Minimal surfaces in $\mathbb{H}^2\times\mathbb{R}$ with finite total curvature and embedded ends
Year: 2026
Type: preprint
Abstract: In this paper we study the global geometry and classification of complete minimal surfaces with embedded ends and finite total curvature in the product space $\mathbb{H}^2 \times \mathbb{R}$. We describe the structure of embedded ends of such surfaces, constraining the combinatorics of their asymptotic polygons at infinity. Focusing on small total curvature values, we establish a complete classification of the possible asymptotic boundaries for embedded minimal surfaces with total curvature $-4π$ and $-6π$. In the $-4π$ case, we prove the surface is either a horizontal catenoid or simply-connected, with asymptotic boundary belonging to one of four explicit configurations. In the $-6π$ case, we show that any embedded example must be simply-connected and classify its possible ideal boundaries. We further construct, via an asymptotic Plateau problem solved by area-minimizing surfaces and successive Schwarz reflections, new 1- and 2-parameter families of (possibly non-embedded) simply-connected minimal surfaces with total curvature $-6π$ (and, more generally, $-2(2k-1)π$) whose ends are embedded and realize asymptotic configurations not previously known to occur.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212080094