# School of Mathematics

## Prof John C Wood

Professor of Pure Mathematics

Pure Mathematics

### Contact details

Room: 8.18

Tel: +44 (0)113 3435106

Email: J.C.Wood @ leeds.ac.uk

### Keywords

Harmonic map

Harmonic morphism

Differential geometry

Twistor theory

Laplace's equation

## Research interests

My main interests are harmonic maps and harmonic morphisms in differential geometry.

Harmonic maps are mappings (`transformations') between Riemannian manifolds which extremize a natural `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 and robotics. I work in the construction and classification of harmonic maps, especially from surfaces to symmetric spaces, and, with others, have given a completely explicit construction for the important case of maps from the Riemann sphere to the unitary group and other classical groups and symmetric spaces.

Harmonic morphisms are mappings of Riemannian manifolds which preserve solutions of Laplace's equation; elementary examples are conformal transformations of the complex plane. The first serious study was by Jacobi who, in 1848, 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]. Harmonic morphisms can be characterized as harmonic maps which satisfy an additional condition called `horizontally weakly conformality' or `semiconformality' [B. Fuglede, Harmonic morphisms between Riemannian manifolds Ann. Inst. Fourier (Grenoble) 28 (1978), 107--144] and [T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (1979), 215--229]. My work has been to understand these mappings, starting with a complete solution to Jacobi's problem. I have also found links with other concepts, such as complex (`Hermitian') structures and the shear-free ray congruences of Mathematical Physics; in particular, I found a twistor theory for harmonic morphisms with values in a surface. In 2003, I wrote with P. Baird the first account in book form of this subject, which has become the standard text.

### Useful links

## Publications

**Svensson M, Wood JC** Harmonic maps into the exceptional symmetric space G 2 /SO(4) *Journal of the London Mathematical Society*, **91**, 291-319, 2015

DOI:10.1112/jlms/jdu073

View abstract

**Svensson M, Wood JC** New constructions of twistor lifts for harmonic maps *Manuscripta Mathematica*, **144**, 457-502, 2014

DOI:10.1007/s00229-014-0659-9

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**Svensson M, Wood JC** Filtrations, Factorizations and Explicit Formulae for Harmonic Maps *Communications in Mathematical Physics*, **310**, 99-134, 2012

DOI:10.1007/s00220-011-1398-3

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**Baird P, Wood JC** Harmonic morphisms and bicomplex manifolds *J GEOM PHYS*, **61**, 46-61, 2011

DOI:10.1016/j.geomphys.2010.09.007

**Wood JC ** *Thirty-nine years of harmonic maps*HARMONIC MAPS AND DIFFERENTIAL GEOMETRY, 1-6 2011

**Wood JC ** *Explicit constructions of harmonic maps*HARMONIC MAPS AND DIFFERENTIAL GEOMETRY, 41-73 2011

**Ferreira MJ, Simoes BA, Wood JC** All harmonic 2-spheres in the unitary group, completely explicitly *MATH Z*, **266**, 953-978, 2010

DOI:10.1007/s00209-009-0607-7

**Wood JC** Conformal variational problems, harmonic maps and harmonic morphisms *Rendiconti del Seminario Matematico*, **67**, 395-406, 2009

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**Lemaire L, Wood JC** JACOBI FIELDS ALONG HARMONIC 2-SPHERES IN 3-AND 4-SPHERES ARE NOT ALL INTEGRABLE *TOHOKU MATH J*, **61**, 165-204, 2009

**Baird P, Wood JC** Harmonic morphisms from Minkowski space and hyperbolic numbers *B MATH SOC SCI MATH*, **52**, 195-209, 2009

**Hélein F; Wood JC ** Harmonic maps: Dedicated to the memory of James Eells. , 417-491 2008

DOI:10.1016/B978-044452833-9.50009-7

**Pantilie R, Wood JC** Twistorial harmonic morphisms with one-dimensional fibres on self-dual four-manifolds *Quarterly Journal of Mathematics*, **57**, 105-132, 2006

DOI:10.1093/qmath/hah063

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**Wood JC ** *Harmonic morphisms between Riemannian manifolds (article)*Modern trends in geometry and topology : proceedings of the 7th International Workshop on Differential Geometry and Its Applications 2006

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**Pantilie R, Wood JC** Topological restrictions for circle actions and harmonic morphisms *MANUSCRIPTA MATH*, **110**, 351-364, 2003

DOI:10.1007/s00229-002-0342-4

**Baird P; Wood JC ** *Harmonic morphisms between Riemannian manifolds*. Oxford University Press 2003

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**Lemaire L, Wood JC** Jacobi fields along harmonic 2-spheres in CP2 are integrable *J LOND MATH SOC*, **66**, 468-486, 2002

DOI:10.1112/S0024610702003496

**Pantilie R, Wood JC** A new construction of Einstein self-dual metrics *The Asian Journal of Mathematics*, **6**, 337-348, 2002

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**Pantilie R, Wood JC** Harmonic morphisms with one-dimensional fibres on Einstein manifolds *Transactions of the American Mathematical Society*, **354**, 4229-4243, 2002

DOI:10.1090/S0002-9947-02-03044-1

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**Wood JC** Jacobi fields along harmonic maps *Contemporary Mathematics*, **308**, 329-340, 2002

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**Pantilie R, Wood JC** New results on harmonic morphisms with one-dimensional fibres *Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie*, **43**, 355-365, 2000

**Baird P, Wood JC** Harmonic morphisms, conformal foliations and shear-free ray congruences *Bulletin of the Belgian Mathematical Society, Simon Stevin*, **5**, 549-564, 1998

**Baird P, Wood JC** Weierstrass representations for harmonic morphisms on Euclidean spaces and spheres *Mathematica Scandinavica*, **81**, 283-300, 1997

**Gudmundsson S, Wood JC** Harmonic morphisms between almost Hermitian manifolds *Unione Matematica Italiana. Bollettino. B*, **11 (Suppl)**, 185-197, 1997

**Lemaire L, Wood JC** On the space of harmonic 2-spheres in CP^{2} *International Journal of Mathematics*, **7**, 211-225, 1996

Wood JC *Infinitesimal Deformations of Harmonic Maps and Morphisms*Symmetry in geometry and physics. In honour of Dmitri Alekseevsky

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Baird P; Wood JC *Harmonic morphisms and shear-free ray congruences*

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Ou Y-L; Wood JC On the classification of quadratic harmonic morphisms between Euclidean
spaces.

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