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The Schrödinger equation on a compact manifold : Strichartz estimates and applications

Nicolas Burq, Patrick Gérard, Nikolay Tzvetkov (2001)

Journées équations aux dérivées partielles

We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any riemannian compact manifold. As a consequence we infer global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three-manifolds in the case of quadratic nonlinearities. We also discuss the optimality of these Strichartz estimates on spheres.

The second Yamabe invariant with singularities

Mohammed Benalili, Hichem Boughazi (2012)

Annales mathématiques Blaise Pascal

Let ( M , g ) be a compact Riemannian manifold of dimension n ≥ 3 .We suppose that g is a metric in the Sobolev space H 2 p ( M , T * M ⊗ T * M ) with p > n 2 and there exist a point P ∈ M and δ > 0 such that g is smooth in the ball B p ( δ ) . We define the second Yamabe invariant with singularities as the infimum of the second eigenvalue of the singular Yamabe operator over a generalized class of conformal metrics to g and of volume 1 . We show that this operator is attained by a generalized metric, we deduce nodal solutions to a Yamabe type equation with...

The signature package on Witt spaces

Pierre Albin, Éric Leichtnam, Rafe Mazzeo, Paolo Piazza (2012)

Annales scientifiques de l'École Normale Supérieure

In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction, which is inductive over the ‘depth’ of the singularity, is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index—the analytic signature of  X —is well-defined....

The spectrum of the Laplace operator for a spherical space form

Gr. Tsagas (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Si determina lo spettro di un operatore di Laplace di una «spherical space form» ( M , g ) e si studia l’influenza di tale spettro su ( M , g ) .

The Srní lectures on non-integrable geometries with torsion

Ilka Agricola (2006)

Archivum Mathematicum

This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics—in particular superstring theory—where these naturally appear. Connections with skew-symmetric torsion are exhibited as one of the main tools to understand non-integrable geometries. To this aim a a series of key examples is presented and successively dealt with using the notions of...

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