We consider the Einstein/Yang-Mills equations in 3+1 space time dimensions with<i>SU</i>(2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang-Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime.

Let M be a complete Einstein manifold of negative curvature, and assume that (as in the AdS/CFT correspondence) it has a Penrose compactification with a conformal boundary M of positive scalar curvature. We show that under these conditions, M and in particular M must be connected. These results resolve some puzzles concerning the AdS/CFT correspondence.

In the description of isolated gravitating systems in General Relativity a spacelike timeslice has the structure of a complete Riemannian three-manifold with an asymptotically at end. Let (N; g) denote an end of such a complete Riemannian three-manifold, ie N is diffeomorphic to R3\B1 (0), and the metric on N asymptotically approaches the Euclidean metric near infinity: gij=

We describe the counting of BPS states of Type II on K3 by relating the supersymmetric cycles of genus <i>g</i> to the number of rational curves with <i>g</i> double points on K3. The generating function for the number of such curves is the left-moving partition function of the bosonic string.

We shall consider the following Dirichlet eigenvalue problem on a smooth bounded domain S~ eRn, I where V is a nonnegative function defined on,~. As is well-known, the eigenvalues of problem (1.1) can be interpreted as the energy levels of a particle travelling under an external force field of a potential q in Rn, where and the corresponding eigenfunctions are wave functions of the Schrodinger equation-J~+ qu= lu. Furthermore, the set of eigenvalues {A,} of (1.1) are nonnegative and can be arranged in a nondecreasing order as follows, It is a significant problem to find a lower bound for~, 1 in terms of the geometry of Q. This subject has been studied extensively by many authors. A rather precise bound in the case V== 0 was worked out not only for a bounded domain in but actually valid for a general Riemannian manifold with certain curvature conditions; we refer to [4] for these recent develop-ments. Nevertheless, very little is known about the obvious interesting question of how big the gap is between 2 and l. There are both physical