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Sibirsk. Mat. Zh., 2005, Volume 46, Number 6, Pages 1248–1264 (Mi smj1037)  

This article is cited in 21 scientific papers (total in 21 papers)

Surfaces in three-dimensional Lie groups

D. A. Berdinskiia, I. A. Taimanovb

a Novosibirsk State University
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: We derive the Weierstrass (or spinor) representation for surfaces in the three-dimensional Lie groups Nil,$\widetilde{SL}_2$ , and Sol with Thurston's geometries and establish the generating equations for minimal surfaces in these groups. Using the spectral properties of the corresponding Dirac operators, we find analogs of the Willmore functional for surfaces in these geometries.

Keywords: surface, three-dimensional Lie group, Weierstrass representation, Willmore functional

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English version:
Siberian Mathematical Journal, 2005, 46:6, 1005–1019

Bibliographic databases:

UDC: 514.772.22
Received: 28.04.2005

Citation: D. A. Berdinskii, I. A. Taimanov, “Surfaces in three-dimensional Lie groups”, Sibirsk. Mat. Zh., 46:6 (2005), 1248–1264; Siberian Math. J., 46:6 (2005), 1005–1019

Citation in format AMSBIB
\by D.~A.~Berdinskii, I.~A.~Taimanov
\paper Surfaces in three-dimensional Lie groups
\jour Sibirsk. Mat. Zh.
\yr 2005
\vol 46
\issue 6
\pages 1248--1264
\jour Siberian Math. J.
\yr 2005
\vol 46
\issue 6
\pages 1005--1019

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    This publication is cited in the following articles:
    1. I. A. Taimanov, “Two-dimensional Dirac operator and the theory of surfaces”, Russian Math. Surveys, 61:1 (2006), 79–159  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Inoguchi JI, “Minimal surfaces in 3-dimensional solvable Lie groups II”, Bulletin of the Australian Mathematical Society, 73:3 (2006), 365–374  crossref  mathscinet  zmath  isi  scopus
    3. 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
    4. Fernandez I, Mira P, “Harmonic maps and constant mean curvature surfaces in H-2 x R”, American Journal of Mathematics, 129:4 (2007), 1145–1181  crossref  mathscinet  zmath  isi  scopus
    5. Fernandez I, Mira P, “A characterization of constant mean curvature surfaces in homogeneous 3-manifolds”, Differential Geometry and Its Applications, 25:3 (2007), 281–289  crossref  mathscinet  zmath  isi  scopus
    6. Galvez J.A., Martinez A., Mira P., “The Bonnet problem for surfaces in homogeneous 3–manifolds”, Communications in Analysis and Geometry, 16:5 (2008), 907–935  crossref  mathscinet  zmath  isi
    7. Inoguchi J.-I., Lee S., “A Weierstrass type representation for minimal surfaces in Sol”, Proceedings of the American Mathematical Society, 136:6 (2008), 2209–2216  crossref  mathscinet  zmath  isi  scopus
    8. de Lira J.H.S., Hinojosa J.A., “The Gauss map of minimal surfaces in Berger spheres”, Annals of Global Analysis and Geometry, 37:2 (2010), 143–162  crossref  mathscinet  zmath  isi  scopus
    9. Chen Q., Qiu H., “Weierstrass representation for surfaces in the three–dimensional Heisenberg group”, Chinese Annals of Mathematics Series B, 31:1 (2010), 119–132  crossref  mathscinet  zmath  isi  scopus
    10. D. A. Berdinskii, “Ob odnom obobschenii funktsionala Uillmora dlya poverkhnostei v $\widetilde{SL}_2$”, Sib. elektron. matem. izv., 7 (2010), 140–149  mathnet  mathscinet
    11. D. A. Berdinsky, “On constant mean curvature surfaces in the Heisenberg group”, Siberian Adv. Math., 22:2 (2012), 75–79  mathnet  crossref  mathscinet
    12. Araujo H., Leite M.L., “Surfaces in S-2 x R and H-2 x R with holomorphic Abresch-Rosenberg differential”, Differential Geom Appl, 29:2 (2011), 271–278  crossref  mathscinet  zmath  isi  elib  scopus
    13. 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
    14. Berdinskii D.A., “O minimalnykh poverkhnostyakh v gruppe geizenberga”, Vestnik Kemerovskogo gosudarstvennogo universiteta, 2011, no. 3-1, 34–38  elib
    15. Alias 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  elib  scopus
    16. Inoguchi J.-i., Lopez R., Munteanu M.-I., “Minimal Translation Surfaces in the Heisenberg Group Nil(3)”, Geod. Dedic., 161:1 (2012), 221–231  crossref  mathscinet  zmath  isi  scopus
    17. Daniel B., Mira P., “Existence and Uniqueness of Constant Mean Curvature Spheres in Sol(3)”, J. Reine Angew. Math., 685 (2013), 1–32  crossref  mathscinet  zmath  isi  elib  scopus
    18. Desmonts Ch., “Constructions of Periodic Minimal Surfaces and Minimal Annuli in Sol3”, Pac. J. Math., 276:1 (2015), 143–166  crossref  mathscinet  zmath  isi  scopus
    19. Dorfmeister J.F. Inoguchi J.-I. Kobayashi Sh., “a Loop Group Method For Minimal Surfaces in the Three-Dimensional Heisenberg Group”, Asian J. Math., 20:3 (2016), 409–448  crossref  mathscinet  zmath  isi  scopus
    20. Bayard P., Roth J., Jimenez B.Z., “Spinorial Representation of Submanifolds in Metric Lie Groups”, J. Geom. Phys., 114 (2017), 348–374  crossref  mathscinet  zmath  isi  scopus
    21. Berdinsky D. Vyatkin Yu., “Willmore-Like Functionals For Surfaces in 3-Dimensional Thurston Geometries”, Osaka J. Math., 54:1 (2017), 75–83  mathscinet  zmath  isi
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