×

zbMATH — the first resource for mathematics

Twistorial construction of harmonic maps of surfaces into four-manifolds. (English) Zbl 0627.58019
In the present work the authors consider conformal harmonic maps from a Riemannian surface M into an arbitrary oriented Riemannian 4-manifold N. In the early sections, they consider the fibre bundles \({\mathcal S}_{\pm}\) over N consisting of unit eigenvectors of the Hodge * operator acting on \(\Lambda^ 2TN\). These total spaces admit a natural almost complex structure \(J_ 1\), which is integrable if N is \(\mp\) selfdual. Their parametrization involves a different almost complex structure \(J_ 2\) obtained from \(J_ 1\) by reversing orientation along fibres.
This almost complex structure \(J_ 2\) is never integrable, so its relevance may come as a surprise. However in homogeneous cases \(J_ 2\) has made previous appearances. Having obtained a twistorial description for all conformal harmonic maps, the authors distinguish special classes. If \(\psi\) is both \(J_ 1\) and \(J_ 2\) holomorphic, then it is horizontal, and its projection \(\phi\) is a real isotropic harmonic map. Such maps include Bryant’s superminimal immersions in \(S^ 4\), most of the known minimal surfaces in the complex projective plane \({\mathbb{C}}P^ 2\), and many stable minimal surfaces in Euclidean space \({\mathbb{R}}^ 4\). The authors explain that twistor methods are most valuable when the target manifold is selfdual and Einstein.
Spinor terminology is used to introduce the twistor degrees of a conformal harmonic map \(\phi\) of a compact Riemannian surface into a 4- manifold; these are used to relate analytical and topological properties of \(\phi\). Then the twistor bundle \({\mathbb{C}}P^ 3\) of \(S^ 4\) is considered. This leads to examples of harmonic maps into \({\mathbb{C}}P^ 3\), and more generally to a theory of harmonic maps into Kähler manifolds.
This theory is in turn applied to study maps into Kähler surfaces, for which the authors pay special attention to the notions of real and complex isotropy. They extend the validity of a formula of J. H. Eschenburg, R. de A. Tribuzy and J. V. Guadalupe [Math. Ann., II. Ser. 270, 571-598 (1985; Zbl 0536.53056)]. Conformal harmonic maps \(M\to {\mathbb{C}}P^ 2\) are interpreted in terms of \(J_ 2\) holomorphic curves in a flag manifold. This implies that such maps into \({\mathbb{C}}P^ 2\) really come in triples, and explains the existence of associated harmonic maps.
Reviewer: W.Kugler

MSC:
58E20 Harmonic maps, etc.
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] 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 . MR 200816 | Zbl 0146.11905 · Zbl 0146.11905
[2] 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 . MR 506229 | Zbl 0389.53011 · Zbl 0389.53011
[3] 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 . Numdam | MR 133083 | Zbl 0101.14201 · Zbl 0101.14201
[4] Sém. A. Besse (1978-79), Géométrie riemannienne en dimension 4 , Cedic/ F. Nathan ( 1981 ). MR 769127
[5] Borel A. - Hirzebruch F. , Characteristic classes and homogeneous spaces I , Amer. J. Math. , 80 ( 1958 ), pp. 458 - 538 . MR 102800 | Zbl 0097.36401 · Zbl 0097.36401
[6] Bryant R.L. , Conformal and minimal immersions of compact surfaces into the 4-sphere , J. Differential Geom. , 17 ( 1982 ), pp. 455 - 473 . MR 679067 | Zbl 0498.53046 · Zbl 0498.53046
[7] Burstall F.E. , Non-linear functional analysis and harmonic maps , Ph. D. Thesis, University of Warwick ( 1984 ). · Zbl 0622.58008
[8] Calabi E. , Minimal immersions of surfaces in Euclidean spheres , J. Differential Geom. , 1 ( 1967 ), pp. 111 - 125 . MR 233294 | Zbl 0171.20504 · Zbl 0171.20504
[9] 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] 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 . MR 180926 | Zbl 0136.16701 · Zbl 0136.16701
[11] 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 . MR 362151 | Zbl 0217.47601 · Zbl 0217.47601
[12] Chern S.S. - Wolfson J.G. , Minimal surfaces by moving frames , Amer. J. Math. , 105 ( 1983 ), pp. 59 - 83 . MR 692106 | Zbl 0521.53050 · Zbl 0521.53050
[13] Eells J. , Gauss maps of surfaces , in Perspectives in Mathematics, Anniversary of Oberwolfach 1984 , Birkhäuser . MR 779673 | Zbl 0581.58013 · Zbl 0581.58013
[14] Eells J. , Minimal branched immersions into three-manifolds , to appear. MR 827263 · Zbl 0574.53039
[15] Eells J. - Lemaire L. , A report on harmonic maps , Bull. London Math. Soc. , 10 ( 1978 ), pp. 1 - 68 . MR 495450 | Zbl 0401.58003 · Zbl 0401.58003
[16] Eells J. - Salamon S. , Constructions twistorielles des applications harmoniques , C. R. Acad. Sci. Paris , 296 ( 1983 ), pp. 685 - 687 . MR 705691 | Zbl 0531.58020 · Zbl 0531.58020
[17] Eells J. - Sampson J.H. , Harmonic mappings of Riemannian manifolds , Amer. J. Math. , 86 ( 1964 ), pp. 109 - 160 . MR 164306 | Zbl 0122.40102 · Zbl 0122.40102
[18] Eells J. - Wood J.C. , Maps of minimum energy , J. London Math. Soc. , 23 ( 1981 ), pp. 303 - 310 . MR 609110 | Zbl 0432.58012 · Zbl 0432.58012
[19] Eells J. - Wood J.C. , Harmonic maps from surfaces to complex projective spaces , Adv. in Math. , 49 ( 1983 ), pp. 217 - 263 . MR 716372 | Zbl 0528.58007 · Zbl 0528.58007
[20] 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 . MR 776173 | Zbl 0536.53056 · Zbl 0536.53056
[21] Friedrich T. , On surfaces in four-spaces , Ann. Glob. Analysis and Geometry , 2 ( 1984 ), pp. 257 - 287 . MR 777909 | Zbl 0562.53039 · Zbl 0562.53039
[22] Gray A. , Minimal varieties and almost Hermitian manifolds , Michigan Math. J. , 12 ( 1965 ), pp. 273 - 287 . Article | MR 184185 | Zbl 0132.16702 · Zbl 0132.16702
[23] Gray A. , Riemannian manifolds with geodesic symmetries of order 3 , J. Differential Geom. , 7 ( 1972 ), pp. 343 - 369 . MR 331281 | Zbl 0275.53026 · Zbl 0275.53026
[24] Gulliver R.D. - Osserman R. - Royden H.L. , A theory of branched immersions of surfaces , Amer. J. Math. , 95 ( 1973 ), pp. 750 - 812 . MR 362153 | Zbl 0295.53002 · Zbl 0295.53002
[25] Hirsch M.W. , Immersions of manifolds , Trans. Amer. Math. Soc. , 93 ( 1959 ), pp. 242 - 276 . MR 119214 | Zbl 0113.17202 · Zbl 0113.17202
[26] Hirzebruch F. , Topological Methods in Algebraic Geometry , 3 rd edition, Springer , 1966 . MR 1335917 | Zbl 0138.42001 · Zbl 0138.42001
[27] Hitchin N.J. , Compact four-dimensional Einstein manifolds , J. Differential Geometry , 9 ( 1974 ), pp. 435 - 441 . MR 350657 | Zbl 0281.53039 · Zbl 0281.53039
[28] Hitchin N.J. , Kählerian twistor spaces , Proc. London Math. Soc. , 43 ( 1981 ), pp. 133 - 150 . MR 623721 | Zbl 0474.14024 · Zbl 0474.14024
[29] Hitchin N.J. , Monopoles and geodesics , Comm. Math. Phys. , 83 ( 1982 ), pp. 579 - 602 . Article | MR 649818 | Zbl 0502.58017 · Zbl 0502.58017
[30] Kodaira K. , On the structure of compact complex analytic surfaces II , Amer. J. Math. , 88 ( 1966 ), pp. 682 - 721 . MR 205280 | Zbl 0193.37701 · Zbl 0193.37701
[31] Koszul J.-L. - Malgrange B. , Sur certaines structures fibrées complexes , Arch. Math. , 9 ( 1958 ), pp. 102 - 109 . MR 131882 | Zbl 0083.16705 · Zbl 0083.16705
[32] Lashof R. - Smale S. , On the immersion of manifolds in Euclidean space , Ann. Math. , 68 ( 1958 ), pp. 562 - 583 . MR 103478 | Zbl 0097.38805 · Zbl 0097.38805
[33] Lawson H.B. , Complete minimal surfaces in S3 , Ann. of Math. , 92 ( 1970 ), pp. 335 - 374 . MR 270280 | Zbl 0205.52001 · Zbl 0205.52001
[34] Lawson H.B. , Lectures on Minimal Submanifolds , 2 nd edition, Publish or Perish , 1980 . MR 576752 | Zbl 0434.53006 · Zbl 0434.53006
[35] Lebrun C.R. , Twistor CR manifolds and 3-dimensional conformal geometry , Trans. Amer. Math. Soc. 284 ( 1984 ), pp, 601 - 616 . MR 743735 | Zbl 0513.53006 · Zbl 0513.53006
[36] Lichnerowicz A. , Applications harmoniques et variétés kählériennes , Symposia Mathematica , 3 ( 1970 ), pp. 341 - 402 . MR 262993 | Zbl 0193.50101 · Zbl 0193.50101
[37] Micallef M.J. , Stable minimal surfaces in Euclidean space , J. Differential Geom. , 19 ( 1984 ), pp. 57 - 84 . MR 739782 | Zbl 0527.32016 · Zbl 0527.32016
[38] Naitoh H. , Isotropic submanifolds with parallel second fundamental form in pm(c), Osaka J . Math. , 18 ( 1981 ), pp. 427 - 464 . MR 628843 | Zbl 0471.53036 · Zbl 0471.53036
[39] Newlander A. - Nirenberg L. , Complex analytic coordinates in almost complex manifolds , Ann. of Math. , 65 ( 1957 ), pp. 391 - 404 . MR 88770 | Zbl 0079.16102 · Zbl 0079.16102
[40] Nijenhuis A. - Woolf W.B. , Some integration problems in almost-complex and complex manifolds , Ann. of Math. , 77 ( 1963 ), pp. 424 - 489 . MR 149505 | Zbl 0115.16103 · Zbl 0115.16103
[41] Obata M. , The Gauss map of immersions of Riemannian manifolds in spaces of constant curvature , J. Differential Geom. , 2 ( 1968 ), pp. 217 - 223 . MR 234388 | Zbl 0181.49801 · Zbl 0181.49801
[42] O’Neill B. , The fundamental equations of a submersion , Michigan Math. J. , 13 ( 1966 ), pp. 459 - 469 . Article | MR 200865 | Zbl 0145.18602 · Zbl 0145.18602
[43] Poon Y.S. , Minimal surfaces in four-dimensional manifolds , M. Sc. Thesis , University of Oxford , 1983 .
[44] Rawnsley J.H. , f-structures, f-twistor spaces and harmonic maps , in Geometry Seminar Luigi Bianchi II , to appear. MR 829229 | Zbl 0592.58009 · Zbl 0592.58009
[45] Ruh E.A. , Minimal immersions of 2-spheres in S4 , Proc. Amer. Math. Soc. , 28 ( 1971 ), pp. 219 - 222 . MR 271880 | Zbl 0212.54003 · Zbl 0212.54003
[46] Ruh E.A. - Vilms J. , The tension field of the Gauss map , Trans. Amer. Math. Soc. , 149 ( 1970 ), pp. 569 - 573 . MR 259768 | Zbl 0199.56102 · Zbl 0199.56102
[47] Salamon S. , Quaternionic Kähler manifolds , Invent. Math. , 67 ( 1982 ), pp. 143 - 171 . MR 664330 | Zbl 0486.53048 · Zbl 0486.53048
[48] Salamon S. , Topics in four-dimensional Riemannian geometry , in Geometry Seminar Luigi Bianchi , Lecture Notes in Mathematics 1022 , Springer ( 1983 ), pp. 33 - 124 . MR 728393 | Zbl 0532.53035 · Zbl 0532.53035
[49] Salamon S. , Harmonic and holomorphic maps , in Geometry Seminar Luigi Bianchi II , to appear. MR 829230 | Zbl 0591.53031 · Zbl 0591.53031
[50] Smale S. , A survey of some recent developments in differential topology , Bull. Amer. Math. Soc. , 69 ( 1963 ), pp. 131 - 145 . Article | MR 144351 | Zbl 0133.16507 · Zbl 0133.16507
[51] Sacks J. - Uhlenbeck K. , The existence of minimal immersions of two-spheres , Ann. of Math. , 113 ( 1981 ), pp. 1 - 24 . MR 604040 | Zbl 0462.58014 · Zbl 0462.58014
[52] Wolf J.A. , Complex homogeneous contact manifolds and quaternionic symmetric spaces , J. Math. Mech. , 14 ( 1965 ), pp. 1033 - 1047 . MR 185554 | Zbl 0141.38202 · Zbl 0141.38202
[53] Wolf J.A. - Gray A. , Homogeneous spaces defined by Lie group automorphisms I, II , J. Differential Geom. , 2 ( 1968 ), pp. 77 - 159 . MR 236328 | Zbl 0182.24702 · Zbl 0182.24702
[54] Wolfson J.G. , Minimal surfaces in complex manifolds , Ph. D. Thesis, University of California Berkley ( 1982 ).
[55] Gauduchon P. - Lawson H.B. , Topologically nonsingular minimal cones . Zbl 0612.53007 · Zbl 0612.53007
[56] Gromov , M. , Pseudo holomorphic curves in symplectic manifolds . Zbl 0592.53025 · Zbl 0592.53025
[57] Webster S.M. , Minimal surfaces in a Kähler surface . Zbl 0561.53054 · Zbl 0561.53054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.