Curvature restrictions on convex, timelike surfaces in Minkowski 3-space

Lin, Senchun

doi:10.1090/S0002-9939-99-05533-1

Curvature restrictions on convex, timelike surfaces in Minkowski 3-space
HTML articles powered by AMS MathViewer

by Senchun Lin PDF

Proc. Amer. Math. Soc. 128 (2000), 1459-1466 Request permission

Abstract:

Suppose that $K$ and $H$ are Minkowski Gauss curvature and Minkowski mean curvature respectively on a timelike surface $S$ that is $C^{2}$ immersed in Minkowski 3-space $E^{3}_{1}$. Suppose also that $0\not \equiv K < 0$ and that $S$ is complete as a surface in the underlying Euclidean 3-space $E^{3}$. It is shown that neither $K$ nor $H$ can be bounded away from zero on such a surface $S$.

References

Herbert Busemann, Convex surfaces, Interscience Tracts in Pure and Applied Mathematics, no. 6, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. MR 0105155
Eugenio Calabi, Examples of Bernstein problems for some nonlinear equations, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 223–230. MR 0264210
S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 385749, DOI 10.1002/cpa.3160280303
—, Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. (2)104(1976), 407-419.
Frank J. Flaherty, Surfaces d’onde dans l’espace de Lorentz-Minkowski, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 9, A521–A523. MR 431062
Philip Hartman and Louis Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901–920. MR 126812, DOI 10.2307/2372995
Tilla Klotz and Robert Osserman, Complete surfaces in $E^{3}$ with constant mean curvature, Comment. Math. Helv. 41 (1966/67), 313–318. MR 211332, DOI 10.1007/BF02566886
Peter Kohlmann, Splitting and pinching for convex sets, Geom. Dedicata 60 (1996), no. 2, 125–143. MR 1384423, DOI 10.1007/BF00160618
Senchun Lin, Splitting and pinching for complete convex surfaces in $E^3$, Geom. Dedicata 69 (1998), no. 1, 83–90. MR 1607544, DOI 10.1023/A:1004972306407
—, A mean cuvature pinching theorem for convex hypersurfaces in $E^{n+1}$. Preprint.
—, Spacelike and timelike convex hypersurfaces in Minkowski space with $H\ge c>0$, Preprint.
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
Tilla Klotz Milnor, A conformal analog of Bernstein’s theorem for timelike surfaces in Minkowski $3$-space, The legacy of Sonya Kovalevskaya (Cambridge, Mass., and Amherst, Mass., 1985) Contemp. Math., vol. 64, Amer. Math. Soc., Providence, RI, 1987, pp. 123–132. MR 881459, DOI 10.1090/conm/064/881459
Tilla Klotz Milnor, Entire timelike minimal surfaces in $E^{3,1}$, Michigan Math. J. 37 (1990), no. 2, 163–177. MR 1058389, DOI 10.1307/mmj/1029004123
O’Neill B,. Semi-Riemannian Geometry with Applications to Relativity, Academic Press (1983).
Richard Sacksteder, On hypersurfaces with no negative sectional curvatures, Amer. J. Math. 82 (1960), 609–630. MR 116292, DOI 10.2307/2372973
Andrejs E. Treibergs, Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math. 66 (1982), no. 1, 39–56. MR 652645, DOI 10.1007/BF01404755
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Tilla Weinstein, An introduction to Lorentz surfaces, De Gruyter Expositions in Mathematics, vol. 22, Walter de Gruyter & Co., Berlin, 1996. MR 1405166, DOI 10.1515/9783110821635
H. Wu, The spherical images of convex hypersurfaces, J. Differential Geometry 9 (1974), 279–290. MR 348685