On the isotropy of compact minimal surfaces in \({\mathbb{C}}P^ n\)
type: Item
instance of: scholarly article
title: On the isotropy of compact minimal surfaces in \({\mathbb{C}}P^ n\)
zbMATH Open document ID: 0667.53045
DOI: 10.1007/BF01230279
author: Gary R. Jensen
author: M. Rigoli
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173999
review text: Let f: \(M\to {\mathbb{C}}P^ n\) be a conformal harmonic map from the compact Riemann surface M into complex projective space. The set of points Z where df is of type (1,0) or (0,1), respectively, is discrete, and therefore, finite. Even more, to each point in Z one can associate an (integer) order and, indicating with \(\sigma_ n\) and \(\sigma_ 0\) the sum of the orders of the points where df is of type (1,0) and (0,1), respectively. The authors of this paper prove the following theorem: Let M be a compact Riemann surface of genus g and let f: \(M\to {\mathbb{C}}P^ n\) be a (weakly) conformal, non-holomorphic, full harmonic map. If \(| \sigma_ 0-\sigma_ n| >2(n+1)(g-1),\) then f is isotropic. This theorem contains a result of \textit{J. Eells} and \textit{J. C. Woods} [cf. Adv. Math. 49, 217-263 (1983; Zbl 0528.58007)]. The proof of the theorem depends on a refinement and a careful analysis of the procedure determining the Frenet frame along the surface f: \(M\to {\mathbb{C}}P^ n\) [cf. \textit{S.-S. Chern} and \textit{J. G. Wolfson}, Am. J. Math. 105, 59-83 (1983; Zbl 0521.53050)].
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 4092292
zbMATH Keywords: minimal surface
zbMATH Keywords: isotropic map
zbMATH Keywords: conformal harmonic map
zbMATH Keywords: Riemann surface
zbMATH Keywords: Frenet frame
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4092292