Abstract
We prove that for any open Riemann surface \({\mathcal{N}}\), natural number N ≥ 3, non-constant harmonic map \({h:\mathcal{N} \to \mathbb{R}}\) N−2 and holomorphic 2-form \({\mathfrak{H}}\) on \({\mathcal{N}}\) , there exists a weakly complete harmonic map \({X=(X_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}\) with Hopf differential \({\mathfrak{H}}\) and \({(X_j)_{j=3,\ldots,{\sc N}}=h.}\) In particular, there exists a complete conformal minimal immersion \({Y=(Y_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}\) such that \({(Y_j)_{j=3,\ldots,{\sc N}}=h}\) . As some consequences of these results (1) there exist complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect N hyperplanes of \({\mathbb{CP}^{{\sc N}-1}}\) in general position. (2) There exist complete non-proper embedded minimal surfaces in \({\mathbb{R}^{\sc N},}\) \({\forall\,{\sc N} >3 .}\)
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Ahlfors L.V.: The theory of meromorphic curves. Acta Soc. Sci. Fennicae. Nova Ser. A 3, 1–31 (1941)
Alarcón A., Fernández I.: Complete minimal surfaces in \({\mathbb {R}^3}\) with a prescribed coordinate function. Differ. Geom. Appl. 29(1 suppl), S9–S15 (2011)
Alarcón, A., Fernández, I., López, F.J.: Complete minimal surfaces and harmonic functions. Comment. Math. Helv. (2010, in press)
Alarcón A., López F.J.: Minimal surfaces in \({\mathbb {R}^3}\) properly projecting into \({\mathbb {R}^2}\) . J. Differ. Geom. (2012, in press)
Alarcón A., López F.J.: Null curves in \({\mathbb {C}^3}\) and Calabi-Yau conjectures. Math. Ann. (2012, in press)
Chern S.S.: Minimal Surfaces in an Euclidean Space of N Dimensions. Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 187–198. Princeton Univ. Press, Princeton (1965)
Chern S.S., Osserman R.: Complete minimal surfaces in euclidean n-space. J. Anal. Math. 19, 15–34 (1967)
Colding T.H., Minicozzi W.P.: The Calabi-Yau conjectures for embedded surfaces. Ann. Math. 167(2), 211–243 (2008)
Fujimoto H.: Extensions of the big Picard’s theorem. Tohoku Math. J. 24, 415–422 (1972)
Fujimoto H.: On the Gauss map of a complete minimal surface in \({\mathbb {R}^{m} }\) . J. Math. Soc. Jpn. 35, 279–288 (1983)
Fujimoto H.: Modified defect relations for the Gauss map of minimal surfaces. II. J. Differ. Geom. 31, 365–385 (1990)
Fujimoto H.: Examples of complete minimal surfaces in \({\mathbb {R}^m}\) whose Gauss maps omit m(m + 1)/2 hyperplanes in general position. Sci. Rep. Kanazawa Univ. 33, 37–43 (1988)
Jones P.W.: A complete bounded complex submanifold of \({\mathbb {C}^3}\) . Proc. Am. Math. Soc. 76, 305–306 (1979)
Jorge L.P.M., Xavier F.: A complete minimal surface in \({\mathbb {R}^3}\) between two parallel planes. Ann. Math. 112(2), 203–206 (1980)
Klotz Milnor T.: Mapping surfaces harmonically into E n. Proc. Am. Math. Soc. 78, 269–275 (1980)
Meeks III, W.H., Pérez, J., Ros A.: The embedded Calabi-Yau conjectures for finite genus (Preprint)
Osserman R.: Global properties of minimal surfaces in E 3 and E n. Ann. Math. 80(2), 340–364 (1964)
Osserman, R.: A Survey of Minimal Surfaces. Second edition. Dover Publications, Inc., New York, vi+207 pp. (1986)
Ru M.: On the Gauss map of minimal surfaces immersed in \({\mathbb {R}^n}\) . J. Differ. Geom. 34, 411–423 (1991)
Wu, H.: The Equidistribution Theory of Holomorphic Curves. Annals of Mathematics Studies, No. 64, Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1970)
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Alarcón, A., Fernández, I. & López, F.J. Harmonic mappings and conformal minimal immersions of Riemann surfaces into \({\mathbb {R}^{\rm N}}\) . Calc. Var. 47, 227–242 (2013). https://doi.org/10.1007/s00526-012-0517-0
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DOI: https://doi.org/10.1007/s00526-012-0517-0