Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space
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The Gauss map of a spacelike constant mean curvature hypersurface of Minkowski space is a harmonic map to hyperbolic space.The properties of such hypersurfaces are interpreted in terms of the harmonic mapping.Given an arbitrary closed set in the ideal boundary at infinity of hyperbolic space, there exists a complete entire spacelike constant mean curvature hypersurface whose Gauss map is a diffeomorphism onto the interior of the hyperbolic space convex hull of the set.Identifying ideal infinity with the light cone, this set corresponds to the lightlike directions of the hypersurface.In terms of this extrinsic data we give conditions for the hyperbolicity or parabolicity of this hypersurface.For example, if the set of lightlike directions has nonempty interior in the unit sphere, then this hypersurface can be constructed so as to admit nontrivial bounded harmonic functions.This gives many new examples of harmonic maps of the disk and the complex plane to the hyperbolic plane, which are of full rank.
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The Calabi-Simons identities (18)-(20) hold for K ∈ S^2_TT(T*Σ, g).
“Lemma 3.1. The following identities hold for K ∈ S 2 TT ( T ∗ Σ , g ) .”
From On the global evolution problem in 2 + 1 gravity (1997) · cites this paper as [1]
In §1 (page unavailable), the paper gives a commutation formula for the second fundamental form and a Bochner-type formula it calls a Simons identity. These support the general connection to Simons identities, but the passages concern the second fundamental form of a spacelike hypersurface, not the stated identities (18)–(20) for an arbitrary transverse-traceless tensor K. (LLM, conf=0.83)
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Research works printed but not linked: 8
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On the global evolution problem in 2 + 1 gravity
Anderson, L., Moncrief, V., & Tromba, A. J. (1997) · Journal of Geometry and Physics · doi:10.1016/s0393-0440(97)87804-7 · Cited by 72 · see citing works
- Authors
- Choi, H. I., & Treibergs, A.
- Venue
- Journal of Differential Geometry
- Publisher
- International Press of Boston
- Published
- 1990-01-01
- Volume
- 32
- Issue
- 3
- Pages
- 775-817
- ISSN
- 0022-040X (ISSN-L)
- OpenAlex
- W1832933695
- Cited by
- 88
- References
- 0
- Citations by year
- 2025: 2 · 2024: 5 · 2022: 2 · 2021: 1 · 2019: 2 · 2018: 2 · 2017: 1 · 2016: 4 · 2015: 3 · 2014: 1 · 2013: 3
- FWCI
- 6.79
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Journal of Differential Geometry (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Funders
- National Science Foundation (8700783, 8503327)
- Alfred P. Sloan Foundation
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- Gauss map, spacelike hypersurfaces, constant mean curvature, Minkowski space, harmonic maps, hyperbolic space, convex hull, hyperbolicity, parabolicity, bounded harmonic functions
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