Conformal Compactification of Asymptotically Locally Hyperbolic Metrics

In this paper we study the extent to which conformally compact asymptotically hyperbolic metrics may be characterized intrinsically. Building on the work of the first author in (Bahuaud, Pac. J. Math. 239(2): 231–249, 2009), we prove that decay of sectional curvature to ?1 and decay of covariant derivatives of curvature outside an appropriate compact set yield Hölder regularity for a conformal compactification of the metric. In the Einstein case, we prove that the estimate on the sectional curvature implies the control of all covariant derivatives of the Weyl tensor, permitting us to strengthen our result.     

Boundary Regularity for Conformally Compact Einstein Metrics in Even Dimensions

Sufficient conditions are derived for a symmetric hyperbolic system with large variable-coefficient terms to be uniformly well posed. Examples of systems satisfying those conditions are presented.     

Sectional Curvature Rigidity of Asymptotically Locally Hyperbolic Manifolds

This paper presents a rigidity result of real hyperbolic quotients inanalogy to Min-Oo's result in Math. Ann. 285(4) (1989),527–539, but without the spin condition. In order to prove this,we use special Killing forms on the exterior form bundle. Moreover, wemake an assumption on the sectional curvature to obtain the necessaryeigenvalue estimates of the curvature endomorphism in theBochner–Weitzenböck formula of ^k M.     

Asymptotically Euclidean Ends of Ricci Flat Manifolds,and Conformal Inversions

We prove that a Ricci flat end of a Riemannian manifold is asymptotically Euclidean if it is obtained from a smooth metric by a conformal inversion. A number of consequences are discussed.     

Locally finite graphs with ends: A topological approach, II. Applications

This paper is the second of three parts of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form from the ArXiv [20]. They are complemented by a third part [31], which looks at the theory from an algebraic-topological point of view.The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. This second part of the survey concentrates on these applications, many of which solve problems or extend earlier work of Thomassen on infinite graphs. Numerous new problems are suggested.