A space of minimal tori with one end and cyclic symmetry
The author considers \(X:M\to{\mathbb R}^3\), a complete, 2-dimensional, minimal surface with one end, where \(M\) is a conformal torus with one point removed. It is assumed that (1) the total curvature of \(X\) equals \(-4\pi m\) for some positive integer \(m\), (2) the branch order, at the end, of the meromorphic function in the Weierstrass representation of \(X\) is \(m-1\) (the maximal value it can have), and (3) \(X(M)\) is invariant under a rotation of \(90^\circ\) around a vertical axis followed by a reflection through a horizontal plane. The author explicitly gives the defining equation of the moduli space of such surfaces. As a consequence of this defining equation, the author is able to show that, up to homotheties and isometries of \({\mathbb R}^3\), there is a unique such surface with total curvature \(-12\pi\).
scientific article; zbMATH DE number 5124584