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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Harmonic maps and classical surface theory in Minkowski $ 3$-space


Author: Tilla Klotz Milnor
Journal: Trans. Amer. Math. Soc. 280 (1983), 161-185
MSC: Primary 58E20; Secondary 53C42, 53C50
MathSciNet review: 712254
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Abstract: Harmonic maps from a surface $ S$ with nondegenerate prescribed and induced metrics are characterized, showing that holomorphic quadratic differentials play the same role for harmonic maps from a surface with indefinite prescribed metric as they do in the Riemannian case. Moreover, holomorphic quadratic differentials are shown to arise as naturally on surfaces of constant $ H$ or $ K$ in $ {M^3}$ as on their counterparts in $ {E^3}$. The connection between the sine-Gordon, $ \sinh$-Gordon and $ \cosh$-Gordon equations and harmonic maps is explained. Various local and global results are established for surfaces in $ {M^3}$ with constant $ H$, or constant $ K \ne 0$. In particular, the Gauss map of a spacelike or timelike surface in $ {M^3}$ is shown to be harmonic if and only if $ H$ is constant. Also, $ K$ is shown to assume values arbitrarily close to $ {H^2}$ on any entire, spacelike surface in $ {M^3}$ with constant $ H$, except on a hyperbolic cylinder.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0712254-7
PII: S 0002-9947(1983)0712254-7
Article copyright: © Copyright 1983 American Mathematical Society