Richard Schoen

On the proof of the positive mass conjecture in general relativity

LetM be a space-time whose local mass density is non-negative everywhere. Then we prove that the total mass ofM as viewed from spatial infinity (the ADM mass) must be positive unlessM is the flat Minkowski space-time. (So far we are making the reasonable assumption of the existence of a maximal spacelike hypersurface. We will treat this topic separately.) We can generalize our result to admit wormholes in the initial-data set. In fact, we show that the total mass associated with each asymptotic regime is non-negative with equality only if the space-time is flat.

Proof of the positive mass theorem. II

The positive mass theorem states that for a nontrivial isolated physical system, the total energy, which includes contributions from both matter and gravitation is positive. This assertion was demonstrated in our previous paper in the important case when the space-time admits a maximal slice. Here this assumption is removed and the general theorem is demonstrated. Abstracts of the results of this paper appeared in [11] and [13].

Conformally flat manifolds, Kleinian groups and scalar curvature

On trouve une classe extensive de varietes localement conformement plates dont les applications developpantes sont injectives

Harmonic maps into singular spaces and

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Publisher Summary This chapter discusses some recent results by Richard Schoen and Shing-Tung Yau on the structure of manifolds with positive scalar curvature. The chapter presents theorems which are felt to provide a more complete picture of manifolds with positive scalar curvature: (1) let M be a compact four-dimensional manifold with positive scalar curvature. Then there exists no continuous map with non-zero degree onto a compact K(π,1). (2) Let M be n-dimensional complete manifold with non-negative scalar curvature. Then any conformed immersion of M into Sn is one to one. In particular, any complete conformally flat manifold with non-negative scalar curvature is the quotient of a domain in Sn by a discrete subgroup of the conformal group. (3.) Let M be a compact manifold whose fundamental group is not of exponential growth. Then unless M is covered by Sn, Sn–1 x S1 or the torus, M admits no conformally flat structure.

-adic superrigidity for lattices in groups of rank one

A fundamental nonlinear object in differential geometry is a map between manifolds. If the manifolds have Riemannian metrics, then it is natural to choose representaives for maps which respect the metric structures of the manifolds. Experience suggests that one should choose maps which are minima or critical points of variational integrals. Of the integrals which have been proposed, the energy has attracted most interest among analysts, geometers, and mathematical physicists. Its critical points, the harmonic maps, are of some geometric interest. They have also proved to be useful in applications to differential geometry. Particularly one should mention the important role they play in the classical minimal surface theory. Secondly, the applications to Kahler geometry given in [S], [SiY] illustrate the usefulness of harmonic maps as analytic tools in geometry. It seems to the author that there is good reason to be optimistic about the role which the techniques and results related to this problem can play in future developments in geometry.

On the structure of manifolds with positive scalar curvature

Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Bohm and Wilking.

Analytic Aspects of the Harmonic Map Problem

defined on M. To ensure a uniqueness property on equation (0.I) it is necessary to res t r ic t f to lie in a suitable function space. Some of the most natural spaces are those consisting of L p functions on M, denoted by LP(M), where integration is defined with respect to the Riemannian measure. In this setting, uniqueness of (0.1) means that if fELP(M) for some O<p~ <oo, t h e n f m u s t be identically constant. We remark that when p = oo all constant functions satisfy (0.1) and belong to L| On the other hand, for O<p<oo, while all constant functions satisfy (0.1), they belong to LP(M) iff M has finite volume, unless the constant is zero. For the sake of simplicity we say a manifold satisfies property ~p for p E (0, oo] if every L p harmonic function on M is constant. We also say that M satisfies property 5ep if every nonnegative L p subharmonic function on M is constant. Observing that the absolute value of a harmonic function is a nonnegative subharmonic function (in the weak sense), M satisfying 5ep implies it also satisfies ~p. The first result towards understanding the uniqueness of (0.1) was due to

Manifolds with 1/4-pinched curvature are space forms

This paper considers the prescribed scalar curvature problem onSn forn>-3. We consider the limits of solutions of the regularization obtained by decreasing the critical exponent. We characterize those subcritical solutions which blow up at the least possible energy level, determining the points at which they can concentrate, and their Morse indices. We then show that forn=3 this is the only blow up which can occur for solutions. We use this in combination with the Morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed scalar curvature problem onS3.

Lp and mean value properties of subharmonic functions on Riemannian manifolds

Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking’s results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology S2 × S1. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).