This erratum corrects an error arising in the proof of Proposition 6.4 in the article “On the density of geometrically finite Kleinian groups”

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.

We define a notion of stability for chiral ring of four dimensional N= 1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N= 1 field theory derived from D3 branes probing a three-fold singularity X, and show that the K stability which implies the existence of Ricci-flat conic metric on X is equivalent to the stability of chiral ring of the corresponding field theory.

We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al.[3] for bounded degree graphs, and for three-connected graphs of fixed genus g . Using this we can show that for a three-connected graph satisfying a certain volume-growth condition, the multiplicity of the g th Laplacian eigenvalue is at most $2\left [

We modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.