Complete minimal surfaces derived from Chen-Gackstatter genus two example
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-Gackstatter genus two surface is the only complete minimal immersion in \(\mathbb{R}^3\) of genus two, total curvature \(-12\pi\) and with eight symmetries.
scientific article; zbMATH DE number 1060837