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Local characterizations of complex curves in $C^{2}$ and sphere Schwarz maps

Naoya Ando

Graduate School of Science and Technology, Kumamoto University, 2–39–1 Kurokami, Kumamoto 860–8555, Japan

E-mail Address: ando@sci.kumamoto-u.ac.jp

https://doi.org/10.1142/S0129167X16500671Cited by:2 (Source: Crossref)

Abstract

In this paper, we obtain a local characterization of a complex curve in $C^{2}$ in terms of the induced metric and a holomorphic cubic differential $Φ$ , and we see that any complex curve in $C^{2}$ is locally represented as the image by the composition of an affine Schwarz map and a parallel translation in $C^{2}$ . We obtain a local characterization of a sphere Schwarz immersion in terms of a positive-valued function $ρ$ .

Keywords:

AMSC: 53C42, 32H02, 35N10, 53B25

References

1. E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom. 1 (1967) 111–125. Crossref, Google Scholar
2. S. Fujimori, M. Noro, K. Saji, T. Sasaki and M. Yoshida, Schwarz maps for the hypergeometric differential equation, Internat. J. Math. 26 (2015) 31, Article ID: 1541002. Link, Web of Science, Google Scholar
3. R. Kobayashi, T. Nishizaka, S. Shinzato and M. Yoshida, Affine Schwarz map for the hypergeometric differential equation, Funkcial. Ekvac. 51 (2008) 281–305. Crossref, Google Scholar
4. M. J. Micallef, Stable minimal surfaces in Euclidean space, J. Differential Geom. 19 (1984) 57–84. Crossref, Web of Science, Google Scholar
5. T. Sasaki, K. Yamada and M. Yoshida, The hyperbolic Schwarz map for the hypergeometric differential equation, Exp. Math. 17 (2008) 269–282. Crossref, Web of Science, Google Scholar
6. T. Sasaki, K. Yamada and M. Yoshida, Derived Schwarz map of the hypergeometric differential equation and a parallel family of flat fronts, Internat. J. Math. 19 (2008) 847–863. Link, Web of Science, Google Scholar
7. M. Yoshida, Affine and sphere Schwarz maps for the hypergeometric differential equation, Kumamoto J. Math. 29 (2016) 35–54. Google Scholar