The Minding formula and its applications
type: Item
instance of: scholarly article
title: The Minding formula and its applications
zbMATH Open document ID: 0958.53042
author: Yi Fang
published in: Archiv der Mathematik
publication date: 2001-04-02T00:00:00Z
review text: The author defines \(M\)-surfaces to be complete open 2-dimensional Riemannian manifolds that are conformal to compact Riemann surfaces with a finite number of points removed, and are allowed to have isolated branch points, and must satisfy certain asymptoticity conditions at its ends and branch points. The author shows that examples of \(M\)-surfaces include: 1) meromorphic surfaces (those whose metric \(ds^2\) is the finite sum of terms of the form \(|\phi_j|^2,\) where \(\phi_j\) is a meromorphic 1-form); 2) complete minimal surfaces with finite total curvature in Euclidean \(n\)-space; 3) complete constant mean curvature 1 surfaces with finite total curvature in hyperbolic 3-space. The author provides a proof of the Minding formula, and then combines the Minding formula with the Gauss-Bonnet formula to get a formula for the total Gaussian curvature of an \(M\)-surface that depends only on the branching orders of the \(M\)-surface at the ends and branch points. The author then shows the applicability of this formula by using it to reprove the Chern-Osserman inequality and the Jorge-Meeks equality for the total curvature of minimal surfaces. The author also uses this formula to reprove two inequalities of Umehara and Yamada for constant mean curvature 1 surfaces in hyperbolic 3-space: one is the strict inequality in the Cohn-Vossen inequality, and the other is the analogue of the Chern-Osserman inequality for constant mean curvature 1 surfaces in hyperbolic 3-space.
reviewed by: Wayne Rossman
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 1333337
zbMATH Keywords: ends
zbMATH Keywords: branch points
zbMATH Keywords: meromorphic surfaces
zbMATH Keywords: complete minimal surfaces
zbMATH Keywords: finite total curvature
zbMATH Keywords: constant mean curvature
zbMATH Keywords: Minding formula
zbMATH Keywords: Cohn-Vossen inequality
zbMATH Keywords: Chern-Osserman inequality
MaRDI profile type: Publication
full work available at URL: s000130050358
OpenAlex ID: W2091550596
DOI: 10.1007/S000130050358
description: scientific article; zbMATH DE number 1333337