Abstract. Let M be a space-time whose local mass density is non-negative everywhere. Then we prove that the total mass of M as viewed from spatial infinity (the ADM mass) must be positive unless M is the fiat 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 nonnegative with equality only if the space-time :is fiat. O. IntroductionThis is the second part of our paper on scalar curvature of a three-dimensional manifold and its relation to general relativity. The problem in general relativity that we address is the following: An isolated gravitating system having nonnegative local mass density must have non-negative total mass, measured gravitationally at spatial infinity.Mathematically, the positive mass conjecture can be described as follows : Let N be a three dimensional Riemannian manifold with metric tensor gi~. Then an initial set consists of N and a symmetric tensor field hi~ so that /~> ~aJaJa 1/2 where/~ and J are defined by [~=½(R--Ehabhab+(~ahaa)2) a,bwhere R is the scalar curvature of our metric.If N is a spacelike hypersurface in a space time so that gi~ is the induced metric and hlj is the second fundamental form, then the above condition says that the apparent energy-momentum of the matter be timelike.

The positive mass theorem states that for a nontriviat 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 [113 and [131. #> (~i JiJ) j/2An initial data set will be said to be asymptotically flat if for some compact set C, N\C consists of a finite number of components N 1 ..... Np such that each N i is diffeomorphic to the complement of a compact set in R 3. Under such diffeomorphisms, the metric tensor will be required to be written in the form gij = 6~5 + O(r-i)and the scalar curvature of N will be assumed to be O(r-4).With each N k we associate a total mass M k defined by the flux integral 1 Mk= t-~ ~ .~. (giJ'j-gjj'i)dai LJwhich is the limit of surface integrals taken over large two spheres in N k.This number M k is called the ADM mass of N k (see Arnowitt, Deser, and Misner [1]). Classically it was assumed that the first term in the asymptotic expansion of gii is spherical. It was pointed out by York [11] that physically it is more desirable to relax this assumption to the one mentioned above. The method in this paper will work assuming only this general asymptotic condition of York.In order for the total mass to be a conserved quantity, one assumes Pij = O(r-2) and ~Pu = O(r-3).

No abstract

No abstract

vanced Study for its support. This paper was completed while they were visitors, during the academic year 1992-93. We thank Bruce Kleiner for several helpful discussions concerning the geometry of metric spaces.Note added in proof. After this paper was written we received a preprint from J. Jost in which he also obtains some existence results for harmonic maps to (NPC) spaces. CONTENTS N. J. KOREVAAR AND R. M. SCHOEN1.5. The energy-density measure 1.6. Lower semicontinuity, and consistency when X = M 581 1.7. Directional energies 1.8. The calculus of directional energies 1.9. Differentiability theory for directional energies 591 1.10. Absolute continuity of de for p > 1 601 1.11. The calculus of energy-density functions 1.12. Trace theory for Lipschitz domains 605 1.13. Precompactness 612 2. Harmonic Maps into Non-Positively Curved Metric Spaces 2.1. Non-positively curved metric spaces 616 2.2. The solution to the Dirichlet Problem 622 2.3. The pull-back inner product TT 2.4. Geodesic homotopies and interior Lipschitz continuity 628 2.5. Center of mass constructions 639 2.6. Equivariant mapping problems 643 2.7. Homotopy problems 656 References 658

The purpose of this paper is to study minimal surfaces in three-dimensional manifolds which, on each compact set, minimize area up to second order. If M is a minimal surface in a Riemannian three-manifold N, then the condition that M be stable is expressed analytically by the requirement that o n any compact domain of M, the first eigenvalue of the operator A+Ric(v)+(AI* be positive. Here Ric (v) is the Ricci curvature of N in the normal direction to M and (A)' is the square of the length of the second fundamental form of M.In the case that N is the flat R3, we prove that any complcte stable minimal surface M is a plane (Corollary 4). The earliest result of this type was due to S. Bernstein [2] who proved this in the case that M is the graph of a function (stability is automatic in this case). The Bernstein theorem was generalized by R. Osserman [lo] who showed that the statement is true provided the image of the Gauss map of M omits an open set on the sphere. The relationship of the stable regions on M to the area of their Gaussian image has been studied by Barbosa and do Carmo [l] (cf. Remark 5 ) . The methods of Schoen-Simon-Yau [ 113 give a proof of this result provided the area growth of a geodesic ball of radius r in M is not larger than r6. An interesting feature of our theorem is that it does not assume that M is of finite type topologically, or that the area growth of M is suitably small.The theorem for R3 is a special case of a classification theorem which we prove for stable surfaces in three-dimensional manifolds N having scalar curvature SZO. We use an observation of Schoen-Yau [8] to rearrange the stability operator so that S comes into play (see formula (12)). Using this, and the study of certain differential operators on the disc (Theorem 2), we are

No abstract

The contents of this paper correspond roughly to that of the author's lecture series given at Montecatini in July 1987. This paper is divided into five sections. In the first we present the Einstein-Hilbert variationM problem on the space of Riemannian metrics on a compact closed manifold M. We compute the first and secol~d variation and observe the distinction which arises between conformal directions and their orthogonal complements. We discuss varia- In §2 we give a general discussion of the Yamabe problem and its resolution. We also give a detailed analysis of the solutions of the Yamabe equation for the product conformal structure on SI(T) x S~-1(1), a circle of radius T crossed with a sphere of radius one. These exhibit interesting variational fea,tures such a.s symmetry breaking and the existence of solutions with high Morse index. Since the time of the summer school in Montecatini, the beautiful survey paper of J. Lee and T. Parker [15] has appeared. This gives a detailed discussion of the Yamabe problem along with a new argument unifying the work of T. Aubin [1] with that of the author. §3 contains an a priori estimate on arbitrary (nonminimizing) solutions of the Yamabe problem in terms of a bound on the energy. The estimate applies uniformly to solutions of the subcritical equation, and implies that solutions of the subcritical equation converge in C 2 norm to solutions of the Yamabe equation. These estimates hold on manifolds which are not conformally diffeomorphic to the standard sphere. We present here the result for locally conformally flat metrics. This estimate has not appeared in print prior to this paper although