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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Harmonic maps and classical surface theory in Minkowski $3$-space
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by Tilla Klotz Milnor PDF
Trans. Amer. Math. Soc. 280 (1983), 161-185 Request permission

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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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 280 (1983), 161-185
  • MSC: Primary 58E20; Secondary 53C42, 53C50
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0712254-7
  • MathSciNet review: 712254