Harmonic maps and classical surface theory in Minkowski 3-space

Abstract

Harmonic maps from a surface S 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 H or K K in M 3 {M^3} as on their counterparts in E 3 {E^3} . The connection between the sine-Gordon, sinh \sinh -Gordon and cosh \cosh -Gordon equations and harmonic maps is explained. Various local and global results are established for surfaces in M 3 {M^3} with constant H H , or constant K ≠ 0 K \ne 0 . In particular, the Gauss map of a spacelike or timelike surface in M 3 {M^3} is shown to be harmonic if and only if H H is constant. Also, K K is shown to assume values arbitrarily close to H 2 {H^2} on any entire, spacelike surface in M 3 {M^3} with constant H H , except on a hyperbolic cylinder.