arXiv subjects

Simon Raulot

Publications and source records attributed to Simon Raulot.

At least 19 recordsLinked to original sources

arXiv2026

Dirac eigenvalues and spacetime mean curvature estimates for DEC spin fill-ins

We establish spacetime counterparts of fill-in inequalities for spin initial data sets satisfying the dominant energy condition in dimensions three through seven, controlling the minimum and the integral of the norm of the spacetime mean curvature vector by intrinsic boundary data. The proof relies on a Reilly--MIT estimate for complete spin manifolds with compact boundary. As further consequences, under suitable nonnegativity assumptions, Gromov's hyperspherical-radius fill-in inequality and B\"ar's recent total mean-curvature estimate extend to complete, possibly noncompact spin manifolds with compact boundary.

math.DG↗

arXiv2026

Charged parallel spinors and applications to mass--charge inequalities

We investigate the equality case of the spin positive mass theorem with charge in the Riemannian setting. This leads naturally to the notion of charged parallel spinor, which plays a central role in the analysis of extremal charged manifolds. As an application, we characterize the equality case of the mass--charge inequality in terms of the extremal Reissner--Nordstr\"om geometry for asymptotically flat manifolds with connected boundary and for manifolds with a single asymptotically cylindrical end.

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arXiv2026

Positive energy-momentum theorems for asymptotically AdS spin initial data sets with charge

For complete spin initial data sets with an asymptotically anti--de Sitter end, we introduce a charged energy--momentum defined as a linear functional arising from the Einstein--Maxwell constraints. Under a dominant energy condition adapted to the presence of a negative cosmological constant, we establish positive energy--momentum theorems, showing in particular that this functional is non--negative on a natural real cone. We place particular emphasis on the case where the manifold carries a compact inner boundary. In the time--symmetric setting, this yields a mass--charge inequality for asymptotically hyperbolic manifolds with charge.

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arXiv2025

Positive Energy Theorems for Spin Initial Data with Charge

We establish positive energy theorems for complete spin initial data sets with charge in dimensions $n \geq 4$, under a dominant energy condition and assuming the existence of at least one asymptotically flat end. Our results, formulated in the purely electric case, extend the classical theorems of Gibbons--Hull~\cite{GibbonsHull}, Gibbons--Hawking--Horowitz--Perry~\cite{GibbonsHawkingHorowitzPerry}, and Bartnik--Chru\'sciel~\cite{BartnikChrusciel}.

math.DG↗

arXiv2025

On the ADM mass of critical area-normalized capacitors

In this note, we prove mass-capacity inequalities for asymptotically flat manifolds whose boundary capacity potential satisfies an overdetermined problem, referred to as critical area-normalized capacitors. As a consequence, we obtain uniqueness results for the Schwarzschild metric, from which improvements in the uniqueness theorems for spin asymptotically flat spacetimes containing a connected photon surface, as well as for spin asymptotically flat static manifolds with boundary are obtained.

math.DG↗

arXiv2024

The Buckling and Clamped Plate Problems on Differential Forms

We extend the buckling and clamped-plate problems to the context of differential forms on compact Riemannian manifolds with smooth boundary. We characterize their smallest eigenvalues and prove that, in the case of bounded Euclidean domains, their spectra without multiplicities on forms coincide with the spectra of the corresponding problems on functions. We obtain various estimates involving the first eigenvalues of the mentioned problems and the ones of the Hodge Laplacian with respect to Dirichlet and absolute boundary conditions on forms. These estimates generalize previous ones in the case of functions.

math.DG↗

arXiv2023

A Poincar\'e formula for differential forms and applications

We prove a new general Poincar\'e-type inequality for differential forms on compact Riemannian manifolds with nonempty boundary. When the boundary is isometrically immersed in Euclidean space, we derive a new inequality involving mean and scalar curvatures of the boundary only and characterize its limiting case in codimension one. A new Ros-type inequality for differential forms is also derived assuming the existence of a nonzero parallel form on the manifold.

math.DG↗

arXiv2022

Total mean curvature and first Dirac eigenvalue

In this note, we prove an optimal upper bound for the first Dirac eigenvalue of some hypersurfaces in Euclidean space by combining a positive mass theorem and the construction of quasi-spherical metrics. As a direct consequence of this estimate, we obtain an asymptotic expansion for the first eigenvalue of the Dirac operator on large spheres in three dimensional asymptotically flat manifolds. We also study this expansion for small geodesic spheres in a three dimensional Riemannian manifold. We finally discuss how this method can be adapted to yield similar results in the hyperbolic space.

math.DG↗

arXiv2022

First boundary Dirac eigenvalue and boundary capacity potential

We derive new lower bounds for the first eigenvalue of the Dirac operator of an oriented hypersurface $\Sigma$ bounding a noncompact domain in a spin asymptotically flat manifold (M n , g) with nonnegative scalar curvature. These bounds involve the boundary capacity potential and, in some cases, the capacity of $\Sigma$ in (M n , g) yielding several new geometric inequalities. The proof of our main result relies on an estimate for the first eigenvalue of the Dirac operator of boundaries of compact Riemannian spin manifolds endowed with a singular metric which may have independent interest.

math.DG↗

arXiv2022

Nonexistence of DEC spin fill-ins

In this note, we show that a closed spin Riemannian manifold does not admit a spin fill-in satisfying the dominant energy condition (DEC) if a certain generalized mean curvature function is point-wise large.

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arXiv2020

A spinorial proof of the rigidity of the Riemannian Schwarzschild manifold

We revisit and generalize a recent result of Cederbaum [C2, C3] concerning the rigidity of the Schwarzschild manifold for spin manifolds. This includes the classical black hole uniqueness theorems [BM, GIS, Hw] as well as the more recent uniqueness theorems for pho-ton spheres [C1, CG1, CG2].

math.DG↗

arXiv2019

Scalar Flat Compactifications Of Poincar{é}-einstein Manifolds And Applications

We derive an integral inequality between the mean curvature and the scalar curvature of the boundary of any scalar flat conformal compactifications of Poincar{é}-Einstein manifolds. As a first consequence , we obtain a sharp lower bound for the first eigenvalue of the conformal half-Laplacian of the boundary of such manifolds. Secondly, a new upper bound for the renormalized volume is given in the four dimensional setting. Finally, some estimates on the first eigenvalues of Dirac operators are also deduced.

math.DG↗

arXiv2019

An Alexandrov theorem in Minkowski spacetime

In this paper, we generalize a theorem {à} la Alexandrov of Wang, Wang and Zhang [WWZ] for closed codimension-two spacelike submanifolds in the Minkowski spacetime for an adapted CMC condition .

math.DG↗

arXiv2019

The Cheeger constant of an asymptotically locally hyperbolic manifold and the Yamabe type of its conformal infinity

Let (M, g) be an (n + 1)-dimensional asymptotically locally hyperbolic (ALH) manifold with a conformal compactification whose conformal infinity is ($\partial$M, [$γ$]). We will first observe that Ch(M, g) $\le$ n, where Ch(M, g) is the Cheeger constant of M. We then prove that, if the Ricci curvature of M is bounded from below by --n and its scalar curvature approaches --n(n+1) fast enough at infinity, then Ch(M, g) = n if and only Y($\partial$M, [$γ$]) $\ge$ 0, where Y($\partial$M, [$γ$]) denotes the Yamabe invariant of the conformal infinity. This gives an answer to a question raised by J. Lee [L].

math.DG↗

arXiv2018

A remark on the rigidity of conformally compact Poincar{é}-Einstein manifolds

In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poin-car{é}-Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof of the rigidity of the hyper-bolic space as the only conformally compact Poincar{é}-Einstein manifold with the round sphere as its conformal infinity.

math.DG↗

arXiv2015

Uniqueness of the de Sitter spacetime among static vacua with positive cosmological constant

We prove that, among all (n + 1)-dimensional spin static vacua with positive cosmological constant, the de Sitter spacetime is characterized by the fact that its spatial Killing hori-zons have minimal modes for the Dirac operator. As a consequence, the de Sitter spacetime is the only vacuum of this type for which the induced metric tensor on some of its Killing horizons is at least equal to that of a round (n -- 1)-sphere. This extends unique-ness theorems shown by Boucher-Gibbons-Horowitz and Chruciel to more general horizon metrics and to the non-single horizon case.

math.DG↗

arXiv2015

A holographic principle for the existence of imaginary Killing spinors

Suppose that $Σ=\partialΩ$ is the $n$-dimensional boundary, with positive (inward) mean curvature $H$, of a connected compact $(n+1)$-dimensional Riemannian spin manifold $(Ω^{n+1},g)$ whose scalar curvature $R\ge -n(n+1)k^2$, for some $k\textgreater{}0$. If $Σ$ admits an isometric and isospin immersion $F$ into the hyperbolic space ${\mathbb{H}^{n+1}\_{-k^2}}$, we define a quasi-local mass and prove its positivity as well as the associated rigidity statement. The proof is based on a holographic principle for the existence of an imaginary Killing spinor. For $n=2$, we also show that its limit, for coordinate spheres in an Asymptotically Hyperbolic (AH) manifold, is the mass of the (AH) manifold.

math.DG↗

arXiv2015

On a Liu--Yau type inequality for surfaces

Let $Ω$ be a compact and mean-convex domain with smooth boundary $Σ:=\partialΩ$, in an initial data set $(M^3,g,K)$, which has no apparent horizon in its interior. If $Σ$ is spacelike in a spacetime $(\E^4,g\_\E)$ with spacelike mean curvature vector $\mathcal{H}$ such that $Σ$ admits an isometric and isospin immersion into $\mathbb{R}^3$ with mean curvature $H\_0$, then: \begin{eqnarray*} \int\_Σ|\mathcal{H}|dΣ\leq\int\_Σ\frac{H\_0^2}{|\mathcal{H}|}dΣ. \end{eqnarray*} If equality occurs, we prove that there exists a local isometric immersion of $Ω$ in $\mathbb{R}^{3,1}$ (the Minkowski spacetime) with second fundamental form given by $K$. In Theorem liu-yau-minkowski, we also examine, under weaker conditions, the case where the spacetime is the $(n+2)$-dimensional Minkowski space $\mathbb{R}^{n+1,1}$ and establish a stronger rigidity result.

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