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On the differential form spectrum of hyperbolic manifolds

Gilles Carron, Emmanuel Pedon (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We give a lower bound for the bottom of the L 2 differential form spectrum on hyperbolic manifolds, generalizing thus a well-known result due to Sullivan and Corlette in the function case. Our method is based on the study of the resolvent associated with the Hodge-de Rham laplacian and leads to applications for the (co)homology and topology of certain classes of hyperbolic manifolds.

On the Djrbashian kernel of a Siegel domain

Elisabetta Barletta, Sorin Dragomir (1998)

Studia Mathematica

We establish an inversion formula for the M. M. Djrbashian A. H. Karapetyan integral transform (cf. [6]) on the Siegel domain Ω n = ζ ∈ ℂ n : ϱ ( ζ ) > 0 , ϱ ( ζ ) = I m ( ζ 1 ) - | ζ ' | 2 . We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the ϱ α -Bergman kernels, α > -1, (in the sense of Z. Pasternak-Winiarski [20] of Ω n . We build an anti-holomorphic embedding of Ω n in the complex projective Hilbert space ℂ ℙ ( H α 2 ( Ω n ) ) and study (in connection with work by A. Odzijewicz [18] the corresponding transition probability amplitudes....

On the embedding of 1-convex manifolds with 1-dimensional exceptional set

Lucia Alessandrini, Giovanni Bassanelli (2001)

Annales de l’institut Fourier

In this paper we show that a 1-convex (i.e., strongly pseudoconvex) manifold X , with 1- dimensional exceptional set S and finitely generated second homology group H 2 ( X , ℤ ) , is embeddable in ℂ m × ℂ ℙ n if and only if X is Kähler, and this case occurs only when S does not contain any effective curve which is a boundary.

On the energy of unit vector fields with isolated singularities

Fabiano Brito, Paweł Walczak (2000)

Annales Polonici Mathematici

We consider the energy of a unit vector field defined on a compact Riemannian manifold M except at finitely many points. We obtain an estimate of the energy from below which appears to be sharp when M is a sphere of dimension >3. In this case, the minimum of energy is attained if and only if the vector field is totally geodesic with two singularities situated at two antipodal points (at the 'south and north pole').

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