Least Area Planes in Hyperbolic 3-Space are Properly Embedded

TL;DR

This paper proves that embedded least area planes in hyperbolic 3-space with certain smooth boundary conditions are necessarily properly embedded, advancing understanding of minimal surfaces in hyperbolic geometry.

Contribution

It establishes a new condition under which least area planes in hyperbolic 3-space are guaranteed to be properly embedded, linking boundary smoothness to properness.

Findings

Least area planes with smooth boundary points are properly embedded

Boundary smoothness influences the properness of minimal surfaces

Provides new insights into minimal surface behavior in hyperbolic space

Abstract

We show that if P is an embedded least area (area minimizing) plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve with at least one smooth point, then P is properly embedded.