Minimal surfaces near short geodesics in hyperbolic $3$-manifolds

Abstract

If $M$ is a finite volume complete hyperbolic $3$-manifold, the quantity ${\mathcal{A}}_1(M)$ 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 ${\mathcal{A}}_1$ with respect to the geometric convergence of hyperbolic manifolds. We prove that it is lower semi-continuous and even continuous if ${\mathcal{A}}_1(M)$ 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

35 pages, 4 figures

References in corpus (3)

• General curvature estimates for stable H-surfaces in 3-manifolds and applications
• Complex length of short curves and Minimal Fibration in hyperbolic $3$-Manifolds fibering over the circle
• Minimal Fibrations of Hyperbolic 3-manifolds