We prove -bounds for the Riesz transforms associated to the Hodge-Laplacian equipped with absolute and relative boundary conditions in a Lipschitz subdomain of a (smooth) Riemannian manifold for in a certain interval depending on the Lipschitz character of the domain.
Assume that is a complete Riemannian manifold with Ricci curvature bounded from below and that satisfies a Sobolev inequality of dimension . Let be a complete Riemannian manifold isometric at infinity to and let . The boundedness of the Riesz transform of then implies the boundedness of the Riesz transform of
When is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results.
Any Kähler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dimension.
All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce a series of invariant differential forms and vector fields which allow us to construct numerous scalar differential invariants...
These notes summarize the papers [8, 9] on the analysis of resolvent, Eisenstein series and scattering operator for geometrically finite hyperbolic quotients with rational non-maximal rank cusps. They complete somehow the talk given at the PDE seminar of Ecole Polytechnique in october 2005.
The Schiffer Problem as originally stated for Euclidean spaces (and later for some symmetric spaces) is the following: Given a bounded connected open set Ω with a regular boundary and such that the complement of its closure is connected, does the existence of a solution to the Overdetermined Neumann Problem (N) imply that Ω is a ball? The same question for the Overdetermined Dirichlet Problem (D). We consider the generalization of the Schiffer problem to an arbitrary Riemannian manifold and also...