Harmonic maps and classical surface theory in Minkowski 3-space

Milnor, Tilla Klotz

doi:10.1090/S0002-9947-1983-0712254-7

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