimensional representations of the group of isometries of hyperbolic space. We describe these invariants in terms of wave-harmonic polynomials and polynomial solutions of the linearized Einstein equations in Minkowski space.

math.DG↗
arXiv2026

A new approach towards the construction of initial data in general relativity with positive Yamabe invariant and arbitrary mean curvature

This paper revisits the classical construction of initial data using the conformal method, as originally proposed by Holst, Nagy, and Tsogtgerel and later refined by Maxwell. We demonstrate that the existence of the solution can be proven using the Banach fixed point theorem, whereas the original proof relied on the Schauder fixed point theorem. This new approach has two main advantages: it guarantees the uniqueness of the solution to the equations of the conformal method as soon as one imposes a bound on the physical volume of it and it provides an explicit construction of the solution.

gr-qc↗
arXiv2025

A note on Poincaré-Sobolev type inequalities on compact manifolds

We prove a Poincaré-Sobolev type inequality on compact Riemannian manifolds where the deviation of a function from a biased average, defined using a density, is controlled by the unweighted Lebesgue norm of its gradient. Unlike classical weighted Poincaré inequalities, the density does not enter the measure or the Sobolev norms, but only the reference average. We show that the associated Poincaré constant depends quantitatively on the Lebesgue norm of the density. This framework naturally arises in the analysis of coupled elliptic systems and seems not to have been addressed in the existing literature.

math.AP↗
arXiv2025

A definition of the mass aspect function for weakly regular asymptotically hyperbolic manifolds

In contrast to the well-known and unambiguous notion of ADM mass for asymptotically Euclidean manifolds, the notion of mass for asymptotically hyperbolic manifolds admits several interpretations. Historically, there are two approaches to defining the mass in the asymptotically hyperbolic setting: the mass aspect function of Wang defined on the conformal boundary at infinity, and the mass functional of Chruściel and Herzlich which may be thought of as the closest asymptotically hyperbolic analogue of the ADM mass. In this paper we unify these two approaches by introducing an ADM-style definition of the mass aspect function that applies to a broad range of asymptotics and in very low regularity. Additionally, we show that the mass aspect function can be computed using the Ricci tensor. Finally, we demonstrate that this function exhibits favorable covariance properties under changes of charts at infinity, which includes a proof of the asymptotic rigidity of hyperbolic space in the context of weakly regular metrics.

math.DG↗
arXiv2019

Prescribed non positive scalar curvature on asymptotically hyperbolic manifolds with application to the Lichnerowicz equation

We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following earlier results, we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.

math.DG↗
arXiv2016

A new point of view on the solutions to the Einstein constraint equations with arbitrary mean curvature and small TT-tensor

In this short note, we give a construction of solutions to the Einstein constraint equations using the well known conformal method. Our method gives a result similar to the one in [15, 16, 24], namely existence when the so called TT-tensor $σ$ is small and the Yamabe invariant of the manifold is positive. The method we describe is however much simpler than the original method and allows easy extensions to several other problems. Some non-existence results are also considered.

gr-qc↗
arXiv2016

Bifurcating solutions of the Lichnerowicz equation

We give an exhaustive description of bifurcations and of the number of solutions of the vacuum Lichnerowicz equation with positive cosmological constant on $S^1\times S^2$ with $U(1)\times SO(3)$-invariant seed data. The resulting CMC slicings of Schwarzschild-de Sitter and Nariai are described.

gr-qc↗
arXiv2013

Asymptotically hyperbolic manifolds with small mass

For asymptotically hyperbolic manifolds of dimension $n$ with scalar curvature at least equal to $-n(n-1)$ the conjectured positive mass theorem states that the mass is non-negative, and vanishes only if the manifold is isometric to hyperbolic space. In this paper we study asymptotically hyperbolic manifolds which are also conformally hyperbolic outside a ball of fixed radius, and for which the positive mass theorem holds. For such manifolds we show that the conformal factor tends to one as the mass tends to zero.

math.DG↗
arXiv2012

On the asymptotic behavior of Einstein manifolds with an integral bound on the Weyl curvature

In this paper we consider the geometric behavior near infinity of some Einstein manifolds $(X^n, g)$ with Weyl curvature belonging to a certain $L^p$ space. Namely, we show that if $(X^n, g)$, $n \geq 7$, admits an essential set and has its Weyl curvature in $L^p$ for some $1<p<\frac{n-1}{2}$, then $(X^n, g)$ must be asymptotically locally hyperbolic. One interesting application of this theorem is to show a rigidity result for the hyperbolic space under an integral condition for the curvature.

math.DG↗
arXiv2012

Penrose type inequalities for asymptotically hyperbolic graphs

In this paper we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space $\bH^n$. The graphs are considered as subsets of $\bH^{n+1}$ and carry the induced metric. For such manifolds the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence we estimate the mass by an integral over an inner boundary. In case the inner boundary satisfies a convexity condition this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam's article concerning the asymptotically Euclidean case.

math.DG↗
arXiv2011

Conformal compactification of asymptotically locally hyperbolic metrics II: Weakly ALH metrics

In this paper we pursue the work initiated in \cite{Bahuaud, BahuaudGicquaud}: study the extent to which conformally compact asymptotically hyperbolic metrics can be characterized intrinsically. We show how the decay rate of the sectional curvature to -1 controls the Hölder regularity of the compactified metric. To this end, we construct harmonic coordinates that satisfy some Neumann-type condition at infinity. Combined with a new integration argument, this permits us to recover to a large extent our previous result without any decay assumption on the covariant derivatives of the Riemann tensor. We believe that our result is optimal.

math.DG↗
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↗