Non-properly Embedded Minimal Planes in Hyperbolic 3-Space

TL;DR

This paper constructs a non-properly embedded minimal plane in hyperbolic 3-space, providing a counterexample to the Calabi-Yau conjecture in negatively curved 3-manifolds.

Contribution

It demonstrates the existence of non-properly embedded minimal surfaces with finite topology in hyperbolic space, challenging previous assumptions.

Findings

Existence of non-properly embedded minimal planes in hyperbolic space

Counterexample to Calabi-Yau conjecture in negative curvature

Construction method for such minimal surfaces

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.