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Twistor constructions for harmonic maps

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Differential Geometry and Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1255))

Abstract

These notes are intended to give an introduction to the ideas of twistor constructions for harmonic maps which are, at the present time, developing very fast. We explain the ubiquitous "J2" structure in twistor theory by showing how it naturally arises from a generalization of S.S. Chern's fundamental theorem on the antiholomorphicity of the Gauss map of a minimal immersion. We show how twistor methods sometimes lead to the classification theorems for harmonic maps, old and new and finish by outlining the research still in progress.

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References

  1. A.R. AITHAL, Harmonic maps from S2 to G2(ℂ5), J. London Math. Soc. 32, (1985), 572–576.

    Article  MathSciNet  MATH  Google Scholar 

  2. A.R. AITHAL, Harmonic maps from S2 to IHP2, Osaka J. Math. (to appear).

    Google Scholar 

  3. M.F. ATIYAH, N.J. HITCHIN and I.M. SINGER, Self duality in four dimensional Riemannian geometry, Proc. R. Soc. London A 362, (1978) 425–461.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. BAIRD, Harmonic maps with symmetry, harmonic morphisms and deformations of metrics, Research Notes in Mathematics,87, Pitman (1983).

    Google Scholar 

  5. J. BARBOSA, On minimal immersions of S2 into S2m, Trans. Amer. Math. Soc. 210, (1975) 75–106.

    MathSciNet  Google Scholar 

  6. L. BERARD-BERGERY and T.O. OCHAI, On some generalizations of the construction of twistor spaces in Global Riemannian Geometry, pp. 52–59, E. Horwood/Wiley, (1984).

    Google Scholar 

  7. H.J. BORCHERS and W.D. GARBER, Local theory of solutions for the O(2k+1) σ-model, Comm. Math. Phys. 72, (1980) 77–102.

    Article  MathSciNet  MATH  Google Scholar 

  8. R.L. BRYANT, Conformal and minimal immersions of compact surfaces into the 4-sphere, J. Diff. Geom. 17, (1982) 455–474.

    MathSciNet  MATH  Google Scholar 

  9. R.L. BRYANT, Submanifolds and special structures on the octonians, J. Diff. Geom. 17 (1982), 185–232.

    MathSciNet  MATH  Google Scholar 

  10. R.L. BRYANT, Lie groups and twistor spaces, preprint, Rice University (1984). Duke Math. J.82 (1985), 223–261.

    Article  Google Scholar 

  11. D. BURNS, Harmonic maps from ℂP1 to ℂPn, in Harmonic Maps, Proceedings, New Orleans 1980, 48–56, Lecture Notes in Mathematics 949, Springer-Verlag, Berlin, 1982.

    Google Scholar 

  12. F.E. BURSTALL, Twistor fibrations of flag manifolds and harmonic maps of the 2-sphere into a Grassmannian, in Differential Geometry, ed. L.A. Cordero, Research Notes in Mathematics, 131, Pitman (1985), 7–16.

    Google Scholar 

  13. F.E. BURSTALL, A twistor description of harmonic maps of a 2-sphere into a Grassmannian, Math. Ann. 274 (1986), 61–74.

    Article  MathSciNet  MATH  Google Scholar 

  14. F.E. BURSTALL, Twistor methods for harmonic maps, in "Differential Geometry with Applications", Nordic Summer School, Lynby 1985, preprint, University of Bath.

    Google Scholar 

  15. F.E. BURSTALL and J. RAWNSLEY, Sphères harmoniques dans les groupes de Lie compacts et courbes holomorphes dans les espaces homogenes, C.R. Acad. Sci. (Paris)302(1986) 709–712.

    MathSciNet  MATH  Google Scholar 

  16. F.E. BURSTALL and S. SALAMON, Tournaments flags and harmonic maps, preprint, Universities of Bath and Oxford, 1985.

    Google Scholar 

  17. F.E. BURSTALL and J.C. WOOD, The construction of harmonic maps into complex Grassmannians, J. Diff. Geom. 23 (1986), 255–298.

    MathSciNet  MATH  Google Scholar 

  18. E. CALABI, Minimal immersions of surfaces in Euclidean spheres, J. Diff. Geom. 1 (1967) 111–125.

    MathSciNet  MATH  Google Scholar 

  19. E. CALABI, Quelques applications de l'analyse complexe aux surfaces d'aire minima, in Topics in Complex Manifolds, pp. 59–81, University of Montréal (1967).

    Google Scholar 

  20. S.S. CHERN, Minimal surfaces in an Euclidean space of N dimensions, in Diff. and Comb. Topology, pp. 187–198, Princeton (1965).

    Google Scholar 

  21. S.S. CHERN, On the minimal immersions of the two-sphere in a space of constant curvature, in Problems in Analysis, pp. 27–40, Princeton Univ. Press, Princeton, N.J. (1970).

    Google Scholar 

  22. S.S. CHERN, On minimal spheres in the four sphere, pp. 137–150, Studies and essays presented to Y.W. Chen, Taiwan (1970).

    Google Scholar 

  23. S.S. CHERN and J.G. WOLFSON, Minimal surfaces by moving frames, Amer. J. Math. 105 (1983) 59–83.

    Article  MathSciNet  MATH  Google Scholar 

  24. S.S. CHERN and J.G. WOLFSON, Harmonic maps of S2 into a complex Grassmann manifold, Proc. Natl. Acad. Sci. U.S.A. 82 (1985), 2217–2219.

    Article  MathSciNet  MATH  Google Scholar 

  25. S.S. CHERN and J.G. WOLFSON, Harmonic maps of S2 into a complex Grassmann manifold, II, preprint (1984).

    Google Scholar 

  26. A.M. DIN and W.J. ZAKRZEWSKI, General classical solutions in the ℂPn+1 model, Nucl. Phys. B. 174 (1980) 397–406.

    Article  MathSciNet  Google Scholar 

  27. M. DUBOIS-VIOLETTE, Progress in Mathematics 37, Birkhäuser (1983).

    Google Scholar 

  28. J. EELLS, Gauss maps of surfaces, in Perspectives in Mathematics, Anniversary of Oberwolfach 1984, pp. 111–129, Birkhäuser (1984).

    Google Scholar 

  29. J. EELLS, Minimal branched immersions into three manifolds, in Proc. Special year in geometry and topology, (College Pk., Md. 1983/84), Lecture Notes in Math. 1167, Springer-Verlag (1985), 81–94.

    Google Scholar 

  30. J. EELLS and L. LEMAIRE, A report on harmonic maps, Bull. London Math. Soc. 10 (1978) 1–68.

    Article  MathSciNet  MATH  Google Scholar 

  31. J. EELLS and L. LEMAIRE, On the construction of harmonic and holomorphic maps between surfaces, Math. Ann. 252 (1980) 27–52.

    Article  MathSciNet  MATH  Google Scholar 

  32. J. EELLS and L. LEMAIRE, Selected topics in harmonic maps, C.B.M.S. Regional Conference Series 50, American Math. Soc. (1983).

    Google Scholar 

  33. J. EELLS and S. SALAMON, Constructions twistorielles des applications harmoniques, C.R. Acad. Sc. Paris I, 296 (1983) 685–687.

    MathSciNet  MATH  Google Scholar 

  34. J. EELLS and S. SALAMON, Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Scuola Norm. Pisa, Cl.di Sci. (4) 12 (1985) 589–640.

    MathSciNet  MATH  Google Scholar 

  35. J. EELLS and J.H. SAMPSON, Harmonic mappings of Riemannian manifolds, American J. Math. 86 (1964) 109–160.

    Article  MathSciNet  MATH  Google Scholar 

  36. J.-H. ESCHENBURG, I.V. GUADALUPE and R. TRIBUZY, The fundamental equations of minimal surfaces in ℂP2,Math.Ann 270 (1985) 571–598.

    Article  MathSciNet  MATH  Google Scholar 

  37. J. EELLS and J.C. WOOD, Restrictions on harmonic maps of surfaces, Topology 15 (1976) 263–266.

    Article  MathSciNet  MATH  Google Scholar 

  38. J. EELLS and J.C. WOOD, Maps of minimum energy, J. London Math. Soc. (2) 23 (1981) 303–310.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. EELLS and J.C. WOOD, Harmonic maps from surfaces to complex projective spaces, Adv. in Math. 49 (1983) 217–263.

    Article  MathSciNet  MATH  Google Scholar 

  40. S. ERDEM, Harmonic maps from surfaces into pseudo-Riemannian spheres and hyperbolic spaces, Math. Proc. Camb. Phil Soc. 94 (1983) 483–494.

    Article  MathSciNet  MATH  Google Scholar 

  41. S. ERDEM and J.F. GLAZEBROOK, Harmonic maps of Riemann surfaces to indefinite complex hyperbolic and projective spaces, Proc. London Math. Soc. (3) 47 (1983), 547–562.

    Article  MathSciNet  MATH  Google Scholar 

  42. S. ERDEM and J.C. WOOD, On the construction of harmonic maps into a Grassmannian, J. London Math. Soc. (2) 28 (1983) 161–174.

    Article  MathSciNet  MATH  Google Scholar 

  43. P. GAUDUCHON, Pseudo-immersions minimales d'une surface de Riemann dans une variété riemannienne de dimension 4, preprint (1985).

    Google Scholar 

  44. V. GLASER and R. STORA, Regular solutions of the â„‚Pn model and further generalizations, preprint, CERN (1980).

    Google Scholar 

  45. J. GLAZEBROOK, The construction of a class of harmonic maps to quaternionic projective space, J. London Math. Soc. (2) 30 (1984) 151–159.

    Article  MathSciNet  MATH  Google Scholar 

  46. A. GRAY, Vector cross products, harmonic maps and the Cauchy-Riemann equations, in Harmonic Maps, Proceedings, New Orleans, 1980, 57–74, Lecture Notes in Mathematics 949, Springer-Verlag, Berlin (1982).

    MATH  Google Scholar 

  47. M.A. GUEST, Geometry of maps between generalized flag manifolds, preprint, Tulane University, 1985.

    Google Scholar 

  48. R.D. GULLIVER, R. OSSERMAN, H.L. ROYDEN, A theory of branched minimal immersions of surfaces, Amer. J. Math. 95 (1973) 750–812.

    Article  MathSciNet  MATH  Google Scholar 

  49. N.J. HITCHIN, Kählerian twistor spaces, Proc. London Math. Soc. 43 (1981) 133–150.

    Article  MathSciNet  MATH  Google Scholar 

  50. G.R. JENSEN, Minimal surfaces in quaternionic projective spaces, preprint.

    Google Scholar 

  51. G.R. JENSEN and M. RIGOLI, Minimal surfaces in spheres Conf. on diff. geom. on homogenous spaces (Turin 1983) Rend. Sem. Mat. Univ. Politec, Torino 1983, Special issue, 75–98 (1984).

    Google Scholar 

  52. J. JOST and S.T. YAU, Harmonic Mappings and Kähler manifolds, Math. Ann. 262 (1983) 145–166.

    Article  MathSciNet  MATH  Google Scholar 

  53. J. JOST and S.T. YAU, A strong rigidity theorem for a certain class of compact complex analytic surfaces, Math. Ann. 271 (1985) 143–152.

    Article  MathSciNet  MATH  Google Scholar 

  54. M. KALKA, Harmonic maps in Kähler geometry and deformation theory, in Harmonic Maps, Proceedings, New Orleans 1980, 75–85, Lecture Notes in Mathematics 949, Springer-Verlag, Berlin (1982).

    MATH  Google Scholar 

  55. J.L. KOSZUL and B. MALGRANGE, Sur certaines structures fibrées complexes, Arch. Math. 9 (1958) 102–109.

    Article  MathSciNet  MATH  Google Scholar 

  56. H.B. LAWSON, Surfaces minimales et la construction de Calabi-Penrose Sém-Bourbaki (1983/84) No. 624, Astérisque 121 (1985) 197–211.

    Google Scholar 

  57. C.R. LeBRUN, Twistor CR manifolds and 3-dimensional conformal geometry, Trans. Amer. Math. Soc. 284 (1984), 601–616.

    Article  MathSciNet  MATH  Google Scholar 

  58. A. LICHNEROWICZ, Applications harmoniques et variétés kähleriennes, Symp. Math. III Bologna (1970) 341–402.

    Google Scholar 

  59. N. MOK, The holomorphic or antiholomorphic character of harmonic maps into irreducible compact quotients of polydiscs, Math. Ann. 272 (1985), 197–216.

    Article  MathSciNet  MATH  Google Scholar 

  60. N. O'BRIAN and J. RAWNSLEY, Twistor spaces, Ann. Global Analysis and Geometry 3 (1985), 29–58.

    Article  MathSciNet  MATH  Google Scholar 

  61. R. PENROSE, The twistor programme, Reports on Math. Phys. 12 (1977) 65–76.

    Article  MathSciNet  Google Scholar 

  62. J. RAMANATHAN, Harmonic maps from S2 to G2,4, J. Diff. Geom. 19 (1984) 207–219.

    MathSciNet  MATH  Google Scholar 

  63. J. RAWNSLEY, f-structures, f-twistor spaces and harmonic maps, Geometry Seminar Luigi Bianchi II-1984, Lecture Notes in Mathematics 1164, Springer-Verlag (1985), 86–159.

    Google Scholar 

  64. J. RAWNSLEY, Twistor methods, in "Differential Geometry with Applications", Nordic Summer School, Lyngby 1985", preprint, University of Warwick.

    Google Scholar 

  65. S. SALAMON, Quaternionic Kähler manifolds, Invent. Math. 67 (1982) 143–171.

    Article  MathSciNet  MATH  Google Scholar 

  66. S. SALAMON, Harmonic and holomorphic maps, Geometry Seminar Luigi Bianchi II-1984, Lecture Notes in Mathematics 1164, Springer-Verlag (1985), 162–224.

    Google Scholar 

  67. Y.-T. SIU, The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. 112 (1980) 73–111.

    Article  MathSciNet  MATH  Google Scholar 

  68. Y.-T. SIU, Some remarks on the complex analyticity of harmonic maps, S.E. Asian Bull. Math. 3 (1979) 240–253.

    MathSciNet  MATH  Google Scholar 

  69. Y.-T. SIU, Complex analyticity of harmonic maps, vanishing and Lefschetz theorems, J. Diff. Geom. 17 (1982) 55–138.

    MathSciNet  MATH  Google Scholar 

  70. Y.-T. SIU and S.-T. YAU, Compact Kähler manifolds of positive bisectional curvature, Invent. Math. 59 (1980) 189–204.

    Article  MathSciNet  MATH  Google Scholar 

  71. G. TOTH, Harmonic and minimal maps, Ellis Horwood/John Wiley (1984).

    Google Scholar 

  72. K.K. UHLENBECK, Harmonic maps into Lie groups (Classical solutions of the chiral model), J. Diff. Geom. (to appear).

    Google Scholar 

  73. G. VALLI, On the energy spectrum of harmonic 2-spheres in unitary groups, preprint, University of Warwick (1985).

    Google Scholar 

  74. J.C. WOOD, Harmonic maps and complex analysis, in Proc. Summer Course in Complex Analysis, Trieste (1975) Vol. III, 289–308.

    Google Scholar 

  75. J.C. WOOD, Holomorphicity of certain harmonic maps from a surface to complex projective n-space, J. London Math. Soc. (2) 20 (1979) 137–192.

    Article  MathSciNet  MATH  Google Scholar 

  76. J.C. WOOD, Holomorphic differentials and classification theorems for harmonic maps and minimal immersions, in Global Riemannian Geometry, ed. T.J. Willmore and N. Hitchin, E. Horwood, Chichester (1984) 168–176.

    Google Scholar 

  77. J.C. WOOD, On the explicit construction and parametrization of all harmonic maps from the two-sphere to a complex Grassmannian, Proc. Colloq. "Applications harmoniques", Luminy, 1986, Travaux en Cours, Hermann, Paris (to appear).

    Google Scholar 

  78. J.C. WOOD, The explicit construction and parametrization of all harmonic maps from the two-sphere to a complex Grassmannian, preprint, University of Leeds (1986).

    Google Scholar 

  79. H.H. WU, The equidistribution theory of holomorphic curves, Ann. Math. Studies 64, Princeton Univ. Press (1970).

    Google Scholar 

  80. K. YANO, On a structure defined by a tensor field f of type (1,1) satisfying f3+f=0, Tensor 14 (1963) 99–109.

    MathSciNet  MATH  Google Scholar 

  81. F.E. BURSTALL, Non-linear functional analysis and harmonic maps, thesis, University of Warwick (1984).

    Google Scholar 

  82. F.E. BURSTALL, Recent developments in twistor methods for harmonic maps, Proc. Colloq. "Applications harmoniques", Luminy, 1986, Travaux en Cours, Hermann, Paris (to appear).

    Google Scholar 

  83. F.E. BURSTALL and J.H. RAWNSLEY, Twistors, harmonic maps and homogeneous geometry, preprint, Universities of Bath and Warwick (1986).

    Google Scholar 

  84. M.A. GUEST, The energy of harmonic maps obtained by the twistor construction, Proc. Colloq. "Applications harmoniques", Luminy, 1986, Travaux en cours, Hermann, Paris (to appear).

    Google Scholar 

  85. S. HILDEBRANDT, Nonlinear elliptic systems and harmonic mappings, Proc. 1980 Beijing Symp. Diff. Geometry and Diff. Equations, 481–616, Gordon and Breech/Science Press, Beijing (1982).

    MATH  Google Scholar 

  86. J. JOST, Harmonic maps between surfaces, Lecture Notes in Mathematics, 1062, Springer-Verlag, (1984).

    Google Scholar 

  87. J. JOST, Lectures on harmonic maps (with applications to conformal mappings and minimal surfaces), Harmonic mappings and minimal immersions (Montecatini 1984), Lecture Notes in Math. 1161, Springer-Verlag (1985).

    Google Scholar 

  88. S. SALAMON, Minimal surfaces and symmetric spaces, in Differential Geometry, ed. L.A. Cordero, Research Notes in Mathematics, 131, Pitman (1985), 103–114.

    Google Scholar 

  89. J. WOLFSON, Harmonic sequences and harmonic maps of surfaces into complex Grassman manifolds, preprint, MPI/SFB Bonn (1985).

    Google Scholar 

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  1. J. C. Wood
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Chaohao Gu Marcel Berger Robert L. Bryant

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Wood, J.C. (1987). Twistor constructions for harmonic maps. In: Gu, C., Berger, M., Bryant, R.L. (eds) Differential Geometry and Differential Equations. Lecture Notes in Mathematics, vol 1255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077687

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  • DOI: https://doi.org/10.1007/BFb0077687

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