Corpus ID: 16234535

Volume and Area Renormalizations for Conformally Compact Einstein Metrics

  title={Volume and Area Renormalizations for Conformally Compact Einstein Metrics},
  author={C. Robin Graham},
Let $X$ be the interior of a compact manifold $\overline X$ of dimension $n+1$ with boundary $M=\partial X$, and $g_+$ be a conformally compact metric on $X$, namely $\overline g\equiv r^2g_+$ extends continuously (or with some degree of smoothness) as a metric to $X$, where $r$ denotes a defining function for $M$, i.e. $r>0$ on $X$ and $r=0$, $dr\ne 0$ on $M$. The restrction of $\overline g$ to $TM$ rescales upon changing $r$, so defines invariantly a conformal class of metrics on $M$, which… Expand
Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds
To any smooth compact manifold $M$ endowed with a contact structure $H$ and partially integrable almost CR structure $J$, we prove the existence and uniqueness, modulo high-order error terms andExpand
On Uniqueness And Existence of Conformally Compact Einstein Metrics with Homogeneous Conformal Infinity
In this paper we show that for a generalized Berger metric $\hat{g}$ on $S^3$ close to the round metric, the conformally compact Einstein (CCE) manifold $(M, g)$ with $(S^3, [\hat{g}])$ as itsExpand
On uniqueness of conformally compact Einstein metrics with homogeneous conformal infinity
  • Gang Li
  • Mathematics
  • Advances in Mathematics
  • 2018
In this paper we show that for a Berger metric $\hat{g}$ on $S^3$, the non-positively curved conformally compact Einstein metric on the $4$-ball $B_1(0)$ with $(S^3, [\hat{g}])$ as its conformalExpand
We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\unicode[STIX]{x2202}M$ has dimension $n$ even. Its definitionExpand
Heat kernel expansions, ambient metrics and conformal invariants
The conformal powers of the Laplacian of a Riemannian metric which are known as the GJMS-operators admit a combinatorial description in terms of the Taylor coefficients of a natural second-orderExpand
Yang–Mills connections on conformally compact manifolds
We study a boundary value problem for Yang–Mills connections on Hermitian vector bundles over a conformally compact manifold $$\overline{M}$$ M ¯ . Our main result is the following: for everyExpand
Inverse scattering on conformally compact manifolds
We study inverse scattering for $\Delta_g+V$ on $(X,g)$ a conformally compact manifold with metric $g,$ with variable sectional curvature $-\alf^2(y)$ at the boundary and $V\in C^\infty(X)$ notExpand
CR-Invariants and the Scattering Operator for Complex Manifolds with Boundary
The purpose of this paper is to describe certain CR-covariant differential operators on a strictly pseudoconvex CR manifold $M$ as residues of the scattering operator for the Laplacian on an ambientExpand
Residue Family Operators on Spinors and Spectral Theory of Dirac operator on Poincaré-Einstein Spaces
We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian $Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their Poincar\'e-Einstein metrics $(X,g_+)$. OurExpand
Asymptotically simple spacetimes and mass loss due to gravitational waves
The cosmological constant $\Lambda$ used to be a freedom in Einstein's theory of general relativity, where one had a proclivity to set it to zero purely for convenience. The signs of $\Lambda$ orExpand


Anti-de Sitter space and holography
Recently, it has been proposed by Maldacena that large $N$ limits of certain conformal field theories in $d$ dimensions can be described in terms of supergravity (and string theory) on the product ofExpand
Einstein metrics with prescribed conformal infinity on the ball
In this paper we study a boundary problem for Einstein metrics. Let A4 be the interior of a compact (n + l)-dimensional manifold-with-boundary I@, and g a Riemannian metric on M. If 2 is a metric onExpand
Regularity at infinity for area-minimizing hypersurfaces in hyperbolic space
equipped with the hyperbolic metric y-Z(dxZ+dy2). A standard compactification of IH involves adding the boundary (R"• {0})w {*} so that I[-I is simply the one point compactification of the EuclideanExpand
Invariants of conformal Laplacians
The conformal Laplacian D = d*d + (n - 2)s/4(n - 1), acting on functions on a Riemannian manifold Mn with scalar curvature s, is a conformally invariant operator. In this paper we will use D toExpand
The Mathematical Heritage of Hermann Weyl
On induced representations by R. Bott Differentiable structures on fractal-like sets, determined by intrinsic scaling functions on dual Cantor sets by D. Sullivan Representation theory and arithmeticExpand
Complete minimal varieties in hyperbolic space
It is a basic problem in the subject of minimal varieties to determine the class of complete minimal and area-minimizing varieties M k in a given Riemannian manifold _~n. Besides being of their ownExpand
The Large-N Limit of Superconformal Field Theories and Supergravity
We show that the large-N limits of certainconformal field theories in various dimensions includein their Hilbert space a sector describing supergravityon the product of anti-de Sitter spacetimes,Expand
The holographic Weyl anomaly
We calculate the Weyl anomaly for conformal field theories that can be described via the adS/CFT correspondence. This entails regularizing the gravitational part of the corresponding supergravityExpand
Gauge Theory Correlators from Non-Critical String Theory
Abstract We suggest a means of obtaining certain Green's functions in 3+1-dimensional N =4 supersymmetric Yang-Mills theory with a large number of colors via non-critical string theory. TheExpand
Operator product expansion for Wilson loops and surfaces in the large N limit
The operator product expansion for {open_quotes}small{close_quotes} Wilson loops in N=4, d=4 SYM theory is studied. The OPE coefficients are calculated in the large {ital N} and g{sub YM}{sup 2}NExpand