One of the natural generalizations of conformal structure on a two dimensional surface is a conformally flat structure on an n-manifold. In higher dimensions, not every manifold admits such a structure and it is a difficult problem to give a good classification of conformally flat manifolds. Recall that conformally flat Riemannian manifolds are manifolds whose metrics are locally conformally equivalent to the Euclidean metric. Kuiper [Kul, Ku2] was the first to study the global properties of these manifolds. He classified those compact conformally flat manifolds with abelian fundamental group. For dimensions greater than two, the Liouville theorem tells us that the conformal transformations of S" are determined locally and are given by M6bius transformations. Hence by a standard monodromy argument, a simply connected conformally flat manifold (with dimension> 3) has a conformal immersion into S" which is unique up to

In this paper, we study the question of which compact manifolds admit a metric with positive scalar curvature. Scalar curvature is perhaps the weakest invariant among all the well-known invariants constructed from the curvature tensor. It measures the deviation of the Riemannian volume of the geodesic ball from the euclidean volume of the geodesic ball. As a result, it does not tell us much of the behavior of the geodesics in the manifold. Therefore it was remarkable that in 1963, Lichnerowicz [i] was able to prove the theorem that on a compact spin manifold with positive scalar curvature, there is no harmonic spinor. Applying

Let M be a compact Riemannian manifold with a fixed conformal structure. Then we introduce the concept of conformal volume of M in the following manner. For each branched conformal immersion q9 of M into the unit sphere S n, we consider the set of all branched conformal immersions obtained by composition of qo with the conformal automorphisms of S". We let Vc (n, qg) be the maximum volume of these branched immersions. The conformal volume of M is defined to be the infimum of V.(n, q0) where qo ranges over all branched conformal immersions of M into the unit sphere S". In this paper, we study the case when M is a compact surface and we call the conformal volume of M to be the conformal area of M. We demonstrate that this conformal invariant is non-trivial. In fact, we prove that if there exists a minimal immersion of M into S" where coordinate functions are first eigenfunctions, then the conformal area of M is given by the area of M with respect to the induced metric. This enables us to compute the conformal area for several surfaces. For example, the conformal area of RP z is 6n and the conformal area of the square torus is 27c 2. We believe that the computation of the conformal area for general surfaces will be very important in studying the geometry of compact surfaces. We demonstrate this claim by applying the concept of conformal area to two different branches of surface theory. The first application is to study the total curvature of a compact surface in R". This problem has a long history. Fenchel and Fary [9] proved that for a closed curve o in R", Slk [> 27c where k is its curvature. Then Milnor [12] r proved that 5lk]> 4~ z if cr is

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].

Let<i>M</i> be a space-time whose local mass density is non-negative everywhere. Then we prove that the total mass of<i>M</i> as viewed from spatial infinity (the ADM mass) must be positive unless<i>M</i> 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.