We give a generalizations of lower Ricci curvature bound in the framework of graphs. We prove that the Ricci curvature in the sense of Bakry and Emery is bounded below by 1 on locally finite graphs. The Ricci flat graph in the sense of Chung and Yau is proved to be a graph with Ricci curvature bounded below by zero. We also get an estimate for the eigenvalue of Laplace operator on finite graphs: 1 dD (exp (dD+ 1) 1)

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.

When enough matter is condensed in a small region, gravitational effects will be strong enough to cause collapse and a black hole will be formed. We formulate and prove here such a statement in the language of general relativity. (This is Theorem 2 of this paper.)