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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Mapping surfaces harmonically into $E^{n}$
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by Tilla Klotz Milnor PDF
Proc. Amer. Math. Soc. 78 (1980), 269-275 Request permission

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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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 78 (1980), 269-275
  • MSC: Primary 53A05; Secondary 53A05, 58E20
  • DOI: https://doi.org/10.1090/S0002-9939-1980-0550511-0
  • MathSciNet review: 550511