John C. Wood
Harmonic maps are mappings f : (M,g) ---> (N,h) between Riemannian manifolds which extremize the `Dirichlet' energy functional
They include geodesics (paths of shortest distance such as great circles on a sphere), minimal surfaces (soap films) and non-linear sigma models in the physics of elementary particles. They also have applications to the theory of liquid crystals (see below) and robotics (see, for example [Y-J Dai, M. Shoji, H. Urakawa, Harmonic maps into Lie groups and homogeneous spaces, Differential Geom. Appl. 7 (1997), 143--160; MR 98e:58054, Zbl.0882.53029].
Click here for a list of books and papers on harmonic maps.
Harmonic morphisms are mappings of Riemannian manifolds which preserve solutions of Laplace's equation; elementary examples are conformal transformations of the complex plane. The concept can be traced back to Jacobi [C.G.J. Jacobi, Über eine Lösung der partiellen Differentialgleichung , Crelle Journal für die Reine und Angewandte Mathematik 36 (1848), 113--134] who wanted to find all complex-valued solutions f to Laplace's equation on Euclidean 3-space such that any analytic function of f is still a solution---such maps are precisely the harmonic morphisms. This problem was also posed by probabilists working in stochastic processes, harmonic morphisms being the Brownian path-preserving transformations, see, for example [A. Bernard, E.A. Campbell and A.M. Davie, Brownian motion and generalized analytic and inner functions, Ann. Inst. Fourier (Grenoble) 29 (1979), 207--228; MR 81b:30088, Zbl.394.30040]. Harmonic morphisms can be characterized as harmonic maps which satisfy an additional condition called `horizontally weakly conformality' or `semiconformality', which is dual to the condition of weak conformality [B. Fuglede, Harmonic morphisms between Riemannian manifolds Ann. Inst. Fourier (Grenoble) 28 (1978), 107--144; MR 80h:58023, Zbl.369.53044] and [T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (1979), 215--229; MR 80k:58045, Zbl.421.31006]. The author has recently completed with P. Baird the first account in book form of this subject: click here for details.
A description of some of my work and its applications.
Click here for a list of my papers.
In  there is a classification of the critical points (points where f has rank less than two) of a smooth harmonic map between surfaces. This has applications to the study the Gauss curvature of constant mean curvature surfaces [S.I. Goldberg and G. Toth, On closed surfaces immersed in E3 with constant mean curvature, J. London Math. Soc. (2) 38 (1989), 333--340; MR 89m:53013, Zbl.0661.53038] and to finding meron solutions explicitly [G. Ghika and M. Visinescu, Meron solutions of the sigma-model and singular harmonic maps, Z. Phys. C, 11, (1982) 353--357; MR 83g:81068].
In  it is proved that there is no harmonic map from the torus to the sphere of degree one; for non-existence results for the Dirichlet problem see [18,19]. This is of interest in the theory of liquid crystals, see [K.S. Chou and X.P. Zhu, Some constancy results for nematic liquid crystals and harmonic maps, Ann. Inst. H. Poincaré --- Anal. non Lineaire, 12 (1995), 99--115; MR 95m:58045, Zbl.0843.35027], for other applications, see [J.F. Escobar, A. Freire and M. Minoo, L2 vanishing theorems in positive curvature, Indiana Univ. Math. J., 42 (1993), 1545--1554; MR 95c:58173, Zbl.0794.53026] and [J.F. Grotowski, Y. Shen and S. Yan, On various classes of harmonic maps, Arch. Math. (Basel), 64 (1995), 353--358; MR 96d:58032, Zbl.0817.58009].
In  following work of Glaser--Stora and Din--Zakrzewski, the classification of harmonic maps from the 2-sphere to complex projective space is carefully described; for maps into certain other symmetric spaces, see [16,22,24, 25,27,28,31]. This is the non-linear sigma-model of elementary particle physics. For some applications, see [G. Dunne, Chern--Simons solitons, Toda theories and the chiral model, Comm. Math. Phys., 150 (1992), 519--535; MR 94f:81102, Zbl.0765.35047], [B. Piette and W.J. Zakrzewski, Properties of classical solutions of the U(N) chiral sigma-models in two dimensions, Nuclear Phys. B 300 (1988), 223--237; MR 90f:81064].
In  we describe the space of harmonic 2-spheres in CP2, as a smooth submanifold of the space of all Ck maps (k>1). There remained the question of whether all Jacobi fields along such harmonic maps are integrable, i.e. do they arise from variations through harmonic maps --- see the question session in [Proceedings of the First Brazil-USA Workshop, Campinas, Brazil, June 30 -- July 7, 1996, B.N. Apanasov, S.B. Bradlow, W.A. Rodrigues, K.K. Uhlenbeck (Editors), Walter de Gruyter & Co, Berlin, New York (1997), 317--333; MR 98k:00014]; this has been answered affirmatively by the authors in  for harmonic 2-spheres in the complex projective plane, and negatively for harmonic 2-spheres in the 4-sphere in . The affirmative answer implies that the Jacobi fields form the tangent bundle to each component of the manifold of harmonic maps from S2 to CP2, thus giving the nullity of any such harmonic map; it also has bearing on the behaviour of weakly harmonic E-minimizing maps from a 3-manifold to CP2 near a singularity and the structure of the singular set of such maps from any manifold to CP2.
In  and  we construct all harmonic morphisms from S3, R3, H3 to surfaces; for an alternative proof of the R3 case see [F. Duheille, Une preuve probabiliste élémentaire d'un résultat de P. Baird et J.C. Wood, Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), 283--291; MR 98c:58034, Zbl.881.58014]. For other 3-manifolds see .
In  for maps from an Einstein 4-manifold to a surface it is shown that the existence of a harmonic morphism implies the existence of a Hermitian structure, an inverse construction is given in the anti-self dual case; see [V. Apostolov and P. Gauduchon, The Riemannian Goldberg-Sachs theorem, Internat. J. Math. 8 (1997), 421--439; MR 98g:53080, Zbl.891.53054] for the full converse. For links with shear-free ray congruences, see .
In  a framework for multivalued harmonic morphisms is given which generalizes the idea of multivalued complex analytic functions to higher dimensional domains.
In [44,48] we construct the first examples of local and global complex-valued harmonic morphisms from Euclidean spaces R2m which are holomorphic with respect to a Hermitian structure but not with respect to a Kähler structure. No such global examples can occur for m=2.
In [Harmonic morphisms with fibers of dimension one, Comm. Anal. Geom. 8 (2000), 219--265; MR 2001i:53101, Zbl.498.53046], R. Bryant showed that there are just two types of harmonic morphism with one-dimensional fibres from a space form of dimension 4 or more, the second type being induced from Killing fields. In , we extend Bryant's result to Einstein manifolds of dimension 5 or more. On the other hand, when the domain is an Einstein 4-manifold, R. Pantilie [Harmonic morphisms with 1-dimensional fibres on 4-dimensional Einstein manifolds, Comm. Anal. Geom., 10 (2002, 779-814] shows that there is just one further type described by the Beltrami fields equation. See  for applications to finding new Einstein metrics. Work continues to understand harmonic morphisms from self-dual $4$-manifolds, where a further type appears . All the new types harmonic morphisms satisfy a monopole-type equation.
[ University of Leeds ] [ School of Maths. ] [ Pure Maths ] [ J.C. Wood ]
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Last Updated 7 November 2003