Riemann bilinear relations on minimal surfaces
The classical Riemann bilinear relations on the Weierstrass one-form of a complete minimal surface with finite total curvature in \(\mathbb{R}^3\), \(\mathbb{R}^3/T\) or \(\mathbb{T} \times \mathbb{R}\), where \(T\) is a nontrivial translation and \(\mathbb{T} \times \mathbb{R}\) is the quotient of \(\mathbb{R}^3\) by a lattice generated by two independent translations, give certain balancing conditions concerning the distribution of its ends and other physically related objects such as forces and torques. This leads to information about the rank of the force map of properly embedded minimal surface in \(\mathbb{R}^3\) with finite total curvature, a characterization of the Riemannian minimal surfaces as the only properly embedded minimal tori in \(\mathbb{R}^3/T\) with two planar ends which are symmetric by the reflection in a plane, and the relative position of the ends of a doubly periodic minimal surface.
minimal surface with finite total curvature
scientific article; zbMATH DE number 1143003