Complete minimal surfaces derived from Chen-Gackstatter genus two example
type: Item
instance of: scholarly article
title: Complete minimal surfaces derived from Chen-Gackstatter genus two example
zbMATH Open document ID: 0921.53003
published in: Pacific Journal of Mathematics
publication date: 1997-09-16T00:00:00Z
review text: In 1982, \textit{C. C. Chen} and \textit{F. Gackstatter} [Math. Ann. 259, 359-369 (1982; Zbl 0468.53008)] discovered two minimal surfaces with finite total curvature in \(\mathbb{R}^3\) and nontrivial topology. These examples have only one end and the symmetry of Enneper's surface. In this paper, the authors use the ideas of Hoffman and Meeks' generalization of Costa's surface [\textit{D. Hoffman} and \textit{W. H. Meeks III}, Ann. Math., II. Ser. 131, 1-34 (1990; Zbl 0695.53004)] to show a new family of complete minimal surfaces \(M_k\) of genus \(k\), \(k\geq 2\), \(k\) even, that generalizes the Chen-Gackstatter genus two example. Furthermore, their main achievement is to prove a uniqueness result for these surfaces under some symmetry assumptions. Summarizing, the surfaces \(M_k\) have the following properties: (i) \(M_k\) is conformally equivalent to \(\overline{M}_k-\{P\}\), where \(\overline{M}_k\) is a compact Riemann surface of genus \(k\) and \(P\in\overline{M}_k\). (ii) \(M_k\) has total curvature \(-4(2k-1)\pi\). (iii) \(M_k\) has \(4k\) symmetries. (iv) \(M_k\) intersects the \((x_1,x_2)\)-plane at \(k\) straight lines meeting at equal angles at the origin. Moreover, the symmetry group \(\text{Sym}(M_k)\) is generated by a rotation of angle \(\pi/k\) around the \(x_3\)-axis followed by a symmetry with respect to the \((x_1,x_2)\)-plane and a symmetry with respect to the \((x_1,x_3)\)-plane. (v) \(M_2\) is the genus two Chen-Gackstatter example. Moreover, if \(M\) is a complete minimal surface with one end, genus \(k\neq 6,15\), finite total curvature and \(\left(\text{Sym} (M)\right)=4k\), then the total curvature \({\mathcal C}(M)\) of \(M\) satisfies \({\mathcal C}(M)\leq -4(2k- 1)\pi\), and the equality holds if and only if \(k\) is even and \(M\) is, up to rigid motions and scaling, the surface \(M_k\). The most important consequence of this is the following uniqueness theorem for the Chen-Gackstatter genus two examples: Theorem. The Chen-Gackstatte…
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1060837
zbMATH Keywords: finite total curvature
zbMATH Keywords: complete minimal surfaces
zbMATH Keywords: symmetries
author: Francisco J. López
author: Francisco Martín
author: Domingo Rodríguez
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1060837