Embedded constant mean curvature surfaces with special symmetry
Let \(M_{g,\kappa}\) be the space of complete embedded surfaces in \(\mathbb{R}^3\) of genus \(g\), with \(\kappa\) ends and constant (nonzero) mean curvature (CMC), where two surfaces are identified as points in \(M_{g,\kappa}\) if there is a rigid motion of \(\mathbb{R}^3\) carrying one surface to the other. It is known that \(M_{g,\kappa}\) is a finite dimensional real analytic variety, and in a neighborhood of a surface with no \(L^2\)-Jacobi fields it is a \((3\kappa-6)\)-dimensional manifold for each \(\kappa\geq 3\), [\textit{R. Kusner}, \textit{R. Mazzeo} and \textit{D. Pollack}, Geom. Funct. Anal. 6, 120-137 (1996; Zbl 0966.58005)]. These spaces are known explicitly for \(\kappa=0,1,2\). The Kapouleas construction [\textit{N. Kapouleas}, Ann. Math. 131, 239-330 (1990; Zbl 0699.53007)] shows that the set \(M_{g,\kappa}\) is non-empty for all \(\kappa\geq 3\) and each \(g\). Furthermore, each end of a surface in \(M_{g,\kappa}\) is asymptotically a Delaunay undoloid and this defines the neckradii and axes of the ends [\textit{N. Korevaar}, \textit{R. Kusner} and \textit{B. Solomon}, J. Differ. Geom. 30, 465-503 (1989; Zbl 0726.53007)]. The authors of this paper give sharp, necessary conditions on complete embedded CMC surfaces with 3 ends and an extra reflection symmetry. The respective submoduli space is a two-dimensional variety in \(M_{g,\kappa}\). These surfaces are called isosceles triundoloids. Fundamental domains of these surfaces are characterized by associated great circle polygons in the three-sphere.
scientific article; zbMATH DE number 1310202