Mapping surfaces harmonically into 𝐸ⁿ

Milnor, Tilla Klotz

doi:10.1090/S0002-9939-1980-0550511-0

Mapping surfaces harmonically into $E^{n}$
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by Tilla Klotz Milnor

Proc. Amer. Math. Soc. 78 (1980), 269-275

DOI: https://doi.org/10.1090/S0002-9939-1980-0550511-0

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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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Abstract Weingarten surfaces

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