Rigidity of superminimal immersions of compact Riemann surfaces into \(CP^ 2\)
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.
scientific article; zbMATH DE number 4180270
scientific article