On the geometry of Riemannian manifolds with a Lie structure at infinity

A manifold with a ``Lie structure at infinity'' is a non-compact manifold $M_0$ whose geometry is described by a compactification to a manifold with corners M and a Lie algebra of vector fields on M, subject to constraints only on $M \smallsetminus M_0$. The Lie structure at infinity on $M_0$ determines a metric on $M_0$ up to bi-Lipschitz equivalence. This leads to the natural problem of understanding the Riemannian geometry of these manifolds. We prove, for example, that on a manifold with a Lie structure at infinity the curvature tensor and its covariant derivatives are bounded. We also study a generalization of the geodesic spray and give conditions for these manifolds to have positive injectivity radius. An important motivation for our work is to study the analysis of geometric operators on manifolds with a Lie structure at infinity. For example, a manifold with cylindrical ends is a manifold with a Lie structure at infinity. The relevant analysis in this case is that of totally characteristic operators on a compact manifold with boundary equipped with a ``b-metric.'' The class of conformally compact manifolds, which was recently proved of interest in the study of Einstein's equation, also consists of manifolds with a Lie structure at infinity.Comment: LaTe

The Ricci flow approach to homogeneous Einstein metrics on flag manifolds

We give the global picture of the normalized Ricci flow on generalized flag manifolds with two or three isotropy summands. The normalized Ricci flow for these spaces descents to a parameter depending system of two or three ordinary differential equations, respectively. We present here the qualitative study of these system's global phase portrait, by using techniques of Dynamical Systems theory. This study allows us to draw conclusions about the existence and the analytical form of invariant Einstein metrics on such manifolds, and seems to offer a better insight to the classification problem of invariant Einstein metrics on compact homogeneous spaces.Comment: 17 pages, 2 figure

Pseudodifferential operators on manifolds with a Lie structure at infinity

Several examples of non-compact manifolds $M_0$ whose geometry at infinity is described by Lie algebras of vector fields $V \subset \Gamma(TM)$ (on a compactification of $M_0$ to a manifold with corners $M$) were studied by Melrose and his collaborators. In math.DG/0201202 and math.OA/0211305, the geometry of manifolds described by Lie algebras of vector fields -- baptised "manifolds with a Lie structure at infinity" there -- was studied from an axiomatic point of view. In this paper, we define and study the algebra \Psi_{1,0,\VV}^\infty(M_0), which is an algebra of pseudodifferential operators canonically associated to a manifold $M_0$ with the Lie structure at infinity $V \subset\Gamma(TM)$. We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to $\Psi_{1,0,V}^\infty(M_0)$. We also consider the algebra \DiffV{*}(M_0) of differential operators on $M_0$ generated by $V$ and \CI(M), and show that $\Psi_{1,0,V}^\infty(M_0)$ is a ``microlocalization'' of \DiffV{*}(M_0). Finally, we introduce and study semi-classical and ``suspended'' versions of the algebra $\Psi_{1,0,V}^\infty(M_0)$. Our construction solves a problem posed by Melrose in his talk at the ICM in Kyoto

Holonomy and submanifold geometry

We survey applications of holonomic methods to the study of submanifold geometry, showing the consequences of some sort of extrinsic version of de Rham decomposition and Berger's Theorem, the so-called Normal Holonomy Theorem. At the same time, from geometric methods in submanifold theory we sketch very strong applications to the holonomy of Lorentzian manifolds. Moreover we give a conceptual modern proof of a result of Kostant for homogeneous space

Deformations and stability in complex hyperbolic geometry

This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into $PU(n,1)$ and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sullivan. Despite Goldman-Millson-Yue rigidity results for such complex manifolds of infinite volume, we present different classes of such manifolds that allow non-trivial (quasi-Fuchsian) deformations and point out that such flexible manifolds have a common feature being Stein spaces. While deformations of complex surfaces from our first class are induced by quasiconformal homeomorphisms, non-rigid complex surfaces (homotopy equivalent to their complex analytic submanifolds) from another class are quasiconformally unstable, but nevertheless their deformations are induced by homeomorphisms