On the isotropy of compact minimal surfaces in \({\mathbb{C}}P^ n\)
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)].
scientific article; zbMATH DE number 4092292
scientific article