Sebastián Montiel

On some real hypersurfaces of a complex hyperbolic space

Given a real hypersurface of a complex hyperbolic space #x2102;?Hn,we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H12n+1.Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of ℂHn.

Willmore two-spheres in the four-sphere

Genus zero Willmore surfaces immersed in the three-sphere S3 correspond via the stereographic projection to minimal surfaces in Euclidean three-space with finite total curvature and embedded planar ends. The critical values of the Willmore functional are 4πk, where k ∈ N∗, with k 6= 2, 3, 5, 7. When the ambient space is the four-sphere S4, the regular homotopy class of immersions of the two-sphere S2 is determined by the self-intersection number q ∈ Z; here we shall prove that the possible critical values are 4π(|q|+ k + 1), where k ∈ N. Moreover, if k = 0, the corresponding immersion, or its antipodal, is obtained, via the twistor Penrose fibration P3 → S4, from a rational curve in P3 and, if k 6= 0, via stereographic projection, from a minimal surface in R4 with finite total curvature and embedded planar ends. An immersion lies in both families when the rational curve is contained in some P2 ⊂ P3 or (equivalently) when the minimal surface of R4 is complex with respect to a suitable complex structure of R4.

Eigenvalues of the Dirac Operator on Manifolds¶with Boundary

Abstract: 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. For the local boundary conditions, limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary.

Dirac Operators on Hypersurfaces of Manifolds with Negative Scalar Curvature

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, are given.

SECOND VARIATION OF SUPERMINIMAL SURFACES INTO SELF-DUAL EINSTEIN FOUR-MANIFOLDS

The index of a compact orientable superminimal surface of a selfdual 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 =

A HOLOGRAPHIC PRINCIPLE FOR THE EXISTENCE OF PARALLEL SPINOR FIELDS AND AN INEQUALITY OF SHI-TAM TYPE

4 and M = CIp2 are given, characterizing the standard Veronese immersions and their twistor deformations as those with lowest index.

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

Suppose that

A holographic principle for the existence of imaginary Killing spinors

\Sigma=\partial M

A positive mass theorem for asymptotically hyperbolic manifolds with inner boundary

is the

Supersymmetric rigidity of asymptotically locally hyperbolic manifolds

n