Complete constant Gaussian curvature surfaces in the Minkowski space and harmonic diffeomorphisms onto the hyperbolic plane (Q1876279)

The authors obtain the classification of complete space-like surfaces with negative constant Gaussian curvature in the Minkowski space. Let \(({\mathbb R}^3, g)\) be the Minkowski \(3\)-space endowed with the inner product \(g = dx_1^2 + dx_2^2 - dx_3^2\) and let \({\mathbb H}^2 = \{ (x_1, x_2, x_3)\mid x_1^2 + x_2^2 - x_3^2 = -1\}\) which is a two-sheeted hyperboloid with constant Gaussian curvature \(-1\) with respect to the induced metric, and let \({\mathbb H}^2_+ = {\mathbb H}^2 \cap \{x_3 > 0\}\). Denote by \({\mathcal A}_K\) the set of all complete space-like immersions in \(({\mathbb R}^3, g)\) with constant negative Gaussian curvature \(K < 0\), where congruent immersions are identified by isometries of \(({\mathbb R}^3, g)\), and denote by \({\mathcal B}\) the set of all harmonic diffeomorphisms from the unit disk \(D\) or the complex plane \({\mathbb C}\) onto the hyperbolic plane \({\mathbb H}^2_+\), where two harmonic diffeomorphisms \(h_1, h_2\) are identified if there exist a conformal equivalence \(\varphi\) on \(D\) or \({\mathbb C}\) and an isometry \(f\) on \({\mathbb H}^2_+\) such that \(h_1 = f \circ h_2 \circ \varphi\). The authors prove that if \(x: S \to ({\mathbb R}^3, g)\) is a space-like immersion with negative constant Gaussian curvature, then the Gauss map (unit normal vector field on \(S\)) is harmonic into the hyperbolic plane \({\mathbb H}^2_+\). They also prove the converse of this result. Namely, if \(N : S \to {\mathbb H}^2_+\) is a harmonic local diffeomorphism, then there exists, up to translations, a unique immersion with negative constant Gaussian curvature \(K\) such that \(N\) (or \(-N\)) is its Gauss map and its conformal structure is induced by the second fundamental form of the immersion. Using these results, the authors obtain their main result which says that there is an one-to-one correspondence between \({\mathcal A}_K\) and \({\mathcal B}\) for all \(K < 0\). This implies that complete space-like surfaces with constant negative Gaussian curvature are classified in terms of harmonic diffeomorphisms from Riemann surfaces onto the hyperbolic plane.