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We formulate a well-posed initial boundary value problem (IBVP) for the vacuum Einstein equations in harmonic gauge by describing the boundary conditions of a spacetime metric in its associated gauge. This gauge is determined, equivariantly with respect to diffeomorphisms, by the spacetime metric. Further we show that vacuum spacetimes satisfying fixed initial-boundary conditions and corner conditions are geometrically unique near the initial surface. In analogy to the solution to the Cauchy problem, we also construct a unique maximal globally hyperbolic vacuum development of the initial-boundary data., Comment: 48 pages; The Introduction has been rewritten to clarify the exposition and results, more detailed discussion of the corner geometry is added, and minor mistakes in the previous manuscript have been corrected

We prove that compact 3-manifolds $M$ of constant curvature +1 with boundary a minimal surface are locally naturally parametrized by the conformal class of the boundary metric $\gamma$ in the Teichmuller space of $\partial M$, when $genus(\partial M) \geq 2$. Stronger results are obtained in the case of genus 1 boundary, giving in particular a new proof of Brendle's solution of the Lawson conjecture. The results generalize to constant mean curvature surfaces, and surfaces in flat and hyperbolic 3-manifolds., Comment: withdrawn for reconstruction. Error in the "stability argument" on p.22

Let (M, g) be a compact Einstein manifold with non-empty boundary. We prove that Killing fields at the boundary extend to Killing fields of any (M, g) provided the boundary is weakly convex and a simple condition on the fundamental group holds. This gives a new proof of the classical infinitesimal rigidity of convex surfaces in Euclidean space and generalizes the result to Einstein metrics of any dimension., Comment: withdrawn for reconstruction; error in "stability argument" on p.13

We prove the existence of (branched) conformal immersions F: S^2 -> R^3 with mean curvature H > 0 arbitrarily prescribed up to a 3-dimensional affine indeterminacy. A similar result is proved for the space forms S^3, H^3 and partial results for surfaces of higher genus., 20 pages

We discuss the problem of prescribing the mean curvature and conformal class as boundary data for Einstein metrics on 3-manifolds, in the context of natural elliptic boundary value problems for Riemannian metrics., Comment: 12 pages; to appear in the volume in honor of Jeff Cheeger's 65th birthday

We prove a semi-global result on the existence of conformal embeddings of the two-sphere into the round three-sphere S^3(1) with prescribed mean curvature., Comment: This article is being withdrawn. 1) More general results are now proved in arXiv:1204.5225. 2) As pointed out to me by Jose Espinar (thanks Jose), the area bound proof in reference [11] by Ho is incorrect, which affects the proof of the main result here

We survey some aspects of the current state of research on Einstein metrics on compact 4-manifolds. A number of open problems are presented and discussed., 30pp, to appear in Handbook of Geometric Analysis. Final version has only minor changes from v1

This paper is withdrawn., Comment: This paper is withdrawn

Final version in paper linked above., Comment: This paper is now replaced by 0710.1305

We discuss classical gravitational aspects of the AdS/CFT correspondence, with the aim of obtaining a rigorous (mathematical) understanding of the semi-classical limit of the gravitational partition function. The paper surveys recent progress in the area, together with a selection of new results and open problems., Comment: 23pp, for Proceedings of the Strasbourg meeting on AdS/CFT, Sept. 03 v.2: references added, and minor changes made, especially to last section

We discuss several aspects of the relation between asymptotically AdS and asymptotically dS spacetimes including: the continuation between these types of spaces, the global stability of asymptotically dS spaces and the structure of limits within this class, holographic renormalization, and the maximal mass conjecture of Balasubramanian-deBoer-Minic., Comment: 22pp, minor changes, references added

This paper was withdrawn by the author due to an error in the proof of the main result; essentially the parameter R used in the proof may depend on the manifold (M, g), not just on dimension and pinching constant., This paper has been withdrawn

This is a continuation of a previous paper of same title. The degeneration, i.e. curvature blow-up, of sequences of metrics appoaching the Sigma constant, assumed non-positive, is analysed. The degeneration is related to the sphere decomposition of the 3-manifold M, in case M is sigma-tame., Comment: 61pp

This paper surveys aspects of the convergence and degeneration of Riemannian metrics on a given manifold M - the Cheeger-Gromov theory - and extensions thereof to Ricci curvature in place of full curvature. This theory is then applied to study a collection of different issues in mathematical aapects of General Relativity., 23pp, for Proc. 2002 Cargese School on General Relativity

See comment above., Comment: Reworked and reorganized into new paper: see math.DG/0104171

This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V, as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M. In addition, we compute and discuss the differential or variation dV of V, or equivalently the variation of the L^2 norm of the Weyl curvature, on the space of such Einstein metrics., Comment: 15 pages, several corrections made. To appear in Math. Research Letters

We study Yamabe metrics, and the moduli space of Yamabe metrics, on an arbitrary closed 3-manifold M. The main focus is on the boundary behavior of the moduli space, i.e. the behavior of degenerating sequences of unit volume Yamabe metrics on M. It is proved that such degenerations, when non-trivial in a certain sense, are described by solutions of the static vacuum Einstein equations. Natural conditions are given for the non-triviality of degenerating sequences and relations with Palais-Smale sequences for the Einstein-Hilbert functional are explored. A number of new examples, both trivial and non-trivial,of degenerating sequences are constructed. It is also proved that the only complete static vacuum solution without horizon is the flat metric, generalizing a classical result of Lichnerowicz., Comment: 89 pages, no figures. No Table of Contents. Also available at: http://www.math.sunysb.edu/~anderson/