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Michael T. Anderson

Publications and source records attributed to Michael T. Anderson.

53 records · Page 3Linked to original sources
arXiv2004

Orbifold compactness for spaces of Riemannian metrics and applications

This work proves certain general orbifold compactness results for spaces of Riemannian metrics, generalizing earlier results along these lines for Einstein metrics or metrics with bounded Ricci curvature. This is then applied to prove such compactness for spaces of Bach-flat (for example half-conformally flat) metrics on 4-manifolds, and related results for metrics which are critical points of other natural Riemannian functionals on the space of metrics.

math.DG↗
arXiv2002

Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand's inverse boundary problem

This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand making essential use of the results of the first two parts.

math.SP↗
arXiv2002

Regularity for Lorentz Metrics under Curvature Bounds

Let (M, g) be an (n+1) dimensional space-time, with bounded curvature with respect to a bounded framing. If (M, g) is vacuum or satisfies a mild condition on the stress-energy tensor, then we show that (M, g) locally admits coordinate systems in which the Lorentz metric is well-controlled in the (space-time) Sobolev space L^{2,p}, for any finite p.

gr-qc↗
arXiv2002

Boundary regularity, uniqueness and non-uniqueness for AH Einstein metrics on 4-manifolds

This paper studies several aspects of asymptotically hyperbolic Einstein metrics, mostly on 4-manifolds. We prove boundary regularity (at infinity) for such metrics and establish uniqueness under natural conditions on the boundary data. By examination of explicit black hole metrics, it is shown that neither uniqueness nor finiteness holds in general for AH Einstein metrics with a prescribed conformal infinity. We then describe natural conditions which are sufficient to ensure finiteness.

math.DG↗
arXiv2002

Scalar curvature and the existence of geometric structures on 3-manifolds, I

This paper analyses the convergence and degeneration of sequences of metrics on a 3-manifold, and relations of such with Thurston's geometrization conjecture. The sequences are minimizing sequences for a certain (optimal) scalar-curvature type functional and their degeneration is related to the sphere and torus decompositions of the 3-manifold under certain conditions.

math.DG↗
arXiv2001

Remarks on evolution of space-times in 3+1 and 4+1 dimensions

A large class of vacuum space-times is constructed in dimension 4+1 from hyperboloidal initial data sets which are not small perturbations of empty space data. These space-times are future geodesically complete, smooth up to their future null infinity, and extend as vacuum space-times through their Cauchy horizon. Dimensional reduction gives non-vacuum space-times with the same properties in 3+1 dimensions.

gr-qc↗
arXiv2001

L^2 curvature and volume renormalization of AHE metrics on 4-manifolds

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.

math.DG↗
arXiv2001

Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds, II

This paper is a continuation of I, (same title), and is concerned with the existence, regularity and degeneration of metrics minimizing natural curvature functionals on the space of metrics on 3-manifolds. The functionals chosen are designed to be optimal w.r.t. the issue of geometrization of the underlying 3-manifold, in the sense of Thurston. Two of the main results are Theorems B and C, in comparison with Theorem A of I.

math.DG↗
arXiv2000

On the structure of solutions to the static vacuum Einstein equations

A complete characterization is obtained of the asymptotic behavior of solutions of the static vacuum Einstein equations which have a (pseudo)-compact horizon or boundary and are complete away from the boundary. It is proved that the time-symmetric space-like hypersurface has only finitely many ends, each of which is either asymptotically flat (AF) or parabolic, as in the (static) Kasner metric. Examples are given with both types of behavior, together with an extensive discussion and new characterization of Weyl metrics. The asymptotics result allows one in most circumstances to drop the AF assumption from the static black hole uniqueness theorems and replace it with just a completeness assumption.

gr-qc↗
arXiv2000

On stationary vacuum solutions to the Einstein equations

It is proved that the only geodesically complete stationary vacuum solution of the Einstein equations is the empty Minkowski space, or a quotient of it by a discrete group of isometries, generalizing a classical result of Lichnerowicz. In addition, we obtain an apriori bound on the curvature of stationary vacuum solutions away from the boundary or horizon.

gr-qc↗
arXiv2000

On long-time evolution in general relativity and geometrization of 3-manifolds

We describe some relations between the long-time asymptotic behavior of the vacuum Einstein evolution equations and the geometrization of 3-manifolds. These relations are expressed in terms of evolution of CMC hypersurfaces in the vacuum space-time.Some results are also obtained on the singularity avoidance of CMC foliations. In addition, the paper describes a number of open problems relating these two areas.

gr-qc↗
arXiv1999

Extrema of curvature functionals on the space of metrics on 3-manifolds, II

Several rigidity results are proved for critical points of natural Riemannian functionals on the space of metrics on 3-manifolds. Two of these results are as follows. Let (N, g) be a complete Riemannian 3-manifold, satisfying one of the following variational conditions: (i) (N, g) has non-negative scalar curvature and is a critical point for the L^2 norm of the full curvature R among compact perturbations of (N, g). (ii) (N, g) has non-negative scalar curvature, a free isometric S^1 action, and is a critical point of the L^2 norm of R among compact volume non-increasing perturbations of (N, g) with non-negative scalar curvature. In either case, (N, g) is flat. The Schwarzschild metric (on the space-like hypersurface) has an isometric S^1 action and satisfies the other assumptions in (ii), showing that this result is sharp.

math.DG↗
arXiv1999

Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds, I

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.

math.DG↗