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Uspekhi Mat. Nauk, 2006, Volume 61, Issue 1(367), Pages 85–164 (Mi umn1716)  

This article is cited in 31 scientific papers (total in 31 papers)

Two-dimensional Dirac operator and the theory of surfaces

I. A. Taimanov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: 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.

DOI: https://doi.org/10.4213/rm1716

Full text: PDF file (1234 kB)
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English version:
Russian Mathematical Surveys, 2006, 61:1, 79–159

Bibliographic databases:

UDC: 514.76+517.984
MSC: Primary 53A05, 53A10, 34L40; Secondary 53C42, 58E12, 14H55, 35Q53, 35Q51, 37K40, 37K15
Received: 30.11.2005

Citation: I. A. Taimanov, “Two-dimensional Dirac operator and the theory of surfaces”, Uspekhi Mat. Nauk, 61:1(367) (2006), 85–164; Russian Math. Surveys, 61:1 (2006), 79–159

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Matsutani Sh., “Generalized Weierstrass Relations and Frobenius Reciprocity”, Math. Phys. Anal. Geom., 9:4 (2007), 353–369  crossref  mathscinet  isi  scopus
    2. D. A. Berdinskii, I. A. Taimanov, “Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional”, Siberian Math. J., 48:3 (2007), 395–407  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    3. Konopelchenko B.G., “Quasiclassical generalized Weierstrass representation and dispersionless DS equation”, J. Phys. A, 40:46 (2007), F995–F1004  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Grinevich P.G., Taimanov I.A., “Infinitesimal Darboux transformations of the spectral curves of tori in the four-space”, Int. Math. Res. Not. IMRN, 2007, no. 2, Art. ID rnm005, 22 pp.  crossref  mathscinet  zmath  isi  scopus
    5. A. I. Bobenko, Yu. B. Suris, “On organizing principles of discrete differential geometry. Geometry of spheres”, Russian Math. Surveys, 62:1 (2007), 1–43  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. Chen Q., Jost J., Wang G.F., “Nonlinear Dirac equations on Riemann surfaces”, Ann. Global Anal. Geom., 33:3 (2008), 253–270  crossref  mathscinet  zmath  isi  elib  scopus
    7. Inoguchi J.I., Lee S., “A Weierstrass type representation for minimal surfaces in Sol”, Proc. Amer. Math. Soc., 136:6 (2008), 2209–2216  crossref  mathscinet  zmath  isi  scopus
    8. Bohle Ch., Pedit F., Pinkall U., “The spectral curve of a quaternionic holomorphic line bundle over a 2-torus”, Manuscripta Math., 130:3 (2009), 311–352  crossref  mathscinet  zmath  isi  elib  scopus
    9. de Lira J.H.S., Hinojosa J.A., “The Gauss map of minimal surfaces in Berger spheres”, Ann. Global Anal. Geom., 37:2 (2010), 143–162  crossref  mathscinet  zmath  isi  elib  scopus
    10. Zakharov D., “A discrete analogue of the Dirac operator and the discrete modified Novikov–Veselov hierarchy”, Internat. Math. Res. Notices, 2010, no. 18, 3463–3488  crossref  mathscinet  zmath  isi  elib  scopus
    11. Guven J., Vázquez-Montejo P., “Spinor representation of surfaces and complex stresses on membranes and interfaces”, Phys. Rev. E, 82:4 (2010), 041604, 12 pp.  crossref  mathscinet  adsnasa  isi  elib  scopus
    12. Skovoroda A.A., Taimanov I.A., “Role of the mean curvature in the geometry of magnetic confinement configurations”, Plasma Physics Reports, 36:9 (2010), 819–823  crossref  adsnasa  isi  elib  elib  scopus
    13. Başar G., Dunne G.V., “Gross-Neveu models, nonlinear Dirac equations, surfaces and strings”, J. High Energ. Phys., 2011:1 (2011), 1–25  crossref  mathscinet  adsnasa  isi  scopus
    14. I. A. Taimanov, “Singular spectral curves in finite-gap integration”, Russian Math. Surveys, 66:1 (2011), 107–144  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    15. Grundland A.M., Post S., “Soliton surfaces associated with generalized symmetries of integrable equations”, J. Phys. A, 44:16 (2011), 165203, 31 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    16. de Lira J.H.S., Hinojosa J.A., “The Gauss map of minimal surfaces in the Anti-de Sitter space”, J. Geom. Phys., 61:3 (2011), 610–623  crossref  mathscinet  zmath  adsnasa  isi  scopus
    17. D. V. Zakharov, “Weierstrass Representation for Discrete Isotropic Surfaces in $\mathbb{R}^{2,1}$, $\mathbb{R}^{3,1}$, and $\mathbb{R}^{2,2}$”, Funct. Anal. Appl., 45:1 (2011), 25–32  mathnet  crossref  crossref  mathscinet  zmath  isi
    18. Matsutani Sh., Nakano K., Shinjo K., “Surface tension of multi-phase flow with multiple junctions governed by the variational principle”, Math. Phys. Anal. Geom., 14:3 (2011), 237–278  crossref  mathscinet  zmath  isi  elib  scopus
    19. Crane K., Pinkall U., Schröder P., “Spin transformations of discrete surfaces”, ACM Transactions on Graphics, 30:4 (2011), 104  crossref  isi  scopus
    20. Bohle Ch., Peters G.P., “Soliton spheres”, Trans. Amer. Math. Soc, 363:10 (2011), 5419–5463  crossref  mathscinet  zmath  isi  scopus
    21. McIntosh I., “The quaternionic KP hierarchy and conformally immersed 2-tori in the 4-sphere”, Tohoku Math. J. (2), 63:2 (2011), 183–215  crossref  mathscinet  zmath  isi  scopus
    22. Alías L.J., de Lira J.H.S., Hinojosa J.A., “Generalized Weierstrass representation for surfaces in Heisenberg spaces”, Differential Geom. Appl., 30:1 (2012), 1–12  crossref  mathscinet  zmath  isi  scopus
    23. Michel Berthier, “Spin Geometry and Image Processing”, Adv. Appl. Clifford Algebras, 2013  crossref  mathscinet  isi  scopus
    24. I. A. Taimanov, “The Moutard Transformation of Two-Dimensional Dirac Operators and Möbius Geometry”, Math. Notes, 97:1 (2015), 124–135  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    25. I. A. Taimanov, “Blowing up solutions of the modified Novikov–Veselov equation and minimal surfaces”, Theoret. and Math. Phys., 182:2 (2015), 173–181  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    26. I. A. Taimanov, “A fast decaying solution to the modified Novikov-Veselov equation with a one-point singularity”, Dokl. Math, 91:1 (2015), 35  crossref  mathscinet  zmath  isi  scopus
    27. Bohle Ch. Taimanov I.A., “Euclidean Minimal Tori With Planar Ends and Elliptic Solitons”, no. 14, 2015, 5907–5932  crossref  mathscinet  zmath  isi  scopus
    28. R. M. Matuev, I. A. Taimanov, “The Moutard Transformation of Two-Dimensional Dirac Operators and the Conformal Geometry of Surfaces in Four-Dimensional Space”, Math. Notes, 100:6 (2016), 835–846  mathnet  crossref  crossref  mathscinet  isi  elib
    29. Zhu M., “Quantization for a nonlinear Dirac equation”, Proc. Amer. Math. Soc., 144:10 (2016), 4533–4544  crossref  mathscinet  zmath  isi  scopus
    30. Ma H., Mironov A.E., Zuo D., “An Energy Functional For Lagrangian Tori in Cp2”, Ann. Glob. Anal. Geom., 53:4 (2018), 583–595  crossref  mathscinet  zmath  isi  scopus
    31. A. V. Ilina, I. M. Krichever, N. A. Nekrasov, “Dvumernye periodicheskie operatory Shredingera, integriruemye na «sobstvennom» urovne energii”, Funkts. analiz i ego pril., 53:1 (2019), 31–48  mathnet  crossref  elib
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