Sebastian Montiel

University of Granada | UGR · Department of Geometry and Topology

Let (M, g) be an \((n+1)\)-dimensional asymptotically locally hyperbolic manifold with a conformal compactification whose conformal infinity is \((\partial M,[\gamma ])\). We will first observe that \({\mathcal Ch}(M,g)\le n\), where \({\mathcal Ch}(M,g)\) is the Cheeger constant of M. We then prove that, if the Ricci curvature of M is bounded from...

Let (M, g) be an (n + 1)-dimensional asymptotically locally hyperbolic (ALH) manifold with a conformal compactification whose conformal infinity is ($\partial$M, [$\gamma$]). 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...

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

We give a spinorial proof of a Heintze-Karcher-type inequality in the hyperbolic space proved by Brendle The proof relies on a generalized Reilly formula on spinors recently obtained in

In this paper, we study Dirac-type operators on time flat submanifolds in spacetimes satisfying the Einstein equations with non positive cosmological constant. We apply our results to obtain global rigidity results for n-dimensional time flat submanifolds in the Minkowski spacetime as well as in the anti-de Sitter spacetime.

In this paper, we prove an optimal Positive Mass theorem for Asymptotically Hyperbolic spin manifolds with compact inner boundary. This improves a previous result of Chruściel and Herzlich [The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math.212(2) (2003) 231–264].

Suppose that $\Sigma=\partial M$ is the $n$-dimensional boundary of a connected compact Riemannian spin manifold $( M,\langle\;,\;\rangle)$ with non-negative scalar curvature, and that the (inward) mean curvature $H$ of $\Sigma$ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to the conformal metri...

Let $\Omega$ be a compact and mean-convex domain with smooth boundary $\Sigma:=\partial\Omega$, in an initial data set $(M^3,g,K)$, which has no apparent horizon in its interior. If $\Sigma$ is spacelike in a spacetime $(\E^4,g\_\E)$ with spacelike mean curvature vector $\mathcal{H}$ such that $\Sigma$ admits an isometric and isospin immersion into...

Suppose that $\Sigma=\partial\Omega$ is the $n$-dimensional boundary, with positive (inward) mean curvature $H$, of a connected compact $(n+1)$-dimensional Riemannian spin manifold $(\Omega^{n+1},g)$ whose scalar curvature $R\ge -n(n+1)k^2$, for some $k\textgreater{}0$. If $\Sigma$ admits an isometric and isospin immersion $F$ into the hyperbolic s...

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 o...

Let (M, g) be an Asymptotically Locally Hyperbolic (ALH) manifold which is the interior of a conformally compact manifold and (∂M, [γ]) its conformal infinity. Suppose that the Ricci tensor of (M, g) dominates that of the hyperbolic space and that its scalar curvature satisfies a certain decay condition at infinity. If the Yamabe invariant of (∂M,...

Let M be a compact orientable n-dimensional hypersurface, with nowhere vanishing mean curvature H, immersed in a Riemannian spin manifold \({\overline{M}}\) admitting a non trivial parallel spinor field. Then the first eigenvalue \({\lambda_1(D_{M}^{H})}\) (with the lowest absolute value) of the Dirac operator \({D_{M}^{H}}\) corresponding to the c...

We prove that an $(n+1)$-dimensional spin static vacuum with negative cosmological constant whose null infinity has a boundary admitting a non-trivial Killing spinor field is the AdS spacetime. As a consequence, we generalize previous uniqueness results by X. Wang \cite{Wa2} and by Chru{\'s}ciel-Herzlich \cite{CH} and introduce, for this class of s...

From the existence of parallel spinor fields on Calabi-Yau, hyper-Kähler or complex flat manifolds, we deduce the existence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the submanifolds are compact, we obtain sharp estimates on their Betti numbers which generalize those obtained b...

We use the half-space model for the open set of a de Sitter space associated to the steady state space to obtain some sharp a priori estimates for the height and the slope of certain constant mean curvature spacelike graphs. These estimates allow us to prove some existence and uniqueness theorems about complete non-compact constant mean curvature s...

 Under intrinsic and extrinsic curvature assumptions on a Riemannian spin manifold and its boundary, we show that there is an isomorphism between the restriction to the boundary of parallel spinors and extrinsic Killing spinors of non-negative Killing constant. As a corollary, we prove that a complete Ricci-flat spin manifold with mean-convex bound...

We give a sharp extrinsic lower bound for the first eigenvaluesof the intrinsic Dirac operator of certain hypersurfaces boundinga compact domain in a spin manifold of negative scalar curvature.Limiting-cases are characterized by the existence, on the domain,of imaginary Killing spinors. Some geometrical applications, as anAlexandrov type theorem, a...

 On a compact Riemannian spin manifold with mean-convex boundary, we analyse the ellipticity and the symmetry of four boundary conditions for the fundamental Dirac operator including the (global) APS condition and a Riemannian version of the (local) MIT bag condition. We show that Friedrich's inequality for the eigenvalues of the Dirac operator on...

Let Ω be a bounded planar domain which is convex (although not necessarily strictly convex) with area A. We prove that, for each real number H satisfying AH2 < ρ2π, with ρ = (√5 - 1)/2, there exists a graph on Ω with constant mean curvature H and boundary ∂Ω. This existence theorem is deduced as a consequence of an L∞ estimate for compact constant...

New extrinsic lower bounds are given for the classical Dirac operator on the boundary of a compact domain of a spin manifold. The main tool is to solve some boundary problems for the Dirac operator of the domain under boundary conditions of Atiyah-Patodi-Singer type. Spinorial techniques are used to give simple proofs of classical results for compa...

Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. Limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary. Comment: 13 pages, LaTex

We propose the study of a conformally invariant functional for surfaces of complex projective plane which is closely related to the classical Willmore functional. We show that minimal surfaces of complex projective plane are critical for this functional and construct some minima for it via the twistors spaces of complex projective plane. Also, we f...

Genus zero Willmore surfaces immersed in the three-sphere S3 correspond via the stereographic projection to minimal surfaces in Euclidean three-space with nite total curvature and embedded planar ends. The critical values of the Willmore functional are 4k ,w herek2 N ,w ithk6 =2 ; 3; 5; 7. When the ambient space is the four-sphere S4, the regular h...

We give an existence result for constant mean curvature graphs in hyperbolic space ℍn+1. Let Ω be a compact domain of a horosphere in ℍn+1 whose boundary ∂Ω is mean convex, that is, its mean curvature H∂Ω (as a submanifold of the horosphere) is positive with respect to the inner orientation. If H is a number such that -H∂Ω < H < 1, then there exist...

We determine all stable constant mean curvature hypersurfaces in a wide class of complete Riemannian manifolds having a foliation whose leaves are umbilical hypersurfaces. Among the consequences of this analysis we obtain all the stable constant mean curvature hypersurfaces in many nonsimply connected hyperbolic space forms.

The index of a compact orientable superminimal surface of a self- dual Einstein four-manifold M with positive scalar curvature is computed in terms of its genus and area. Also a lower bound of its nullity is obtained. Applications to the cases M = S4 and M = CP2 are given, characterizing the standard Veronese immersions and their twistor deformatio...

We study constant mean curvature compact surfaces in Euclidean space with planar boundary. Two geometric conditions for these surfaces to be graphs are given.

We characterize a class of hyperbolic cylinders of the de Sitter spacetime as the only complete non-compact spacelike hypersurfaces with constant lowest mean curvature and having more than one topological end.

It has been long conjectured that the two spherical caps are then only discs in the Euclidean three-space R3 with non-zero constant mean curvature spanning a round circle. In this work, we prove that it is true when the area of such a disc is less than or equal to that of the big spherical cap.

Given a real hypersurface of a complex hyperbolic space #x2102;?H n ,we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H 1 2n+1 .Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of H n .

In this paper, we will use some techniques in Morse Theory in order to compute the Betti numbers of an indefinite flag manifold. The problem is reduced to compute it for the definite flag manifolds.

In [ 8 ], Fialkow classified Einstein hypersurfaces in indefinite space forms, when the Weingarten endomorphism is diagonalizable. Recently, [ 13 ]–[ 15 ], Magid completed this work when such endomorphism is not diagonalizable. On the other hand, Smyth [ 17 ], classified complex Einstein hypersurfaces in complex space forms. Since Barros & Romero,...

In (2), indefinite Kählerian manifolds have been examined from the point of view of holomorphic sectional curvature. Examples in (2) show that the analogue of Kulkarni's theorem (see (4), p. 173) for the holomorphic sectional curvature is false and the best possible result in the direction is: Theorem 1 ( known , (2)). Let M be a connected indefini...