Non-properly Embedded Minimal Planes in Hyperbolic 3-Space

Abstract

In this paper, we show that there are non-properly embedded minimal surfaces with finite topology in a simply connected Riemannian 3-manifold with nonpositive curvature. We show this result by constructing a non-properly embedded minimal plane in hyperbolic 3-space. Hence, this gives a counterexample to Calabi-Yau conjecture for embedded minimal surfaces in the negative curvature case.

References in corpus (2)

• Asymptotic Plateau Problem
• Least Area Planes in Hyperbolic 3-Space are Properly Embedded

Cited by in corpus (3)

• Minimal Surfaces in Hyperbolic 3-manifolds
• The Slab Theorem for Minimal Surfaces in $\mathbb{E}(-1,\tau )$
• $H$-Surfaces with Arbitrary Topology in Hyperbolic 3-Space