Mapping surfaces harmonically into 𝐸ⁿ

Milnor, Tilla Klotz

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

Mapping surfaces harmonically into $E^{n}$
HTML articles powered by AMS MathViewer

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.

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

Abstract Weingarten surfaces

Lectures on minimal submanifolds

Robert Osserman, A survey of minimal surfaces, Van Nostrand Reinhold Co., New York-London-Melbourne, 1969. MR 0256278