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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Mapping surfaces harmonically into $ E\sp{n}$


Author: Tilla Klotz Milnor
Journal: Proc. Amer. Math. Soc. 78 (1980), 269-275
MSC: Primary 53A05; Secondary 53A05, 58E20
MathSciNet review: 550511
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Abstract: A Weierstrass representation is given for harmonic maps from simply connected surfaces into $ {E^3}$. The main result implies that the normals to a complete, harmonically immersed surface in $ {E^n}$ cannot omit a neighborhood of an (unoriented) direction if the mean curvature vector never vanishes, and the map from given to induced conformal structure is quasiconformal. In particular, the closure of the Gauss map to the complete graph of a harmonic function must be a hemisphere if the mean curvature never vanishes, and vertical projection is quasiconformal.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0550511-0
Article copyright: © Copyright 1980 American Mathematical Society