Harmonic morphisms and circle actions on 3- and 4-manifolds
Annales de l'Institut Fourier, Tome 40 (1990) no. 1, pp. 177-212.

On considère les morphismes harmoniques comme généralisation naturelle des fonctions analaytiques qu’on rencontre dans la théorie des surfaces de Riemann. On montre que chaque variété fermée et analytique à 3 dimensions qui supporte un morphisme harmonique à valeurs dans une surface de Riemann est un espace fibré de Seifert. On étudie les morphismes harmoniques φ:MN définies sur une variété fermée à 4 dimensions et à valeurs dans une variété à 3 dimensions. Ceux-ci déterminent une action du cercle sur M qui est localement différentiable, peut-être avec des points fixes. Par conséquent la topologie de M est limitée. Dans chaque cas, un morphisme harmonique φ:MN défini sur une variété fermée à n+1 dimensions et à valeurs dans une variété à n dimensions (n2, avec M, N analytiques dans le cas où n=2) détermine une action du cercle sur M qui est localement différentiable.

Harmonic morphisms are considered as a natural generalization of the analytic functions of Riemann surface theory. It is shown that any closed analytic 3-manifold supporting a non-constant harmonic morphism into a Riemann surface must be a Seifert fibre space. Harmonic morphisms φ:MN from a closed 4-manifold to a 3-manifold are studied. These determine a locally smooth circle action on M with possible fixed points. This restricts the topology of M. In all cases, a harmonic morphism φ:MN from a closed (n+1)-dimensional manifold to an n-dimensional manifold (n2, with M, N analytic in the case n=2) determines a locally smooth circle action on M.

@article{AIF_1990__40_1_177_0,
     author = {Baird, Paul},
     title = {Harmonic morphisms and circle actions on 3- and 4-manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {177--212},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {40},
     number = {1},
     year = {1990},
     doi = {10.5802/aif.1210},
     zbl = {0676.58023},
     mrnumber = {91e:57025},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1210/}
}
TY  - JOUR
AU  - Baird, Paul
TI  - Harmonic morphisms and circle actions on 3- and 4-manifolds
JO  - Annales de l'Institut Fourier
PY  - 1990
SP  - 177
EP  - 212
VL  - 40
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1210/
DO  - 10.5802/aif.1210
LA  - en
ID  - AIF_1990__40_1_177_0
ER  - 
%0 Journal Article
%A Baird, Paul
%T Harmonic morphisms and circle actions on 3- and 4-manifolds
%J Annales de l'Institut Fourier
%D 1990
%P 177-212
%V 40
%N 1
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1210/
%R 10.5802/aif.1210
%G en
%F AIF_1990__40_1_177_0
Baird, Paul. Harmonic morphisms and circle actions on 3- and 4-manifolds. Annales de l'Institut Fourier, Tome 40 (1990) no. 1, pp. 177-212. doi : 10.5802/aif.1210. https://aif.centre-mersenne.org/articles/10.5802/aif.1210/

[1] P. Baird, Harmonic maps with symmetry, harmonic morphisms and deformations of metrics, Research Notes in Math., 87, Pitman, (1983). | MR | Zbl

[2] P. Baird, Harmonic morphisms onto Riemann surfaces and generalized analytic functions, Ann. Inst. Fourier, Grenoble, 37-1 (1987), 135-173. | Numdam | MR | Zbl

[3] P. Baird and J. Eells, A conservation law for harmonic maps, Geometry Symp. Utrecht (1980), Springer Notes, 894 (1981), 1-25. | MR | Zbl

[4] P. Baird and J.C. Wood, Bernstein theorems for harmonic morphisms from R3 and S3, Math. Ann. 280 (1988), 579-603. | MR | Zbl

[5] P. Baird and J.C. Wood, Harmonic morphisms and conformal foliations of 3-dimensional space forms, preprint. | Zbl

[6] A. Bernard, E.A. Campbell and A.M. Davie, Brownian motion and generalized analytic and inner functions, Ann. Inst. Fourier, Grenoble, 29-1 (1979), 207-228. | Numdam | MR | Zbl

[7] G.E. Bredon, Introduction to Compact Transformation Groups, Academic Press, (1972). | MR | Zbl

[8] M. Brelot, Lectures on Potential Theory, Tata Institute of Fundamental Research, Bombay, (1960). | MR | Zbl

[9] C. Constantinescu and A. Cornea, Compactifications of harmonic spaces, Nagoya Math. J., 25 (1965), 1-57. | MR | Zbl

[10] J. Eells, Regularity of certain harmonic maps, Global Riemannian Geometry, Durham (1982), E. Horwood (1984), 137-147. | Zbl

[11] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc., 10 (1978), 1-68. | MR | Zbl

[12] J. Eells and L. Lemaire, Selected topics in harmonic maps, C.B.M.S. Regional Conference Series 50, A.M.S. (1983). | MR | Zbl

[13] J. Eells and A. Ratto, Harmonic maps between spheres and ellipsoids, preprint, I.H.E.S., (1988).

[14] J. Eells and J.H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. | MR | Zbl

[15] D.B.A. Epstein, Periodic flows on 3-manifolds, Annals of Math., 95 (1972), 68-82. | MR | Zbl

[16] D.B.A. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier, Grenoble 26-1 (1976), 265-282. | Numdam | MR | Zbl

[17] D.B.A. Epstein, Pointwise periodic homeomorphisms, Proc. London Math. Soc., (3) 42 (1981), 415-460. | MR | Zbl

[18] R. Fintushel, Classification of circle actions on 4-manifolds, Trans. Amer. Math. Soc., 242 (1978), 377-390. | MR | Zbl

[19] B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier, Grenoble, 28-2 (1978), 107-144. | Numdam | MR | Zbl

[20] M. Greenberg, Lectures on algebraic topology, Benjamin, 1966.

[21] R.E. Greene and H. Wu, Embeddings of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier, Grenoble, 12 (1962), 415-571. | Numdam

[22] D.A. Hoffman and R. Osserman, The geometry of the generalized Gauss map, Memoirs Amer. Math. Soc., vol 28, n° 236 (1980). | MR | Zbl

[23] T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ., 19 (1979), 215-229. | MR | Zbl

[24] C.G.J. Jacobi, Uber Eine Particuläre Lösing der Partiellen Differential Gleichung ∂2v/∂x2 + ∂2v/∂y2 + ∂2v/∂z2 = 0, Crelle Journal für die reine und angewandte Mathematik, 36 (1847), 113-134.

[25] J. Milnor, Microbundles I, Topology, Vol. 3 Suppl. 1, (1964), 53-80. | MR | Zbl

[26] J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Math. Studies, 61, P.U.P., 1968. | MR | Zbl

[27] J. Milnor, On the 3-dimensional Brieskorn manifolds M(p, q, r), Knots, Groups and 3-manifolds, ed. L.P. Neuwith, Annals of Math. Studies, 84, P.U.P. (1975), 175-225. | MR | Zbl

[28] P.S. Pao, Non linear circle actions on the 4-sphere and twisting spun knots, Topology, 17 (1978), 291-296. | MR | Zbl

[29] A. Ratto, Harmonic maps from deformed spheres to spheres, preprint. | Zbl

[30] D. Rolfsen, Knots and Links, Mathematics Lecture Series 7, Publish or Perish, (1976). | MR | Zbl

[31] P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc., 15 (1983). | MR | Zbl

[32] C.L. Siegel, Topics in Complex Function Theory I, Wiley, 1969. | Zbl

[33] R.T. Smith, Harmonic mappings of spheres, Amer. J. Math., 97 (1975), 364-385. | MR | Zbl

[34] N. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press, Princeton, 1951. | MR | Zbl

[35] W.P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc., 6 (1982), 357-381. | Zbl

[36] I. Vaisman, Conformal foliations, Kodai Math. J., 2 (1979), 26-37. | MR | Zbl

[37] J.C. Wood, Harmonic morphisms, foliations and Gauss maps, Complex Differential Geometry, ed. Y.T. Siu, Contemporary Mathematics, 49, A.M.S., (1986), 145-183. | MR | Zbl

[38] Y.H. Yiu, Quadratic forms between spheres and the non-existence of sums of squares formulae, Math. Proc. Camb. Phil. Soc., 100 (1986), 493-504. | Zbl

Cité par Sources :