Minimal surfaces near short geodesics in hyperbolic 3-manifolds

If M is a finite volume complete hyperbolic 3-manifold, the quantity A1(M) is defined as the infimum of the areas of closed minimal surfaces in M. In this paper we study the continuity property of the functional A1 with respect to the geometric convergence of hyperbolic manifolds. We prove that it is lower semi-continuous and even continuous if A1(M) is realized by a minimal surface satisfying some hypotheses. Understanding the interaction between minimal surfaces and short geodesics in M is the main theme of this paper.