On the periodicity of planes with parallel mean curvature vector in \(\mathbb{C} H^2\)
type: Item
instance of: scholarly article
title: On the periodicity of planes with parallel mean curvature vector in \(\mathbb{C} H^2\)
zbMATH Open document ID: 1081.53053
author: Shinya Hirakawa
published in: Tokyo Journal of Mathematics
publication date: 2005-03-14T00:00:00Z
review text: The paper studies isometric immersions of the Euclidean plane in the complex hyperbolic plane \(\mathbb C H^2\) with non-zero parallel mean curvature vector, and gives explicit conditions under which they are invariant under an action of \(\mathbb Z\times \mathbb Z\) or of \(\mathbb Z\). The proof (and actually the statement of the main result) uses a description of such parallel mean curvature isometric immersions which was given by \textit{K. Kenmotsu} and \textit{D. Zhou} [Am. J. Math. 122, 295--317 (2000; Zbl 0997.53006)]. They showed that such immersions are characterized by two numbers, one of which is the length of the mean curvature vector and the other one is defined in a less simple way in terms of two quadratic holomorphic differentials associated to the immersion. The paper under review shows that some inequalities on those two numbers determine whether the immersion is periodic.
reviewed by: Jean-Marc Schlenker
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 2145307
zbMATH Keywords: isometric
zbMATH Keywords: immersion
zbMATH Keywords: mean curvature
zbMATH Keywords: complex hyperbolic
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208405
OpenAlex ID: W1999548922
cites work: Slant immersions
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cites work: Minimal surfaces with constant curvature and Kähler angle in complex space forms
DOI: 10.3836/TJM/1244208405
description: scientific article; zbMATH DE number 2145307