Timelike minimal surfaces via loop groups. (English) Zbl 1185.53008
Summary: This work consists of two parts. In Part I, we shall give a systematic study of Lorentz conformal structure from structural viewpoints. We study manifolds with split-complex structure. We apply general results on split-complex structure for the study of Lorentz surfaces. In Part II, we study the conformal realization of Lorentz surfaces in the Minkowski 3-space via conformal minimal immersions. We apply loop group theoretic Weierstrass-type representation of timelike constant mean curvature for timelike minimal surfaces. Classical integral representation formula for timelike minimal surfaces will be recovered from loop group theoretic viewpoint.
MSC:
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 58E20 | Harmonic maps, etc. |
| 22E67 | Loop groups and related constructions, group-theoretic treatment |
References:
| [1] | Balan, V. and Dorfmeister, J.: A Weierstrass-type representation for harmonic maps from Riemann surfaces to general Lie groups, Balkan J. Geom. Appl. 5(2000), 7-37 (http://www.emis.de/journals/BJGA/5.1/2.html). · Zbl 0958.58012 |
| [2] | Balan, V. and Dorfmeister, J.: Birkhoff decomposition and Iwasawa decomposition for loop groups, Tôhoku Math. J.53(2001), 593-615. · Zbl 0996.22013 · doi:10.2748/tmj/1113247803 |
| [3] | Balan, V. and Dorfmeister, J.: Generalized Weierstrass type representation for harmonic maps into general symmetric spaces via loop groups, in preparation. · Zbl 1110.53051 |
| [4] | Barbashov, B. M. and Nesterenko, V. V.: Differential geometry and nonlinear field models, Fortschritte Phys.28(1980), 427-464. · doi:10.1002/prop.19800280802 |
| [5] | Barbashov, B. M., Nesterenko, V. V. and Chervjakov, A. M.: Generalization of the relativistic string model in the scope of the geometric approach, Theoret. Mat. Fiz.45(1980), 365-376 (in Russian), English translation: Theoret. Math. Phys.45(1980), 1082-1089. · Zbl 0461.53018 |
| [6] | Bracken, P. and Hayes, J.: On a formulation of certain integrable systems using hyperbolic numbers, Phys. Lett. A301(3-4) (2002), 191-194. · Zbl 0997.37048 · doi:10.1016/S0375-9601(02)00958-1 |
| [7] | Burns, J. T.: Curvature forms for Lorentz 2-manifold, Proc. Amer. Math. Soc.63(1977), 134-136. · Zbl 0349.53051 |
| [8] | Burstall, F. E. and Hertrich-Jeromin, U.: Harmonic maps in unfashionable geometries, Manuscripta Math.108(2002), 171-189. · Zbl 1010.53049 · doi:10.1007/s002290200269 |
| [9] | Brink, L. and Henneaux, M.: Principles of String Theory, Series of the Centro de Estudios Cientificos de Santiago, Plenum Press, New York, 1988. · doi:10.1007/978-1-4613-0909-3 |
| [10] | Clelland, J. N.: A Bäcklund transformation for timelike surfaces of constant mean curvature in ℝ1,2, In: Bäcklund and Darboux Transformations. The Geometry of Solitons (Halifax, NS, 1999), CRM Proc. Lecture Notes 29, Amer. Math. Soc., Providence, RI, 2001, pp. 141-150. · Zbl 0992.53012 |
| [11] | Cruceanu, V., Fortuny, P. and Gadea, P. M.: A survey on paracomplex geometry, Rocky Mountain J. Math.26(1996), 83-115. · Zbl 0856.53049 · doi:10.1216/rmjm/1181072105 |
| [12] | Dajczer, M.; Nomizu, K.; Hano, J. (ed.), On flat surfaces in S31 and H3 1, 71-108 (1981), Boston · doi:10.1007/978-1-4612-5987-9_5 |
| [13] | Deck, T.: A geometric Cauchy problem for timelike minimal surfaces, Ann. Global Anal. Geom.12(1994), 305-312. · Zbl 0830.53049 · doi:10.1007/BF02108303 |
| [14] | Deck, T.: Classical string dynamics with non-trivial topology, J. Geom. Phys.16(1995), 1-14. · Zbl 0830.53050 · doi:10.1016/0393-0440(94)00018-Y |
| [15] | Dorfmeister, J. and Eitner, U.: Weierstraß-type representation of affine spheres, Abh. Math. Sem. Hamburg71(2001), 225-250. · Zbl 1031.53021 · doi:10.1007/BF02941473 |
| [16] | Dorfmeister, J., Gradl, H. and Szmigielski, J.: Systems of PDEs obtained from factorization in loop groups, Acta Appl. Math.53(1998), 1-58. · Zbl 0922.35151 · doi:10.1023/A:1005947719355 |
| [17] | Dorfmeister, J.; Inoguchi, J.; Toda, M., Weierstraß type representations of timelike surfaces with constant mean curvature, 77-99 (2002), Providence, RI · Zbl 1035.53016 · doi:10.1090/conm/308/05313 |
| [18] | Dorfmeister, J., Pedit, F. and Toda, M.: Minimal surfaces via loop groups, Balkan J. Geom. Appl.2(1997), 25-40 (http://www.emis.de/journals/BJGA/2.1/4.html). · Zbl 0932.53006 |
| [19] | Dorfmeister, J., Pedit, F. and Wu, H.: Weierstrass type representations of harmonic maps into symmetric spaces, Comm. Anal. Geom.6(1998), 633-668. · Zbl 0932.58018 · doi:10.4310/CAG.1998.v6.n4.a1 |
| [20] | Dzan, J. J.: Gauss—Bonnet formula for general Lorentzian surfaces, Geom. Dedicata15(1984), 215-231. · Zbl 0535.53051 |
| [21] | Erdem, S.: Harmonic maps of Lorentz surfaces, quadratic differentials and paraholomorphicity, Bei. Alg. Geom.38(1997), 19-32 (http://www.emis.de/journals/BAG/vol.38/no.1/3.html). · Zbl 0879.58015 |
| [22] | Erdem, S.: Paracomplex projective models and harmonic maps into them, Bei. Alg. Geom.40(1999), 385-398 (http://www.emis.de/journals/BAG/vol.40/no.2/10.html). · Zbl 0957.53036 |
| [23] | Ferrández, A., Giménez, A. and Lucas, P.: Null helices in Lorentzian space forms, Int. J. Modern Phys. A16(2001), 4845-4863. · Zbl 1003.53052 · doi:10.1142/S0217751X01005821 |
| [24] | Fujimoto, A.: Theory of G-structures (Japanese), A Report in Differential Geometry, 1966, English translation: Publications of the Study Group of Geometry, Vol. 1, Department of Applied Mathematics, College of Liberal Arts and Science, Okayama University, Okayama, 1972. |
| [25] | Fujioka, A. and Inoguchi, J.: Timelike Bonnet surfaces in Lorentzian space forms, Differential Geom. Appl.18(2003), 103-111. · Zbl 1040.53018 · doi:10.1016/S0926-2245(02)00141-9 |
| [26] | Fujioka, A. and Inoguchi, J.: Timelike surfaces with harmonic inverse mean curvature, In: Surveys on Geometry and Integrable Systems, Advanced Studies in Pure Mathematics, Math. Soc. Japan, to appear. · Zbl 1172.53015 |
| [27] | Gadea, P. M. and Masqué, J. M.: A-differentiability and A-analyticity, Proc. Amer. Math. Soc.124(1996), 1437-1443. · Zbl 0846.30031 · doi:10.1090/S0002-9939-96-03070-5 |
| [28] | Glazebrook, J. F.: Strings, harmonic maps and hyperbolic systems, Comput. Math. Appl.19(8/9) (1990), 85-101. · Zbl 0693.58044 |
| [29] | Graves, L.: Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc.252(1979), 367-392. · Zbl 0415.53041 · doi:10.1090/S0002-9947-1979-0534127-4 |
| [30] | Gu, C. H.: On the motion of a string in a curved space-time, In: N. Hu (ed.), Proc. 3rd Grossmann Meeting on General Relativity, Beijing, 1983, pp. 139-142. |
| [31] | Gu, C. H.: The extremal surfaces in the 3-dimensional Minkowski space, Acta Math. Sinica, New Series1(1985), 173-180. · Zbl 0595.49027 · doi:10.1007/BF02560031 |
| [32] | Hitchin, N.: Monopoles, Minimal Surfaces and Algebraic Curves, Sém. Math. Sup. 105, Presses Univ. Montreal, 1987. · Zbl 0644.53059 |
| [33] | Hong, J. Q.: Timelike surfaces with mean curvature one in anti de Sitter 3-space, Kōdai Math. J.17(1994), 341-350. · Zbl 0812.53052 |
| [34] | Inoguchi, J.: Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo J. Math.21(1998), 141-152. · Zbl 0930.53042 · doi:10.3836/tjm/1270041992 |
| [35] | Kaneyuki, S.: Homogeneous paraKähler manifolds and dipolarizations in Lie algebras, RIMS Kokyuroku1069(1999), 119-139 (Japanese). · Zbl 0938.17008 |
| [36] | Kaneyuki, S. and Kozai, M.: Paracomplex structures and affine symmetric spaces, Tokyo J. Math.8(1985), 81-98. · Zbl 0585.53029 · doi:10.3836/tjm/1270151571 |
| [37] | Kimura, M.: Space of geodesics in hyperbolic spaces and Lorentz numbers, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci.36(2003), 61-67 (http://www.math.shimaneu. ac.jp/common/memoir/36/memoir36.htm). · Zbl 1049.53015 |
| [38] | Klarreich, N.: Smoothability of the conformal boundary of a Lorentz surface inplies ‘global smoothability’, Geom. Dedicata89(2002), 59-107. · Zbl 1003.53053 · doi:10.1023/A:1014283929609 |
| [39] | Kobayashi, S.: Transformation Groups in Differential Geometry, Ergeb. Math. Grenzgeb. 70, Springer-Verlag, 1972, reprinted as Springer Classics in Mathematics, 1995. · Zbl 0246.53031 |
| [40] | Kričever, I. M.: An analogue of the d’Alembert formula for the equation of a principla chiral field and the sine-Gordon equation, Dokl. Akad. Nauk SSSR253(1980), 288-292 (Russian), English translation: Soviet Math. Dokl.22(1980), 79-84 (1981). · Zbl 0496.35073 |
| [41] | Kričever, I. M. and Novikov, S. P.: Virasoro type algebras, Riemann surfaces and strings in Minkowski space, Functional. Anal. i Prilozhen.21(1987), 47-61. · Zbl 0659.17012 |
| [42] | Kulkarni, R.: An analogue of the Riemann mapping theorem for Lorentz metrics, Proc. Roy. Soc. London Ser. A401(1985), 117-130. · Zbl 0574.53040 · doi:10.1098/rspa.1985.0090 |
| [43] | Lelieuvre, M.: Sur les lignes asymptotiques et leur représentation sphérique, Bull. Sci. Math.12(1888), 126-128. · JFM 20.0756.01 |
| [44] | Libermann, P.: Sur le problème d’équivalence de certaines structures infinitésimales, Ann. Mat. Pura Appl.36(1954), 27-120. · Zbl 0056.15401 · doi:10.1007/BF02412833 |
| [45] | Lin, S. and Weinstein, T.: E3-complete timelike surfaces in E13 are globally hyperbolic, Michigan Math. J.44(3) (1997), 529-541. · Zbl 0897.53045 · doi:10.1307/mmj/1029005786 |
| [46] | Lin, S. and Weinstein, T.: A better conformal Bernstein’s theorem, Geom. Dedicata81(1-3) (2000), 53-59. · Zbl 0962.53036 · doi:10.1023/A:1005224005625 |
| [47] | Luo, F. and Stong, R.: Conformal embedding of a disc with a Lorentzian metric into the plane, Math. Ann.309(1997), 359-373. · Zbl 0891.53051 · doi:10.1007/s002080050117 |
| [48] | Magid, M. A.: The Bernstein problem for timelike surfaces, Yokohama Math. J.37(1989), 125-137. · Zbl 0716.53056 |
| [49] | Magid, M. A.: Timelike surfaces in Lorentz 3-space with prescribed mean curvature and Gauss map, Hokkaido Math. J.20(1991), 447-464. · Zbl 0753.53039 · doi:10.14492/hokmj/1381413979 |
| [50] | McNertney, L.: One-parameter families of surfaces with constant curvature in Lorentz 3-space, Ph.D. thesis, Brown University, 1980. |
| [51] | Milnor, T. K.: A conformal analog of Bernstein’s theorem in Minkowski 3-space, In: The Legacy of Sonya Kovalevskaya, Contemp. Math. 64, Amer. Math. Soc., Providence, RI, 1987, pp. 123-130. |
| [52] | Milnor, T. K.: Entire timelike minimal surfaces in E3,1,Michigan Math. J.37(1990), 163-177. · Zbl 0729.53015 · doi:10.1307/mmj/1029004123 |
| [53] | Milnor, T. K.: Associate harmonic immersions in 3-space, Michigan Math. J.37(1990), 179-190. · Zbl 0725.53004 · doi:10.1307/mmj/1029004124 |
| [54] | Mira, P. and Pastor, J. A.: Helicoidal maximal surfaces in Lorentz—Minkowski space, Monatsh. Math.140(2003), 315-334. · Zbl 1050.53014 · doi:10.1007/s00605-003-0111-9 |
| [55] | O’Neill, B. O.: Semi-Riemannian Geometry with Application to Relativity, Pure Appl. Math. 130, Academic Press, Orlando, 1983. · Zbl 0531.53051 |
| [56] | Poincaré, H.: La science et l’hypothèse, 1902. · JFM 34.0080.12 |
| [57] | Porteous, I. R.: Clifford Algebras and the Classical Groups, Cambridge Univ. Press, Cambridge, 1995. · Zbl 0855.15019 · doi:10.1017/CBO9780511470912 |
| [58] | Pressley, A. and Segal, G.: Loop Groups, OxfordMath. Monographs, Oxford Univ. Press, 1986. · Zbl 0618.22011 |
| [59] | Smith, J. W.: Lorentz structures on the plane, Trans. Amer. Math. Soc.95(1960), 226-237. · Zbl 0171.42202 · doi:10.1090/S0002-9947-1960-0117678-1 |
| [60] | Smyth, R. W.: Uncountably many C0 conformally distinct Lorentz surfaces and a finiteness theorem, Proc. Amer. Math. Soc.124(1996), 1559-1566. · Zbl 0863.53043 · doi:10.1090/S0002-9939-96-03558-7 |
| [61] | Smyth, R. W.: Completing the conformal boundary of a simply connected Lorentz surface, Proc. Amer. Math. Soc.130(2002), 841-847. · Zbl 1031.53098 · doi:10.1090/S0002-9939-01-06067-1 |
| [62] | Smyth, R. W. and Weinstein, T.: Conformally homeomorphic Lorentz surfaces need not be conformally diffeomorphic, Proc. Amer. Math. Soc.123(1995), 3499-3506. · Zbl 0863.53044 · doi:10.1090/S0002-9939-1995-1273526-7 |
| [63] | Smyth, R. W. and Weinstein, T.: How many Lorentz surfaces are there, In: S. Grindikin (ed.), Topics in Geometry in Memory of Joseph D’Atri, Progr. Nonlinear Differential Equations Appl. 20, Birkhäuser, Boston, MA, 1996, pp. 315-330. · Zbl 0860.53039 |
| [64] | Toda, M.: Weierstrass-type representation of weakly regular pseudospherical surfaces in Euclidean space, Balkan J. Geom. Appl.7(2002), 87-136 (http://www.emis.de/journals /BJGA/7.2/10.html). · Zbl 1028.53006 |
| [65] | Van deWoestijne, I.; Boyom, M. (ed.), Minimal surfaces of the 3-dimensional Minkowski space, 344-369 (1990), Teaneck, NJ · Zbl 0731.53057 |
| [66] | Weinstein, T.: Inextendiable conformal realization of Lorentz surfaces in Minkowski 3-space, Michigan Math. J.40(1993), 545-559. · Zbl 0807.53047 · doi:10.1307/mmj/1029004837 |
| [67] | Weinstein, T.: An Introduction to Lorentz Surfaces, de Gruyter Exposition in Math. 22, Walter de Gruyter, Berlin, 1996. · Zbl 0881.53001 · doi:10.1515/9783110821635 |
| [68] | Yokota, I.: Realization of involutive automorphism sand Gsof exceptional linear Lie group I, G= G2, F4 and E6, Tsukuba J. Math.14(1990), 185-223. · Zbl 0732.22002 · doi:10.21099/tkbjm/1496161328 |
| [69] | Yokota, I.: Classical Lie Groups (Japanese), Gendai Sugakusha, Kyoto, 1990. · Zbl 0741.22004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
