Minimal surfaces in finite volume non compact hyperbolic $3$-manifolds

Pascal Collin (1) , Laurent Hauswirth (2) , Laurent Mazet (2) , Harold Rosenberg (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 IMPA - Instituto Nacional de Matemática Pura e Aplicada

We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This determines its asymptotic behavior. Some rigidity theorems are obtained.

Domains: Differential Geometry [math.DG]

Pascal Collin, Laurent Hauswirth, Laurent Mazet, Harold Rosenberg. Minimal surfaces in finite volume non compact hyperbolic $3$-manifolds. Transactions of the American Mathematical Society, 2017, 369 (6), pp.4293-4309. ⟨10.1090/tran/6859⟩. ⟨hal-01015056⟩