On the uniqueness of the helicoid and Enneper’s surface in the Lorentz-Minkowski space ℝ₁³

Fernandez, Isabel; Lopez, Francisco J.

doi:10.1090/S0002-9947-2011-05133-0

On the uniqueness of the helicoid and Enneper’s surface in the Lorentz-Minkowski space $\mathbb {R}_1^3$
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by Isabel Fernandez and Francisco J. Lopez PDF

Trans. Amer. Math. Soc. 363 (2011), 4603-4650 Request permission

Abstract:

In this paper we deal with the uniqueness of the Lorentzian helicoid and Enneper’s surface among properly embedded maximal surfaces with lightlike boundary of mirror symmetry in the Lorentz-Minkowski space $\mathbb {R}_1^3.$

References

Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911
Robert Bartnik and Leon Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys. 87 (1982/83), no. 1, 131–152. MR 680653
Tobias H. Colding and William P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. IV. Locally simply connected, Ann. of Math. (2) 160 (2004), no. 2, 573–615. MR 2123933, DOI 10.4007/annals.2004.160.573
Klaus Ecker, Area maximizing hypersurfaces in Minkowski space having an isolated singularity, Manuscripta Math. 56 (1986), no. 4, 375–397. MR 860729, DOI 10.1007/BF01168501
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
Shiu Yuen Cheng and Shing Tung Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. (2) 104 (1976), no. 3, 407–419. MR 431061, DOI 10.2307/1970963
Pascal Collin, Robert Kusner, William H. Meeks III, and Harold Rosenberg, The topology, geometry and conformal structure of properly embedded minimal surfaces, J. Differential Geom. 67 (2004), no. 2, 377–393. MR 2153082
Isabel Fernández, Francisco J. López, and Rabah Souam, The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space, Math. Ann. 332 (2005), no. 3, 605–643. MR 2181764, DOI 10.1007/s00208-005-0642-6
Shoichi Fujimori, Kentaro Saji, Masaaki Umehara, and Kotaro Yamada, Singularities of maximal surfaces, Math. Z. 259 (2008), no. 4, 827–848. MR 2403743, DOI 10.1007/s00209-007-0250-0
Hirotaka Fujimoto, On the number of exceptional values of the Gauss maps of minimal surfaces, J. Math. Soc. Japan 40 (1988), no. 2, 235–247. MR 930599, DOI 10.2969/jmsj/04020235
Hirotaka Fujimoto, Modified defect relations for the Gauss map of minimal surfaces, J. Differential Geom. 29 (1989), no. 2, 245–262. MR 982173
Alexander Grigor′yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 2, 135–249. MR 1659871, DOI 10.1090/S0273-0979-99-00776-4
A. A. Klyachin and V. M. Miklyukov, Traces of functions with spacelike graphs and a problem of extension with constraints imposed on the gradient, Mat. Sb. 183 (1992), no. 7, 49–64 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 76 (1993), no. 2, 305–316. MR 1186974, DOI 10.1070/SM1993v076n02ABEH003415
Osamu Kobayashi, Maximal surfaces in the $3$-dimensional Minkowski space $L^{3}$, Tokyo J. Math. 6 (1983), no. 2, 297–309. MR 732085, DOI 10.3836/tjm/1270213872
Osamu Kobayashi, Maximal surfaces with conelike singularities, J. Math. Soc. Japan 36 (1984), no. 4, 609–617. MR 759417, DOI 10.2969/jmsj/03640609
Peter Li and Jiaping Wang, Finiteness of disjoint minimal graphs, Math. Res. Lett. 8 (2001), no. 5-6, 771–777. MR 1879819, DOI 10.4310/MRL.2001.v8.n6.a7
Francisco J. López and Joaquín Pérez, Parabolicity and Gauss map of minimal surfaces, Indiana Univ. Math. J. 52 (2003), no. 4, 1017–1026. MR 2001943, DOI 10.1512/iumj.2003.52.2250
Joaquín Pérez, Stable embedded minimal surfaces bounded by a straight line, Calc. Var. Partial Differential Equations 29 (2007), no. 2, 267–279. MR 2307776, DOI 10.1007/s00526-006-0069-2
Joaquín Pérez and Antonio Ros, Properly embedded minimal surfaces with finite total curvature, The global theory of minimal surfaces in flat spaces (Martina Franca, 1999) Lecture Notes in Math., vol. 1775, Springer, Berlin, 2002, pp. 15–66. MR 1901613, DOI 10.1007/978-3-540-45609-4_{2}
Jerrold E. Marsden and Frank J. Tipler, Maximal hypersurfaces and foliations of constant mean curvature in general relativity, Phys. Rep. 66 (1980), no. 3, 109–139. MR 598585, DOI 10.1016/0370-1573(80)90154-4
William H. Meeks III and Harold Rosenberg, The uniqueness of the helicoid, Ann. of Math. (2) 161 (2005), no. 2, 727–758. MR 2153399, DOI 10.4007/annals.2005.161.727
Norbert Quien, Plateau’s problem in Minkowski space, Analysis 5 (1985), no. 1-2, 43–60. MR 791491, DOI 10.1524/anly.1985.5.12.43
Joel L. Schiff, Normal families, Universitext, Springer-Verlag, New York, 1993. MR 1211641, DOI 10.1007/978-1-4612-0907-2
J. Pérez: Parabolicity and minimal surfaces. Proceedings of the conference on global theory of minimal surfaces, Berkeley (2003).
Masaaki Umehara and Kotaro Yamada, Maximal surfaces with singularities in Minkowski space, Hokkaido Math. J. 35 (2006), no. 1, 13–40. MR 2225080, DOI 10.14492/hokmj/1285766302