Two-dimensional Dirac operator and the theory of surfaces

I. A. Taimanov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 61 (2006) no. 1, pp. 79-159

Résumé

A survey is given of the Weierstrass representations of surfaces in three- and four-dimensional spaces, their applications to the theory of the Willmore functional, and related problems in the spectral theory of the two-dimensional Dirac operator with periodic coefficients.

Bibliographie

[1] I. A. Taimanov, “Modified Novikov–Veselov equation and differential geometry of surfaces”, Amer. Math. Soc. Transl. Ser. 2, 179 (1997), 133–151 ; http://arxiv.org/dg-ga/9511005 | MR | Zbl

[2] B. G. Konopelchenko, “Induced surfaces and their integrable dynamics”, Stud. Appl. Math., 96:1 (1996), 9–51 | MR | Zbl

[3] I. A. Taimanov, “Predstavlenie Veiershtrassa zamknutykh poverkhnostei v $\mathbb R^3$”, Funkts. analiz i ego pril., 32:4 (1998), 49–62 | MR | Zbl

[4] B. A. Dubrovin, I. M. Krichever, S. P. Novikov, “Uravnenie Shrëdingera v periodicheskom pole i rimanovy poverkhnosti”, Dokl. AN SSSR, 229:1 (1976), 15–19 | MR | Zbl

[5] M. U. Schmidt, A proof of the Willmore conjecture, SFB 256, Preprint no. 546, Technischen Universität Berlin, 2002; http://arxiv.org/math.DG/0203224

[6] I. A. Taimanov, “Predstavlenie Veiershtrassa sfer v $\mathbb R^3$, chisla Uillmora i solitonnye sfery”, Trudy MIAN, 225, 1999, 339–361 | MR | Zbl

[7] D. Ferus, K. Leschke, F. Pedit, U. Pinkall, “Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori”, Invent. Math., 146:3 (2001), 507–593 | DOI | MR | Zbl

[8] F. Pedit, U. Pinkall, “Quaternionic analysis on Riemann surfaces and differential geometry”, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math., Extra Vol. II, 1998, 389–400 | MR | Zbl

[9] B. G. Konopelchenko, “Weierstrass representations for surfaces in 4D spaces and their integrable deformations via DS hierarchy”, Ann. Global Anal. Geom., 16:1 (2000), 61–74 | DOI | MR | Zbl

[10] I. A. Taimanov, “Operatory Diraka i konformnye invarianty torov v trekhmernom prostranstve”, Trudy MIAN, 244, 2004, 249–280 | MR | Zbl

[11] D. A. Berdinskii, I. A. Taimanov, “Poverkhnosti v trekhmernykh gruppakh Li”, Sib. matem. zhurn., 46:6 (2005), 1248–1264 | MR | Zbl

[12] F. E. Burstall, D. Ferus, K. Leschke, F. Pedit, U. Pinkall, Conformal geometry of surfaces in $S^4$ and quaternions, Lecture Notes in Math., 1772, Springer-Verlag, Berlin, 2002 | MR | Zbl

[13] W. Helfrich, “Elastic properties of lipid bilayers: theory and possible experiments”, Z. Naturforsch. C, 28 (1973), 693–703 | MR

[14] E. M. Blokhius, W. F. C. Sager, “Helfrich energy for aggregation and adhesion”, J. Chemical Phys., 110 (1999), 3148–3152 | DOI

[15] C.-H. Lin, M.-H. Lo, Y.-C. Tsai, “Shape energy of fluid membranes”, Progr. Theoret. Phys., 109:4 (2003), 591–618 | DOI | Zbl

[16] L. P. Eisenhart, A Treatise on the differential geometry of curves and surfaces, Ginn, Boston, 1909 | Zbl

[17] T. Friedrich, “On the spinor representation of surfaces in Euclidean $3$-space”, J. Geom. Phys., 28:1–2 (1998), 143–157 | DOI | MR | Zbl

[18] S. Matsutani, “Immersion anomaly of Dirac operator on surface in $\mathbb R^3$”, Rev. Math. Phys., 11:2 (1999), 171–186 | DOI | MR | Zbl

[19] K. Kenmotsu, “Weierstrass formula for surfaces of prescribed mean curvature”, Math. Ann., 245:2 (1979), 89–99 | DOI | MR | Zbl

[20] P. Scott, “The geometries of $3$-manifolds”, Bull. London Math. Soc., 15:5 (1983), 401–487 | DOI | MR | Zbl

[21] W. P. Thurston, “Three-dimensional manifolds, Kleinian groups and hyperbolic geometry”, Bull. Amer. Math. Soc. (N.S.), 6:3 (1982), 357–379 | DOI | MR | Zbl

[22] U. Abresch, “Generalized Hopf differentials”, Meeting on Geometry (Oberwolfach, 2004)

[23] U. Abresch, H. Rosenberg, “A Hopf differential for constant mean curvature surfaces in $S^2\times R$ and $H^2\times R$”, Acta Math., 193:2 (2004), 141–174 | DOI | MR | Zbl

[24] I. Fernandez, P. Mira, A characterization of constant mean curvature surfaces in homogeneous 3-manifolds, http://arxiv.org/math.DG/0512280 | MR

[25] J.-I. Inoguchi, T. Kumamoto, N. Ohsugi, Y. Suyama, “Differential geometry of curves and surfaces in 3-dimensional homogeneous spaces. II”, Fukuoka Univ. Sci. Rep., 30:1 (2000), 17–47 | MR | Zbl

[26] J. Inoguchi, “Minimal surfaces in 3-dimensional solvable Lie groups”, Chinese Ann. Math. Ser. B, 24:1 (2003), 73–84 | DOI | MR | Zbl

[27] B. Daniel, Isometric immersions into 3-dimensional homogeneous manifolds, http://arxiv.org/math.DG/0503500 | MR

[28] A. S. Fokas, I. M. Gelfand, “Surfaces on Lie groups, on Lie algebras, and their integrability”, Comm. Math. Phys., 177:1 (1996), 203–220 | DOI | MR | Zbl

[29] G. Kamberov, P. Norman, F. Pedit, U. Pinkall, Quaternions, spinors, and surfaces, Contemp. Math., 299, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl

[30] G. Kamberov, F. Pedit, U. Pinkall, “Bonnet pairs and isothermic surfaces”, Duke Math. J., 92:3 (1998), 637–644 | DOI | MR | Zbl

[31] J. Richter, Conformal maps of a Riemann surface into the space of quaternions, Ph.D. Thesis, Technische Universität Berlin, 1997

[32] K. Leschke, F. Pedit, U. Pinkall, “Willmore tori in the $4$-sphere with nontrivial normal bundle”, Math. Ann., 332:2 (2005), 381–394 | DOI | MR | Zbl

[33] F. Hélein, P. Romon, “Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions”, Comment. Math. Helv., 75:4 (2000), 668–680 | DOI | MR | Zbl

[34] F. Hélein, On Konopelchenko's representation for surfaces in 4 dimensions, http://arxiv.org/math.DG/0104101

[35] I. A. Taimanov, “Surfaces in the four-space and the Davey–Stewartson equations”, J. Geom. Phys., 56:8 (2006), 1235–1256 | DOI | MR | Zbl

[36] L. V. Bogdanov, “Uravnenie Veselova–Novikova kak estestvennoe dvumernoe obobschenie uravneniya Kortevega–de Friza”, TMF, 70:2 (1987), 309–314 | MR | Zbl

[37] L. V. Bogdanov, “O dvumernoi zadache Zakharova–Shabata”, TMF, 72:1 (1987), 155–159 | MR | Zbl

[38] A. P. Veselov, S. P. Novikov, “Konechnozonnye dvumernye potentsialnye operatory Shrëdingera. Yavnye formuly i evolyutsionnye uravneniya”, Dokl. AN SSSR, 279:1 (1984), 20–24 | MR | Zbl

[39] A. P. Veselov, S. P. Novikov, “Konechnozonnye dvumernye operatory Shrëdingera. Potentsialnye operatory”, Dokl. AN SSSR, 279:4 (1984), 784–788 | MR | Zbl

[40] I. A. Taimanov, “Konechnozonnye resheniya modifitsirovannykh uravnenii Veselova–Novikova, ikh spektralnye svoistva i prilozheniya”, Sib. matem. zhurn., 40:6 (1999), 1352–1363 | MR | Zbl

[41] I. A. Taimanov, “O dvumernykh konechnozonnykh potentsialnykh operatorakh Shrëdingera i Diraka s osobymi spektralnymi krivymi”, Sib. matem. zhurn., 44:4 (2003), 870–882 | MR | Zbl

[42] F. Burstall, F. Pedit, U. Pinkall, “Schwarzian derivatives and flows of surfaces”, Differential Geometry and Integrable Systems (Tokyo, 2000), Contemp. Math., 308, Amer. Math. Soc., Providence, RI, 2002, 39–61 | MR | Zbl

[43] B. G. Konopelchenko, U. Pinkall, “Integrable deformations of affine surfaces via the Nizhik–Veselov–Novikov equation”, Phys. Lett. A, 245:3–4 (1998), 239–245 | DOI | MR

[44] E. V. Ferapontov, “Surfaces with flat normal bundle: an explicit construction”, Differential Geom. Appl., 14:1 (2001), 15–37 | DOI | MR | Zbl

[45] L. V. Bogdanov, E. V. Ferapontov, “Projective differential geometry of higher reductions of the two-dimensional Dirac equation”, J. Geom. Phys., 52:3 (2004), 328–352 | DOI | MR | Zbl

[46] I. A. Taimanov, “Surfaces of revolution in terms of solitons”, Ann. Global Anal. Geom., 12:5 (1997), 419–435 | DOI | MR

[47] J. Langer, “Recursion in curve geometry”, New York J. Math., 5 (1999), 25–51 | MR | Zbl

[48] O. Garay, J. Langer, “Taimanov's surface evolution and Bäcklund transformations for curves”, Conform. Geom. Dyn., 3 (1999), 37–49 | DOI | MR | Zbl

[49] M. V. Keldysh, “O sobstvennykh znacheniyakh i sobstvennykh funktsiyakh nekotorykh klassov nesamosopryazhennykh uravnenii”, Dokl. AN SSSR, 77:1 (1951), 11–14 | MR | Zbl

[50] M. V. Keldysh, “O polnote sobstvennykh funktsii nekotorykh klassov nesamosopryazhennykh lineinykh operatorov”, UMN, 26:4 (1971), 15–41 | MR | Zbl

[51] H. Grauert, R. Remmert, Coherent Analytic Sheaves, Grundlehren Math. Wiss., 265, Springer-Verlag, Berlin, 1984 | MR | Zbl

[52] I. M. Krichever, “Spektralnaya teoriya dvumernykh periodicheskikh operatorov i ee prilozheniya”, UMN, 44:2 (1989), 121–184 | MR | Zbl

[53] P. A. Kuchment, “Teoriya Floke dlya uravnenii v chastnykh proizvodnykh”, UMN, 37:4 (1982), 3–52 | MR | Zbl

[54] P. Kuchment, Floquet theory for partial differential equations, Oper. Theory Adv. Appl., 60, Birkhäuser, Basel, 1993 | MR | Zbl

[55] I. A. Taimanov, “Finite-gap theory of the Clifford torus”, Internat. Math. Res. Notices, 2 (2005), 103–120 | DOI | MR | Zbl

[56] S. P. Novikov, “Periodicheskaya zadacha Kortevega–de Friza”, Funkts. analiz i ego pril., 8:3 (1974), 54–66 | MR | Zbl

[57] S. V. Manakov, “Metod obratnoi zadachi i dvumernye solitonnye uravneniya”, UMN, 31:5 (1976), 245–246 | MR | Zbl

[58] J. Feldman, H. Knörrer, E. Trubowitz, “There is no two-dimensional analogue of Lamé's equation”, Math. Ann., 294:2 (1992), 295–324 | DOI | MR | Zbl

[59] J. Feldman, H. Knörrer, E. Trubowitz, Riemann surfaces of infinite genus, CRM Monogr. Ser., 20, Amer. Math. Soc., Providence, RI, 2003 | MR | Zbl

[60] I. M. Krichever, “Metody algebraicheskoi geometrii v teorii nelineinykh uravnenii”, UMN, 32:6 (1977), 183–208 | MR | Zbl

[61] P. G. Grinevich, M. U. Schmidt, “Conformal invariant functionals of immersions of tori into $\mathbb R^3$”, J. Geom. Phys., 26:1–2 (1998), 51–78 | DOI | MR | Zbl

[62] N. J. Hitchin, “Harmonic maps from a 2-torus to the 3-sphere”, J. Differential Geom., 31:3 (1990), 627–710 | MR | Zbl

[63] K. Pohlmeyer, “Integrable Hamiltonian systems and interactions through constraints”, Comm. Math. Phys., 46:3 (1976), 207–221 | DOI | MR | Zbl

[64] V. E. Zakharov, A. V. Mikhailov, “Relyativistski-invariantnye dvumernye modeli teorii polya, integriruemye metodom obratnoi zadachi rasseyaniya”, ZhETF, 74:6 (1978), 1953–1973 | MR

[65] K. Uhlenbeck, “Harmonic maps into Lie groups: classical solutions of the chiral model”, J. Differential Geom., 30:1 (1989), 1–50 | MR | Zbl

[66] M. A. Guest, Harmonic maps, loop groups, and integrable systems, London Math. Soc. Stud. Texts, 38, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[67] A. P. Fordy, J. C. Wood (eds.), Harmonic maps and integrable systems, Aspects Math., E23, Vieweg, Braunschweig, 1994 | MR | Zbl

[68] F. Hélein, Constant mean curvature surfaces, harmonic maps and integrable systems, Notes taken by Roger Moser, Lectures Math. ETH Zurich, Birkhäuser, Basel, 2001 | MR | Zbl

[69] A. I. Bobenko, “Surface in terms of 2 by 2 matrices. Old and new integrable cases”, Harmonic Maps and Integrable Systems, Aspects Math., E23, eds. A. P. Fordy, J. C. Wood, Vieweg, Braunschweig, 1994, 83–127 | MR | Zbl

[70] A. S. Fokas, I. M. Gel'fand, F. Finkel, Q. M. Liu, “A formula for constructing infinitely many surfaces on Lie algebras and integrable equations”, Selecta Math. (N.S.), 6:4 (2000), 347–375 | DOI | MR | Zbl

[71] E. A. Ruh, J. Vilms, “The tension field of the Gauss map”, Trans. Amer. Math. Soc., 149 (1970), 569–573 | DOI | MR | Zbl

[72] H. Hopf, Differential geometry in the large, Lecture Notes in Math., 1000, Springer-Verlag, Berlin, 1983 | MR | Zbl

[73] A. D. Alexandrov, “A characteristic property of spheres”, Ann. Mat. Pura Appl. (4), 58 (1962), 303–315 | DOI | MR | Zbl

[74] H. C. Wente, “Counterexample to a conjecture of H. Hopf”, Pacific J. Math., 121:1 (1986), 193–243 | MR | Zbl

[75] U. Abresch, “Constant mean curvature tori in terms of elliptic functions”, J. Reine Angew. Math., 374 (1987), 169–192 | MR | Zbl

[76] U. Abresch, “Old and new doubly periodic solutions of the sinh-Gordon equation”, Seminar on new results in nonlinear partial differential equations (Bonn, 1984), Aspects Math., E10, Vieweg, Braunschweig, 1987, 37–73 | MR | Zbl

[77] U. Pinkall, I. Sterling, “On the classification of constant mean curvature tori”, Ann. of Math. (2), 130:2 (1989), 407–451 | DOI | MR | Zbl

[78] A. I. Bobenko, “All constant mean curvature tori in $\mathbb R^3$, $S^3$, $H^3$ in terms of theta-functions”, Math. Ann., 290:2 (1991), 209–245 | DOI | MR | Zbl

[79] A. I. Bobenko, “Poverkhnosti postoyannoi srednei krivizny i integriruemye uravneniya”, UMN, 46:4 (1991), 3–42 | MR | Zbl

[80] N. Kapouleas, “Complete constant mean curvature surfaces in Euclidean three-space”, Ann. of Math. (2), 131:2 (1990), 239–330 | DOI | MR | Zbl

[81] N. Kapouleas, “Constant mean curvature surfaces constructed by fusing Wente tori”, Invent. Math., 119:3 (1995), 443–518 | DOI | MR | Zbl

[82] B. G. Konopelchenko, I. A. Taimanov, “Constant mean curvature surfaces via an integrable dynamical system”, J. Phys. A, 29:6 (1996), 1261–1265 | DOI | MR | Zbl

[83] F. Ehlers, H. Knörrer, “An algebro-geometric interpretation of the Bäcklund transformation for the Korteweg–de Vries equation”, Comment. Math. Helv., 57:1 (1982), 1–10 | DOI | MR | Zbl

[84] T. J. Willmore, “Note on embedded surfaces”, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. (N.S.), 11B (1965), 493–496 | MR | Zbl

[85] G. Thomsen, “Über konforme Geometrie. I: Grundlangen der konformen Flächentheorie”, Abh. Math. Sem. Univ. Hamburg, 3 (1923), 31–56 | DOI | Zbl

[86] W. Blaschke, Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie, III: Differentialgeometrie der Kreise und Kugeln, Springer-Verlag, Berlin, 1929 | Zbl

[87] R. L. Bryant, “A duality theorem for Willmore surfaces”, J. Differential Geom., 20:1 (1984), 23–53 | MR | Zbl

[88] U. Pinkall, “Hopf tori in $S^3$”, Invent. Math., 81:2 (1985), 379–386 | DOI | MR | Zbl

[89] M. Babich, A. Bobenko, “Willmore tori with umbilic lines and minimal surfaces in hyperbolic space”, Duke Math. J., 72:1 (1993), 151–185 | DOI | MR | Zbl

[90] L. Simon, “Existence of surfaces minimizing the Willmore functional”, Comm. Anal. Geom., 1:2 (1993), 281–326 | MR | Zbl

[91] M. Bauer, E. Kuwert, “Existence of minimizing Willmore surfaces of prescribed genus”, Internat. Math. Res. Notices, 10 (2003), 553–576 | DOI | MR | Zbl

[92] M. U. Schmidt, Existence of minimizing Willmore surfaces of prescribed conformal class, http://arxiv.org/math.DG/0403301

[93] C.-K. Peng, L. Xiao, “Willmore surfaces and minimal surfaces with flat ends”, Geometry and Topology of Submanifolds, X (Beijing/Berlin, 1999), World Scientific, River Edge, NJ, 2000, 259–265 | MR | Zbl

[94] R. L. Bryant, “Surfaces in conformal geometry”, The Mathematical heritage of Hermann Weyl (Durham, 1987), Proc. Sympos. Pure Math., 48, Amer. Math. Soc., Providence, RI, 1988, 227–240 | MR | Zbl

[95] K. Shiohama, R. Takagi, “A characterization of a standard torus in $E^3$”, J. Differential Geom., 4 (1970), 477–485 | MR | Zbl

[96] T. J. Willmore, “Mean curvature of Riemannian immersions”, J. London Math. Soc. (2), 3 (1971), 307–310 | DOI | MR | Zbl

[97] U. Hertrich-Jeromin, U. Pinkall, “Ein Beweis des Willmoreschen Vermutung für Kanaltori”, J. Reine Angew. Math., 430 (1992), 21–34 | MR | Zbl

[98] J. Langer, D. Singer, “Curves in the hyperbolic plane and mean curvature of tori in 3-space”, Bull. London Math. Soc., 16:5 (1984), 531–534 | DOI | MR | Zbl

[99] P. Li, S. T. Yau, “A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue for compact surfaces”, Invent. Math., 69:2 (1982), 269–291 | DOI | MR | Zbl

[100] S. Montiel, A. Ros, “Minimal immersions of surfaces by the first eigenfunctions and conformal area”, Invent. Math., 83:1 (1986), 153–166 | DOI | MR | Zbl

[101] A. Ros, “The Willmore conjecture in the real projective space”, Math. Res. Lett., 6:5–6 (1999), 487–493 | MR | Zbl

[102] B. Ammann, “The Willmore conjecture for immersed tori with small curvature integral”, Manuscripta Math., 101:1 (2000), 1–22 | DOI | MR | Zbl

[103] P. Topping, “Towards the Willmore conjecture”, Calc. Var. Partial Differential Equations, 11:4 (2000), 361–393 | DOI | MR | Zbl

[104] J. L. Weiner, “On a problem of Chen, Willmore, et al.”, Indiana Univ. Math. J., 27:1 (1978), 19–35 | DOI | MR | Zbl

[105] B. Palmer, “The conformal Gauss map and the stability of Willmore surfaces”, Ann. Global Anal. Geom., 9:3 (1991), 305–317 | DOI | MR | Zbl

[106] R. Kusner, N. Schmitt, The spinor representation of minimal surfaces, GANG Preprint 3.27, , Center for Geometry, Analysis, Numerics Graphics, Amherst, MA, 1994 http://www.gans.umass.edu

[107] C. J. Costa, “Complete minimal surfaces in ${\mathbb R}^3$ of genus one and four planar embedded ends”, Proc. Amer. Math. Soc., 119:4 (1993), 1279–1287 | DOI | MR | Zbl

[108] E. I. Shamaev, “Ob odnom semeistve minimalnykh torov s ploskimi kontsami v $\mathbb R^3$”, Sib. matem. zhurn., 46:6 (2005), 1407–1426 | MR

[109] D. Ferus, F. Pedit, U. Pinkall, I. Sterling, “Minimal tori in $S^4$”, J. Reine Angew. Math., 429 (1992), 1–47 | MR | Zbl

[110] M. V. Babich, “Willmore surfaces, 4-particle Toda lattice and double coverings of hyperelliptic surfaces”, Amer. Math. Soc. Transl. Ser. 2, 174 (1996), 143–168 | MR | Zbl

[111] F. Hélein, “Willmore immersions and loop groups”, J. Differential Geom., 50:2 (1998), 331–385 | MR | Zbl

[112] R. Kusner, “Comparison surfaces for the Willmore problem”, Pacific J. Math., 138:2 (1989), 317–345 | MR | Zbl

[113] P. Wintgen, “On the total curvature of surfaces in $E^4$”, Colloq. Math., 39:2 (1978), 289–296 | MR | Zbl

[114] P. Wintgen, “Sur l'inégalité de Chen–Willmore”, C. R. Acad. Sci. Paris Ser. A, 288:21 (1979), 993–995 | MR | Zbl

[115] W. P. Minicozzi II, “The Willmore functional of Lagrangian tori: its relation to area and existence of smooth minimizers”, J. Amer. Math. Soc., 8:4 (1995), 761–791 | DOI | MR | Zbl

[116] B.-Y. Chen, “Some conformal invariants of submanifolds and their applications”, Boll. Unione. Mat. Ital. (4), 10 (1974), 380–385 | MR | Zbl

[117] E. Kuwert, R. Schatzle, “Removability of point singularities of Willmore surfaces”, Ann. of Math. (2), 160:1 (2004), 315–357 | DOI | MR | Zbl

[118] U. Pinkall, I. Sterling, “Willmore surfaces”, Math. Intelligencer, 9:2 (1987), 38–43 | DOI | MR | Zbl

[119] C. Bohle, G. P. Peters, U. Pinkall, Constrained Willmore surfaces, http://arxiv.org/math.DG/0411479

[120] L. A. Takhtadzhyan, L. D. Faddeev, Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR | Zbl

[121] T. Friedrich, “Eigenvalues of the Dirac operator on Riemannian manifolds”, Conference on Differential Geometry (Brno, August 1998)

[122] C. Bär, “Lower eigenvalue estimates for Dirac operators”, Math. Ann., 293:1 (1992), 39–46 | DOI | MR | Zbl

[123] T. Friedrich, Dirac operators in Riemannian geometry, Graduate Studies in Math., 25, Amer. Math. Soc., Providence, RI, 2000 | MR | Zbl

[124] H.-B. Lawson Jr., M.-L. Michelsohn, Spin Geometry, Princeton Univ. Press, Princeton, NJ, 1989 | MR | Zbl

[125] H. Baum, T. Friedrich, R. Grunewald, I. Kath, Twistors and Killing spinors on Riemannian manifolds, Teubner-Texte Math., 124, Teubner, Stuttgart, 1991 | MR | Zbl

[126] N. Aronszajn, “A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order”, J. Math. Pures Appl. (9), 36 (1957), 235–249 | MR | Zbl

[127] P. Hartman, A. Wintner, “On the local behavior of solutions of non-parabolic differential equations”, Amer. J. Math., 75 (1953), 449–476 | DOI | MR | Zbl

[128] F. Griffits, Dzh. Kharris, Printsipy algebraicheskoi geometrii, t. 1, 2, Mir, M., 1982 | MR | Zbl

[129] M. Haskins, “The geometric complexity of special Lagrangian $T^2$-cones”, Invent. Math., 157:1 (2004), 11–70 | DOI | MR | Zbl

[130] G. P. Peters, Soliton spheres, Ph.D. Thesis, Technische Universität Berlin, 2004; http://edocs.tu-berlin.de/diss/

[131] C. Bohle, G. P. Peters, Bryant surfaces with smooth ends, http://arxiv.org/math.DG/0411480

[132] R. L. Bryant, “Surfaces of mean curvature one in hyperbolic space”, Théorie des variétés minimales et applications (Palaiseau, 1983–1984), Astérisque, 154–155, 1987, 321–347 | MR

[133] J.-H. Eschenburg, Willmore surfaces and Moebius geometry, Preprint, Augsburg University, 1988

[134] U. Hertrich-Jeromin, Introduction to Möbius differential geometry, London Math. Soc. Lecture Note Ser., 300, Cambridge Univ. Press, Cambridge, 2003 | MR | Zbl

[135] V. E. Zakharov, A. B. Shabat, “Tochnaya teoriya dvumernoi samofokusirovki i odnomernoi avtomodulyatsii voln v nelineinykh sredakh”, ZhETF, 61:1 (1971), 118–134 | MR

[136] L. D. Faddeev, “Svoistva $S$-matritsy odnomernogo operatora Shrëdingera”, Trudy MIAN, 73, 1964, 314–336 | MR | Zbl

[137] V. A. Marchenko, Operatory Shturma–Liuvillya i ikh prilozheniya, Naukova Dumka, Kiev, 1977 | MR | Zbl

[138] M. Ablovits, Kh. Sigur, Solitony i metod obratnoi zadachi, Mir, M., 1987 | MR | Zbl