Abstract
A maximal surface \({\mathcal{S}}\) with isolated singularities in a complete flat Lorentzian 3-manifold
is said to be entire if it lifts to a (periodic) entire multigraph \({\sim\mathcal{S}}\) in \({\mathbb{L}^3}\) . In addition, \({\mathcal{S}}\) is called of finite type if it has finite topology, finitely many singular points and \({\sim\mathcal{S}}\) is a finitely sheeted multigraph. Complete (or proper) maximal immersions with isolated singularities in \({\mathcal{N}}\) are entire, and entire embedded maximal surfaces in \({\mathcal{N}}\) with a finite number of singularities are of finite type. We classify complete flat Lorentzian 3-manifolds carrying entire maximal surfaces of finite type, and deal with the topology, Weierstrass representation and asymptotic behavior of this kind of surfaces. Finally, we construct new examples of periodic entire embedded maximal surfaces in \({\mathbb{L}^3}\) with fundamental piece having finitely many singularities.
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References
Ahlfors L.V. and Sario L. (1960). Riemann surfaces. Princeton University Press, Princeton
Barbot T. and Zeghib A. (2004). Group actions on Lorentz spaces, Mathematical aspects: a survey. In: Chrusciel, P.T. and Friedrich, H. (eds) The Einstein equations in the large scale behaviour of gravitational fields: 50 years of the Cauchy problem in general relativity, pp. Birkhäuser Verlag, Basel
Bartnik R. and Simon L. (1982). Spacelike hypersurfaces with prescribed boundary values and mean curvature. Comm. Math. Phys. 87: 131–152
Bernal A.N. and Sánchez M. (2003). On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Comm. Math. Phys. 243(3): 461–470
Calabi E. (1970). Examples of the Bernstein problem for some nonlinear equations. Proc. Symp. Pure Math. 15: 223–230
Charette V., Drumm T., Goldman W. and Morrill M. (2003). Complete flat affine and Lorentzian manifolds. Geom. Dedicata 97: 187–198
Cheng S.Y. and Yau S.T. (1976). Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces. Ann. Math. 104: 407–419
Fernández I., López F.J., Souam R.: The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz–Minkowski space \({\mathbb{L}^3}\) . Math. Ann. 332(3), 605–643
Fernández, I., López, F.J., Souam, R.: The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz–Minkowski space \({\mathbb{L}^3}\) (preprint) Ar**v e-print archive math.DG/0412190 v1
Goldman W., Margulis G.A.: Flat Lorentz 3-manifolds and cocompact Fuchsian groups. Contemp. Math. 262, Amer. Math. Soc., Providence, 2000, 135–145
Grigor’yan A. (1999). Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull Amer. Math. Soc. 36: 135–249
He Z.-X. and Schramm O. (1993). Fixed points, Koebe uniformization and circle packings. Ann. Math. 137(2): 369–406
Huber A. (1957). On subharmonic functions and differential geometry in the large. Comment. Math. Helv. 32: 13–72
Jorge L. and Meeks W.H. (1983). The topology of complete minimal surfaces of finite total Gaussian curvature. Topology 2: 203–221
Klyachin V.A. and Miklyukov V.M. (2003). Geometric structures of tubes and bands of zero mean curvature in Minkowski space. Ann. Acad. Sci. Fenn. Math. 28: 239–270
Kobayashi O. (1984). Maximal surfaces with conelike singularities. J. Math. Soc. Japan 36(4): 609–617
López, F.J., López, R., Souam, R.: Maximal surfaces of Riemann type in Lorentz–Minkowski space \({\mathbb{L}^3}\) . Mich. J. Math. 47, 469–497 (2000)
Margulis G.A. (1987). Complete affine locally flat manifolds with a free fundamental group. J. Soviet Math. 134: 129–134
Rosenberg H. and Meeks W.H. (1993). The geometry of periodic minimal surfaces. Comment Math. Helv. 68(4): 538–578
Mess, G.: Lorentz spacetimes of constant curvature. Institut des Hautes Études Scientifiques (preprint 1990)
Marsden J.E. and Tipler F.J. (1980). Maximal hypersurfaces and foliations of constant mean curvature in general relativity. Phys. Rep. 66(3): 109–139
O’Neill B. (1983). Semmi-riemannian geometry. Academic Press, New York
Osserman R. (1986). A survey of minimal surfaces. 2nd edn. Dover Publications, New York
Wolf J. (1967). Spaces of Constant curvature. McGraw-Hill, New York
Umehara M. and Yamada K. (2006). Maximal surfaces with singularities in Minkowski space. Hokkaido Math. J. 35(1): 13–40
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Fernández, I., López, F.J. Periodic maximal surfaces in the Lorentz–Minkowski space \({\mathbb{L}^3}\) . Math. Z. 256, 573–601 (2007). https://doi.org/10.1007/s00209-006-0087-y
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DOI: https://doi.org/10.1007/s00209-006-0087-y