Hopf tori in \(S^ 3\)
type: Item
instance of: scholarly article
title: Hopf tori in \(S^ 3\)
zbMATH Open document ID: 0585.53051
DOI: 10.1007/BF01389060
author: Ulrich Pinkall
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143262
review text: Let \(\pi\) : \(S^ 3\to S^ 2\) be the Hopf fibration. The author proves that the inverse image of any closed curve on \(S^ 2\) is an immersed torus in \(S^ 3\). This torus is called a Hopf torus. Using Hopf tori, he obtains: Every compact Riemann surface of genus one can be conformally embedded in the unit sphere \(S^ 3\subset R^ 4\) as a flat torus. The embedding can be chosen as the intersection of \(S^ 3\) with a quartic hypersurface in \(R^ 4\). As a corollary, he gets: Every compact Riemann surface of genus one can be conformally embedded in \(R^ 3\) as an algebraic surface of degree eight. A surface in \(R^ 3\) is called a Willmore surface if it is an extremal surface for the variational functional \(\int H^ 2dA\) (H the mean curvature). The only known examples of compact Willmore surfaces are the stereographic projections of compact minimal surfaces in \(S^ 3\). By showing that Hopf tori of some kind are Willmore surfaces, the author obtains an infinite series of compact Willmore surfaces that do not stem from minimal surfaces in \(S^ 3\).
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 57R40
zbMATH DE Number: 3938007
zbMATH Keywords: conformal embedding
zbMATH Keywords: Hopf fibration
zbMATH Keywords: Hopf torus
zbMATH Keywords: flat torus
zbMATH Keywords: algebraic surface
zbMATH Keywords: Willmore surface
zbMATH Keywords: mean curvature
Wikidata QID: Q57406248
reviewed by: Toru Ishihara
MaRDI profile type: Publication
cites work: A duality theorem for Willmore surfaces
cites work: On the conformal types of algebraic surfaces of euclidean space
cites work: Curve-straightening in Riemannian manifolds
cites work: Complete minimal surfaces in \(S^ 3\)
cites work: The principle of symmetric criticality
cites work: Embeddings of open Riemann surfaces
cites work: Q5548166
description: scientific article; zbMATH DE number 3938007