Related papers: On the renormalized volumes for conformally compac…
We present a renormalized Gauss-Bonnet formula for approximate Kahler-Einstein metrics on compact complex manifolds with pseudo-Einstein CR boundaries. The boundary integral is given explicitly, and it is proved that it gives a…
We develop methods for constructing and computing conformal invariants of submanifolds, with a particular emphasis on conformal submanifold scalars and conformally invariant integrals of natural submanifold scalars. These methods include a…
Given a smooth non-trapping compact manifold with strictly con- vex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. This data consist of the exit directions…
The Gauss-Bonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the Poincar\'e-Einstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and…
Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the…
We discuss smooth metric measure spaces admitting two weighted Einstein representatives of the same weighted conformal class. First, we describe the local geometries of such manifolds in terms of certain Einstein and quasi-Einstein warped…
In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold…
The main result of this paper is that the space of conformally compact Einstein metrics on a given manifold is a smooth, infinite dimensional Banach manifold, provided it is non-empty, generalizing earlier work of Graham-Lee and Biquard. We…
We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a…
We investigate the asymptotic expansion and the renormalized volume of minimal submanifolds, $Y^m$ of arbitrary codimension in Poincare-Einstein manifolds, $M^{n+1}$. In particular, we derive formulae for the first and second variations of…
We obtain a general five dimensional (5D) quasispherical solutions of irrotational dust in Einstein gravity with the Gauss-Bonnet combination of quadratic curvature terms. These solutions are generalization, to Einstein-Gauss-Bonnet…
We define a geometric quantity for asymptotically hyperbolic manifolds, which we call the volume-renormalized mass. It is essentially a linear combination of the ADM mass surface integral and a renormalization of the volume. We show that…
We exhibit the equivalence between the renormalized volume of asymptotically anti-de Sitter (AAdS) Einstein manifolds in four and six dimensions, and their renormalized Euclidean bulk gravity actions. The action is that of Einstein gravity,…
Explicit models for the restricted conformal group of the Einstein static universe of dimension greater than two and for its universal covering group are constructed. Based on these models, as an application we determine all oriented and…
We present a systematic calculation of the volumes of compact manifolds which appear in physics: spheres, projective spaces, group manifolds and generalized flag manifolds. In each case we state what we believe is the most natural scale or…
We show that the renormalized volume of a quasifuchsian hyperbolic 3-manifold is equal, up to an additive constant, to the volume of its convex core. We also provide a precise upper bound on the renormalized volume in terms of the…
We give a general survey of the solution of the Einstein constraints by the conformal method on n dimensional compact manifolds. We prove some new results about solutions with low regularity (solutions in $H_{2}$ when n=3), and solutions…
We look at general braneworlds in six-dimensional Einstein-Gauss-Bonnet gravity. We find the general matching conditions for the Einstein-Gauss-Bonnet braneworld, which remarkably turn out to give precisely the four-dimensional Einstein…
The renormalized volume of hyperbolic manifolds is a quantity motivated by the AdS/CFT correspondence of string theory and computed via a certain regularization procedure. The main aim of the present paper is to elucidate its geometrical…
We renormalize the Chern-Simons invariant for convex-cocompact hyperbolic 3-manifolds by finding the asymptotics along an equidistance foliation. We prove that the metric Chern-Simons invariant has an exponentially divergent term given by…