Hermitian metrics of negative holomorphic sectional curvature on some hyperbolic manifolds

Research partially supported by National Science Foundation     

New examples of manifolds with negative curvature

This lecture is about the global structure of Riemannian manifolds with negative sectional curvature and finite volume. A fundamental question is to find and analyze geometric conditions which ensure that Mⁿ has finitely many ends or, stronger, that Mⁿ is of finite topological type.     

CR manifolds with noncompact connected automorphism groups

The main result of this paper is that the identity component of the automorphism group of a compact, connected, strictly pseudoconvex CR manifold is compact unless the manifold is CR equivalent to the standard sphere. In dimensions greater than 3, it has been pointed out by D. Burns that this result follows from known results on biholomorphism groups of complex manifolds with boundary and the fact that any such CR manifoldM can be realized as the boundary of an analytic variety. WhenM is 3-dimensional, Burns’s proof breaks down because abstract CR 3-manifolds are generically not realizable as boundaries. This paper provides an intrinsic proof of compactness that works in any dimension.     

Conformal deformations of prescribing scalar curvature on Riemannian manifolds with negative curvature

We study the conformal deformation for prescribing scalar curvature function on Cartan-Hadamard manifoldM ⁿ (n≥3) with strongly negative curvature. By employing the supersubsolution method and a careful construction for the supersolution, we obtain the best possible asymptotic behavior for near infinity so that the problem of complete conformal deformation is solvable. In more general cases, we prove an asymptotic estimation on the solutions of the conformal scalar curvature equation. Project partially supported by the NNSF of China     

Isoscattering deformations for complete manifolds of negative curvature

We construct continuous families of nonisometric metrics on simply connected manifolds of dimension n ≥ 9which have the same scattering phase, the same resolvent resonances, and strictly negative sectional curvatures. This situation contrasts sharply with the case of compact manifolds of negative curvature, where Guillemin/Kazhdan, Min-Oo, and Croke/Sharafutdinov showed that there are no nontrivial isospectral deformations of such metrics.     

Some remarks on convex surfaces in simply connected homogeneous three manifolds

Here we extend Hadamards’ Theorem to other homogeneous three manifolds, i.e., we prove that a compact orientable immersed surface Σ in general position whose principal curvatures κ _i , i = 1, 2, satisfy κ _i ≥ τ, is an embedded sphere.     

Limit theorems for probability measures on simply connected nilpotent lie groups

LetX ₁,X ₂,... be i.i.d. random variables with values in a simply connected nilpotent Lie groupG. Assume the laws of to be weakly convergent to a probability measure onG, _n Aut(G), and (k _n)_n strictly increasing in. In this paper we want to characterize the possible limit laws. We obtain that every limit law is continuously embeddable and we prove a kind of functional limit theorem. Further, we study the connections between two different concepts of stability (resp. semistability) and limit laws. Finally, we describe the various domains of attraction of measures (resp. of convolution semigroups).     

Growth of discrete groups of isometries in negative curvature: a gap-property

We prove that a finitely generated group acting without fixed point on a n-dimensional Cartan–Hadamard manifold of pinched sectional curvature $?a^2?K??1$ is either virtually nilpotent or has entropy $\mathrm{Ent}(\Gamma )?C(n,a)>0$. To cite this article: G. Besson et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).