Minimal surfaces near short geodesics in hyperbolic 3-manifolds

Harold Rosenberg (1) , Laurent Mazet (2)

1 IMPA - Instituto Nacional de Matemática Pura e Aplicada
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

If M is a finite volume complete hyperbolic 3-manifold, the quantity 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 A_1 with respect to the geometric convergence of hyperbolic manifolds. We prove that it is lower semi-continuous and even continuous if 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.

Domaines: Géométrie différentielle [math.DG]

Harold Rosenberg, Laurent Mazet. Minimal surfaces near short geodesics in hyperbolic 3-manifolds. Advances in Mathematics, 2020, 372, ⟨10.1016/j.aim.2020.107285⟩. ⟨hal-01546786⟩