## 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.## References

- Lars V. Ahlfors and Leo Sario,
*Riemann surfaces*, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR**0114911**, DOI 10.1515/9781400874538 - Lipman Bers,
*Quasiconformal mappings and Teichmüller’s theorem*, Analytic functions, Princeton Univ. Press, Princeton, N.J., 1960, pp. 89–119. MR**0114898** - Richard L. Bishop and Richard J. Crittenden,
*Geometry of manifolds*, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR**0169148** - Shiing Shen Chern and Samuel I. Goldberg,
*On the volume decreasing property of a class of real harmonic mappings*, Amer. J. Math.**97**(1975), 133–147. MR**367860**, DOI 10.2307/2373664 - Tilla Klotz Milnor,
*Restrictions on the curvatures of $\Phi$-bounded surfaces*, J. Differential Geometry**11**(1976), no. 1, 31–46. MR**461382** - Tilla Klotz Milnor,
*Harmonically immersed surfaces*, J. Differential Geometry**14**(1979), no. 2, 205–214. MR**587548**
—, - Robert Osserman,
*A survey of minimal surfaces*, Van Nostrand Reinhold Co., New York-London-Melbourne, 1969. MR**0256278**

*Abstract Weingarten surfaces*, preprint. H. B. Lawson, Jr.,

*Lectures on minimal submanifolds*, Notas de Math., Inst. de Mat. Pura e Aplicada, Rio de Janeiro, 1974.

## Bibliographic 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