Lower bounds for the extrinsic total curvatures of a space-like codimension $2$ surface in Minkowski space
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- by Marek Kossowski PDF
- Proc. Amer. Math. Soc. 109 (1990), 787-795 Request permission
Abstract:
There are three invariant curvature functions defined on any smooth space-like $2$-surfaces in four-dimensional Minkowski space. (If the surface lies in a Euclidean hyperplane then the functions agree with ${H^2},{K^2}$, and ${\left ( {{H^2} - K} \right )^2}$. For each of these functions we show that there exists a space-like immersion of any oriented compact (or noncompact complete) surface with associated total curvature arbitrarily small.References
- John K. Beem and Paul E. Ehrlich, Global Lorentzian geometry, Monographs and Textbooks in Pure and Applied Mathematics, vol. 67, Marcel Dekker, Inc., New York, 1981. MR 619853
- Marek Kossowski, The $S^2$-valued Gauss maps and split total curvature of a space-like codimension-$2$ surface in Minkowski space, J. London Math. Soc. (2) 40 (1989), no. 1, 179–192. MR 1028922, DOI 10.1112/jlms/s2-40.1.179
- Marek Kossowski, The intrinsic conformal structure and Gauss map of a light-like hypersurface in Minkowski space, Trans. Amer. Math. Soc. 316 (1989), no. 1, 369–383. MR 938920, DOI 10.1090/S0002-9947-1989-0938920-1
- Wolfgang Kühnel and Ulrich Pinkall, On total mean curvatures, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 148, 437–447. MR 868618, DOI 10.1093/qmath/37.4.437
- Tilla Klotz Milnor, Harmonic maps and classical surface theory in Minkowski $3$-space, Trans. Amer. Math. Soc. 280 (1983), no. 1, 161–185. MR 712254, DOI 10.1090/S0002-9947-1983-0712254-7
- Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023 R. Penrose, Battille Recontares, Benjamin, NY, 1968. T. J. Wilmore, Total curvature in Riemannian geometry, Wiley, 1982.
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 109 (1990), 787-795
- MSC: Primary 53C50
- DOI: https://doi.org/10.1090/S0002-9939-1990-1013972-5
- MathSciNet review: 1013972