Renormalization group flows from gravity in anti-de Sitter space versus black hole no-hair theorems

Black hole no-hair theorems are proven using inequalities that govern the r adial dependence of spherically symmetric configurations of matter fields. In this paper, we analyze the analogous inequalities for geometries dual to renormalization group flows via the AdS/CFT correspondence. These inequali ties give much useful information about the qualitative properties of such flows. For Poincare invariant flows, we show that generic flows of relevant or irrelevant operators lead to singular geometries. For the case of irrel evant operators, this leads to an apparent conflict with Polchinski's decou pling theorem, and we offer two possible resolutions to this problem.