Abstract
We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a flat spacetime of dimension \((2+1).\) The proof follows from polyhedral approximation.
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References
Ahlfors, L.V.: Lectures on Quasiconformal Mappings, 2nd edn. University Lecture Series, vol. 38. American Mathematical Society, Providence (2006) (With supplemental chapters by C.J. Earle, I. Kra, M. Shishikura and J.H. Hubbard)
Alexander, S., Kapovitch, V., Petrunin, A.: Invitation to Alexandrov Geometry: CAT[0] Spaces. arXiv:1701.03483
Alexandrov, A.D.: A. D. Alexandrov selected works. Part II. In: Kutateladze, S.S. (ed.) Intrinsic Geometry of Convex Surfaces. Chapman & Hall/CRC, Boca Raton (2006) (Translated from the Russian by S. Vakhrameyev)
Aleksandrov, A.D., Zalgaller, V.A.: Intrinsic Geometry of Surfaces. Translations of Mathematical Monographs, vol. 15. American Mathematical Society, Providence (1967) (Translated from the Russian by J. M. Danskin)
Burago, D., Burago, Yu., Ivanov, S.: A Course in Metric Geometry. American Mathematical Society (AMS), Providence (2001)
Barbot, T., Béguin, F., Zeghib, A.: Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes: application to the Minkowski problem in the Minkowski space. Ann. Inst. Fourier (Grenoble) 61(2), 511–591 (2011)
Bridson, M., Haefliger, A.: Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)
Brunswic, L.: Surfaces de Cauchy polyédrales des espaces-temps plats singuliers. Ph.D. thesis, Université d’Avignon et des pays de Vaucluse (2017)
Burtscher, A.Y.: Length structures on manifolds with continuous Riemannian metrics. N. Y. J. Math. 21, 273–296 (2015)
Buser, P.: Geometry and Spectra of Compact Riemann surfaces. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston (2010) (Reprint of the 1992 edition)
Chen, B.-L., Yin, L.: Isometric embedding of negatively curved complete surfaces in Lorentz–Minkowski space. Pac. J. Math. 276(2), 347–367 (2015)
Fathi, A., Laudenbach, F., Poénaru, V.: Thurston’s Work on Surfaces. Princeton University Press, Princeton (2012) (Transl. from the French by Djun Kim and Dan Margalit)
Fillastre, F.: Fuchsian polyhedra in Lorentzian space-forms. Math. Ann. 350(2), 417–453 (2011)
Fillastre, F.: Fuchsian convex bodies: basics of Brunn–Minkowski theory. Geom. Funct. Anal. 23(1), 295–333 (2013)
Fillastre, F., Izmestiev, I., Veronelli, G.: Hyperbolization of cusps with convex boundary. Manuscr. Math. 150(3–4), 475–492 (2016)
Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton (2012)
Imayoshi, Y., Taniguchi, M.: An introduction to Teichmüller spaces. Springer, Tokyo (1992) (Translated and revised from the Japanese by the authors)
Labourie, F., Schlenker, J.-M.: Surfaces convexes fuchsiennes dans les espaces lorentziens à courbure constante. Math. Ann. 316(3), 465–483 (2000)
Papadopoulos, A.: Metric Spaces, Convexity and Nonpositive Curvature. IRMA Lectures in Mathematics and Theoretical Physics, vol. 6. European Mathematical Society (EMS), Zürich (2005)
Ratcliffe, J.: Foundations of Hyperbolic Manifolds, 2nd edn. Graduate Texts in Mathematics, vol. 149. Springer, New York (2006)
Rivin, I., Hodgson, C.D.: A characterization of compact convex polyhedra in hyperbolic \(3\)-space. Invent. Math. 111(1), 77–111 (1993)
Rivin, I.: On geometry of convex polyhedra in hyperbolic 3-space. Ph.D. thesis, Princeton University (1986)
Rockafellar, T.: Convex Analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997) (Reprint of the 1970 original, Princeton Paperbacks)
Schlenker, J.-M.: Hyperbolic manifolds with polyhedral boundary. arXiv:math/0111136
Schlenker, J.-M.: Surfaces convexes dans des espaces lorentziens à courbure constante. Commun. Anal. Geom. 4(1–2), 285–331 (1996)
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, expanded edn. Encyclopedia of Mathematics and Its Applications, vol. 151. Cambridge University Press, Cambridge (2014)
Sokolov, D.D.: The regularity of convex surfaces with definite metric in three-dimensional pseudo-Euclidean space. In: Problems in Geometry, vol. 8 (Russian), pp. 257–277, 280. Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow (1977)
Troyanov, M.: Les surfaces euclidiennes à singularités coniques. Enseign. Math. (2) 32(1–2), 79–94 (1986)
Acknowledgements
The authors want to thank Giona Veronelli who pointed out an error in a preceding version of this text. Most of this work was achieved when the second author was a post-doc in the AGM institute of the Cergy-Pontoise University. He wants to thank the institution for its support.
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Fillastre, F., Slutskiy, D. Embeddings of non-positively curved compact surfaces in flat Lorentzian manifolds. Math. Z. 291, 149–178 (2019). https://doi.org/10.1007/s00209-018-2076-3
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DOI: https://doi.org/10.1007/s00209-018-2076-3