Abstract
This paper concerns the global theory of properly embedded spacelike surfaces in three-dimensional Minkowski space in relation to their Gaussian curvature. We prove that every regular domain which is not a wedge is uniquely foliated by properly embedded convex surfaces of constant Gaussian curvature. This is a consequence of our classification of surfaces with bounded prescribed Gaussian curvature, sometimes called the Minkowski problem, for which partial results were obtained by Li, Guan-Jian-Schoen, and Bonsante-Seppi. Some applications to minimal Lagrangian self-maps of the hyperbolic plane are obtained.
Highlights
Minkowski 3-space is the connected geodesically complete flat Lorentzian manifold R2,1 = (R3, d x12 + d x22 − d x32)
The aim of the paper is to provide a full classification of properly embedded spacelike surfaces with constant Gaussian curvature (CGC) in Minkowski space in terms of their asymptotic behavior
First we study some special CGC surfaces whose domain of dependence is the future of a spacelike half-line in Minkowski space
Summary
5 an entire CGC-K for which φ is finite at exactly three points, i.e. one whose domain of dependence is the intersection of the future of three null planes This surface and the refined comparison principle are the key new ingredients to prove Theorem F. Using the symmetry of the embedding σF we reduce the problem to showing that the image of a line of symmetry is a properly embedded curve in Minkowski space This is proved by studying the growth of the holomorphic energy of the harmonic map along the curve and its relation with the principal curvature of this isometric immersion. First we study some special CGC surfaces whose domain of dependence is the future of a spacelike half-line in Minkowski space Those surfaces and our comparison principle will be the key ingredients to prove Theorem E.
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