Rigidity of superminimal immersions of compact Riemann surfaces into \(CP^ 2\)
type: Item
instance of: scholarly article
title: Rigidity of superminimal immersions of compact Riemann surfaces into \(CP^ 2\)
zbMATH Open document ID: 0716.53055
author: Quo-Shin Chi
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1990-01-01T00:00:00Z
review text: The rigidity problem for superminimal immersions of compact Riemann surfaces in \(CP^ 2\) has been studied by several authors: \textit{E. Calabi} [Ann. Math., II. Ser. 58, 1-23 (1953; Zbl 0051.131)], \textit{S. Bando} and \textit{Y. Ohnita} [J. Math. Soc. Japan 39, 475-487 (1987; Zbl 0609.53017)] etc. The purpose of this paper is to give some results for this question. The author reduces the study of the rigidity of superminimal immersions in \(CP^ 2\) to those immersions of the form \(\partial g\) for some holomorphic curve g. Rigidity theorem: ``Let g: \(M\to CP^ 2\) be a nondegenerate holomorphic curve with the dual curve \(\hat g.\) If the ramified points of g and \(\hat g\) are disjoint and are all of index \(\geq 2\), then \(\partial g\) is a superminimal immersion which is rigid among all superminimal immersions, i.e. any other superminimal immersion with the same pull-back metric as that of \(\partial g\) must be unitarily equivalent to \(\partial g\) up to complex conjugation.'' Examples of various genera are constructed.
reviewed by: Costache Apreutesei
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 4180270
zbMATH Keywords: Fermat variety
zbMATH Keywords: rational curve
zbMATH Keywords: rigidity
zbMATH Keywords: superminimal immersions
zbMATH Keywords: Riemann surfaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227621
OpenAlex ID: W2072670234
cites work: Q4177796
cites work: On Minimal Immersions of S 2 Into S 2m
cites work: On conformal minimal immersions of \(S^ 2\) into \({\mathbb{C}}P^ n\)
cites work: Minimal 2-spheres with constant curvature in \(P_ n({\mathbb C})\)
cites work: Conformal and minimal immersions of compact surfaces into the 4-sphere
cites work: Q4745011
cites work: Isometric imbedding of complex manifolds
cites work: Minimal Surfaces by Moving Frames
cites work: Harmonic maps of the two-sphere into a complex Grassmann manifold. II
cites work: Rigidity of Pseudo-Holomorphic Curves of Constant Curvature in Grassmann Manifolds
cites work: A rigidity theorem for properly embedded minimal surfaces in \(R^ 3\)
cites work: Gauss parametrizations and rigidity aspects of submanifolds
cites work: Harmonic maps from surfaces to complex projective spaces
cites work: The fundamental equations of minimal surfaces in \({\mathbb{C}}P^ 2\)
cites work: Q5571530
cites work: Q4195061
cites work: Q4128900
cites work: Equivalence problem and automorphism groups of certain compact Riemann surfaces
cites work: Q3220682
cites work: Q3924318
cites work: Rigidity of minimal surfaces in S 3
cites work: Q4771357
cites work: Q5799167
cites work: On Minimal Surfaces in a Kahler Manifold of Constant Holomorphic Sectional Curvature
DOI: 10.2748/TMJ/1178227621
Recommended article: Rigidity of superminimal surfaces in complex projective spaces
Recommended article: A Rigidity Result for Holomorphic Immersions of Surfaces in C P n
Recommended article: Q4219016
Recommended article: Q3973954
Recommended article: Q4305702
Recommended article: Rigidity of conformal minimal immersions of constant curvature from \(S^2\) to \(Q_4\)
Recommended article: Rigidity of superconformal minimal surfaces lying fully in odd-dimensional unit spheres
Recommended article: Rigidity in the class of orientable compact surfaces of minimal total absolute curvature
Recommended article: The Riemannian geometry of superminimal surfaces in complex space forms
Recommended article: On minimal immersions of \(R^ 2\) into \(P^ n(C)\)
description: scientific article; zbMATH DE number 4180270