Riemann bilinear relations on minimal surfaces
type: Item
instance of: scholarly article
title: Riemann bilinear relations on minimal surfaces
zbMATH Open document ID: 0892.53005
author: Joaquín Pérez
published in: Mathematische Annalen
publication date: 1998-08-03T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1143003
zbMATH Keywords: distribution of ends
zbMATH Keywords: minimal surface with finite total curvature
zbMATH Keywords: rank of the force map
reviewed by: Juan Antonio Pérez
MaRDI profile type: Publication
full work available at URL: s002080050150
OpenAlex ID: W2064829573
DOI: 10.1007/S002080050150
description: scientific article; zbMATH DE number 1143003