Simply-connected minimal surfaces with finite total curvature in $\H^2\times\R$

TL;DR

This paper classifies simply-connected minimal surfaces with finite total curvature in , showing that those with total curvature  are Scherk graphs over quadrilaterals, and constructs new examples with higher curvature.

Contribution

It provides a complete classification of simply-connected minimal surfaces with total curvature  in  and introduces new properly embedded examples with larger total curvature.

Findings

Surfaces with total curvature  are exactly Scherk graphs over ideal quadrilaterals.

Constructed new properly embedded simply-connected minimal surfaces with total curvature k for any integer k.

Classified all complete minimal surfaces with total curvature  in .

Abstract

Laurent Hauswirth and Harold Rosenberg developed the theory of minimal surfaces with finite total curvature in $\H^2\times\R$. They showed that the total curvature of one such a surface must be a non-negative integer multiple of $-2\pi$. The first examples appearing in this context are vertical geodesic planes and Scherk minimal graphs over ideal polygonal domains. Other non simply-connected examples have been constructed recently. In the present paper, we show that the only complete minimal surfaces in $\H^2\times\R$ of total curvature $-2\pi$ are Scherk minimal graphs over ideal quadrilaterals. We also construct properly embedded simply-connected minimal surfaces with total curvature $-4k\pi$, for any integer $k\geq 1$, which are not Scherk minimal graphs over ideal polygonal domains.