Twistorial construction of harmonic maps of surfaces into four-manifolds

J. Eells; S. Salamon

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1985)

  • Volume: 12, Issue: 4, page 589-640
  • ISSN: 0391-173X

How to cite

top

Eells, J., and Salamon, S.. "Twistorial construction of harmonic maps of surfaces into four-manifolds." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 12.4 (1985): 589-640. <http://eudml.org/doc/83968>.

@article{Eells1985,
author = {Eells, J., Salamon, S.},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {conformal harmonic maps; 4-manifold; superminimal immersions; Kähler manifolds},
language = {eng},
number = {4},
pages = {589-640},
publisher = {Scuola normale superiore},
title = {Twistorial construction of harmonic maps of surfaces into four-manifolds},
url = {http://eudml.org/doc/83968},
volume = {12},
year = {1985},
}

TY - JOUR
AU - Eells, J.
AU - Salamon, S.
TI - Twistorial construction of harmonic maps of surfaces into four-manifolds
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1985
PB - Scuola normale superiore
VL - 12
IS - 4
SP - 589
EP - 640
LA - eng
KW - conformal harmonic maps; 4-manifold; superminimal immersions; Kähler manifolds
UR - http://eudml.org/doc/83968
ER -

References

top
  1. [A] Almgren F.J., Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem, Ann. of Math., 84 (1966), pp. 277-292. Zbl0146.11905MR200816
  2. [AHS] Atiyah M.F. - Hitchin N.J. - Singer I.M.: Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. Lond. Ser. A, 362 (1978), pp. 425-461. Zbl0389.53011MR506229
  3. [B] Berger M., Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 15 (1961), pp. 179-246. Zbl0101.14201MR133083
  4. [Be] Sém. A. Besse (1978-79), Géométrie riemannienne en dimension 4, Cedic/ F. Nathan (1981). MR769127
  5. [BH] Borel A. - Hirzebruch F., Characteristic classes and homogeneous spaces I, Amer. J. Math., 80 (1958), pp. 458-538. Zbl0097.36401MR102800
  6. [Br] Bryant R.L., Conformal and minimal immersions of compact surfaces into the 4-sphere, J. Differential Geom., 17 (1982), pp. 455-473. Zbl0498.53046MR679067
  7. [Bu] Burstall F.E., Non-linear functional analysis and harmonic maps, Ph. D. Thesis, University of Warwick (1984). 
  8. [C1] Calabi E., Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom., 1 (1967), pp. 111-125. Zbl0171.20504MR233294
  9. [C 2] Calabi E., Quelques applications de l'analyse complexe aux surfaces d'aire minima. Topics in Complex Manifolds, Univ. de Montréal (1967), pp. 59-81. 
  10. [Ch1] Chern S.S., Minimal surfaces in Euclidean space of N dimensions, in Differential and Combinatorial Topology: Symposium in honor of Marston Morse, Princeton University Press (1965), pp. 187-198. Zbl0136.16701MR180926
  11. [Ch 2] Chern S.S., On the minimal immersions of the two-sphere in a space of constant curvature. Problems in Analysis, Princeton University Press (1970), pp. 27-40. Zbl0217.47601MR362151
  12. [CW] Chern S.S. - Wolfson J.G., Minimal surfaces by moving frames, Amer. J. Math., 105 (1983), pp. 59-83. Zbl0521.53050MR692106
  13. [E 1] Eells J., Gauss maps of surfaces, in Perspectives in Mathematics, Anniversary of Oberwolfach1984, Birkhäuser. Zbl0581.58013MR779673
  14. [E 2] Eells J., Minimal branched immersions into three-manifolds, to appear. Zbl0574.53039MR827263
  15. [EL] Eells J. - Lemaire L., A report on harmonic maps, Bull. London Math. Soc., 10 (1978), pp. 1-68. Zbl0401.58003MR495450
  16. [ES] Eells J. - Salamon S., Constructions twistorielles des applications harmoniques, C. R. Acad. Sci. Paris, 296 (1983), pp. 685-687. Zbl0531.58020MR705691
  17. [ESa] Eells J. - Sampson J.H., Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), pp. 109-160. Zbl0122.40102MR164306
  18. [EW1] Eells J. - Wood J.C., Maps of minimum energy, J. London Math. Soc., 23 (1981), pp. 303-310. Zbl0432.58012MR609110
  19. [EW2] Eells J. - Wood J.C., Harmonic maps from surfaces to complex projective spaces, Adv. in Math., 49 (1983), pp. 217-263. Zbl0528.58007MR716372
  20. [ETG] Eschenburg J.H. - Tribuzy R. de A. - Guadalupe I.V., The fundamental equations of minimal surfaces in CP2, Math. Ann., 270 (1985), pp. 571-598. Zbl0536.53056MR776173
  21. [F] Friedrich T., On surfaces in four-spaces, Ann. Glob. Analysis and Geometry, 2 (1984), pp. 257-287. Zbl0562.53039MR777909
  22. [G1] Gray A., Minimal varieties and almost Hermitian manifolds, Michigan Math. J., 12 (1965), pp. 273-287. Zbl0132.16702MR184185
  23. [G2] Gray A., Riemannian manifolds with geodesic symmetries of order 3, J. Differential Geom., 7 (1972), pp. 343-369. Zbl0275.53026MR331281
  24. [GOR] Gulliver R.D. - Osserman R. - Royden H.L., A theory of branched immersions of surfaces, Amer. J. Math., 95 (1973), pp. 750-812. Zbl0295.53002MR362153
  25. [Hi] Hirsch M.W., Immersions of manifolds, Trans. Amer. Math. Soc., 93 (1959), pp. 242-276. Zbl0113.17202MR119214
  26. [Hir] Hirzebruch F., Topological Methods in Algebraic Geometry, 3rd edition, Springer, 1966. Zbl0138.42001MR1335917
  27. [H 1] Hitchin N.J., Compact four-dimensional Einstein manifolds, J. Differential Geometry, 9 (1974), pp. 435-441. Zbl0281.53039MR350657
  28. [H2] Hitchin N.J., Kählerian twistor spaces, Proc. London Math. Soc., 43 (1981), pp. 133-150. Zbl0474.14024MR623721
  29. [H3] Hitchin N.J., Monopoles and geodesics, Comm. Math. Phys., 83 (1982), pp. 579-602. Zbl0502.58017MR649818
  30. [K] Kodaira K., On the structure of compact complex analytic surfaces II, Amer. J. Math., 88 (1966), pp. 682-721. Zbl0193.37701MR205280
  31. [KM] Koszul J.-L. - Malgrange B., Sur certaines structures fibrées complexes, Arch. Math., 9 (1958), pp. 102-109. Zbl0083.16705MR131882
  32. [LS] Lashof R. - Smale S., On the immersion of manifolds in Euclidean space, Ann. Math., 68 (1958), pp. 562-583. Zbl0097.38805MR103478
  33. [L1] Lawson H.B., Complete minimal surfaces in S3, Ann. of Math., 92 (1970), pp. 335-374. Zbl0205.52001MR270280
  34. [L 2] Lawson H.B., Lectures on Minimal Submanifolds, 2nd edition, Publish or Perish, 1980. Zbl0434.53006MR576752
  35. [Le] Lebrun C.R., Twistor CR manifolds and 3-dimensional conformal geometry, Trans. Amer. Math. Soc.284 (1984), pp, 601-616. Zbl0513.53006MR743735
  36. [Li] Lichnerowicz A., Applications harmoniques et variétés kählériennes, Symposia Mathematica, 3 (1970), pp. 341-402. Zbl0193.50101MR262993
  37. [M] Micallef M.J., Stable minimal surfaces in Euclidean space, J. Differential Geom., 19 (1984), pp. 57-84. Zbl0527.32016MR739782
  38. [N] Naitoh H., Isotropic submanifolds with parallel second fundamental form in pm(c), Osaka J. Math., 18 (1981), pp. 427-464. Zbl0471.53036MR628843
  39. [NN] Newlander A. - Nirenberg L., Complex analytic coordinates in almost complex manifolds, Ann. of Math., 65 (1957), pp. 391-404. Zbl0079.16102MR88770
  40. [NW] Nijenhuis A. - Woolf W.B., Some integration problems in almost-complex and complex manifolds, Ann. of Math., 77 (1963), pp. 424-489. Zbl0115.16103MR149505
  41. [O] Obata M., The Gauss map of immersions of Riemannian manifolds in spaces of constant curvature, J. Differential Geom., 2 (1968), pp. 217-223. Zbl0181.49801MR234388
  42. [ON] O'Neill B., The fundamental equations of a submersion, Michigan Math. J., 13 (1966), pp. 459-469. Zbl0145.18602MR200865
  43. [P] Poon Y.S., Minimal surfaces in four-dimensional manifolds, M. Sc. Thesis, University of Oxford, 1983. 
  44. [R] Rawnsley J.H., f-structures, f-twistor spaces and harmonic maps, in Geometry Seminar Luigi Bianchi II, to appear. Zbl0592.58009MR829229
  45. [Ru] Ruh E.A., Minimal immersions of 2-spheres in S4, Proc. Amer. Math. Soc., 28 (1971), pp. 219-222. Zbl0212.54003MR271880
  46. [RV] Ruh E.A. - Vilms J., The tension field of the Gauss map, Trans. Amer. Math. Soc., 149 (1970), pp. 569-573. Zbl0199.56102MR259768
  47. [S1] Salamon S., Quaternionic Kähler manifolds, Invent. Math., 67 (1982), pp. 143-171. Zbl0486.53048MR664330
  48. [S 2] Salamon S., Topics in four-dimensional Riemannian geometry, in Geometry Seminar Luigi Bianchi, Lecture Notes in Mathematics1022, Springer (1983), pp. 33-124. Zbl0532.53035MR728393
  49. [S 3] Salamon S., Harmonic and holomorphic maps, in Geometry Seminar Luigi Bianchi II, to appear. Zbl0591.53031MR829230
  50. [Sm] Smale S., A survey of some recent developments in differential topology, Bull. Amer. Math. Soc., 69 (1963), pp. 131-145. Zbl0133.16507MR144351
  51. [SU] Sacks J. - Uhlenbeck K., The existence of minimal immersions of two-spheres, Ann. of Math., 113 (1981), pp. 1-24. Zbl0462.58014MR604040
  52. [W] Wolf J.A., Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. Mech., 14 (1965), pp. 1033-1047. Zbl0141.38202MR185554
  53. [WG] Wolf J.A. - Gray A., Homogeneous spaces defined by Lie group automorphisms I, II, J. Differential Geom., 2 (1968), pp. 77-159. Zbl0182.24702MR236328
  54. [Wo] Wolfson J.G., Minimal surfaces in complex manifolds, Ph. D. Thesis, University of California Berkley (1982). 
  55. [GL] Gauduchon P. - Lawson H.B., Topologically nonsingular minimal cones. Zbl0612.53007
  56. [Gr] Gromov, M., Pseudo holomorphic curves in symplectic manifolds. Zbl0592.53025
  57. [We] Webster S.M., Minimal surfaces in a Kähler surface. Zbl0561.53054

Citations in EuDML Documents

top
  1. Eric Loubeau, Radu Pantilie, Harmonic morphisms between Weyl spaces and twistorial maps II
  2. Norio Ejiri, Minimal deformation of a non-full minimal surface in S 4 ( 1 )
  3. Simon Salamon, Complex Structures and Conformal Geometry
  4. Paul Gauduchon, Twisteurs et applications harmoniques en dimension 4
  5. P. Gauduchon, Pseudo-immersions superminimales d'une surface de Riemann dans une variété riemannienne de dimension 4
  6. Jingyi Chen, Jiayu Li, Quaternionic maps and minimal surfaces
  7. C. M. Wood, Harmonic almost-complex structures
  8. Guillaume Deschamps, Compatible complex structures on twistor space

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.