Mathematics, Faculty of ScienceSigmundur GudmundssonResearch Interests |
Recent work is mainly devoted to the existence theory of complex-valued harmonic morphisms from Riemannian homogeneous spaces of various types, such as symmetric spaces and semisimple, solvable and nilpotent Lie groups.
Lecture Notes: Harmonic Morphisms - Basics -
Lecture Notes: Harmonic Morphisms - Some Existence Theory -
Harmonic maps are very important both in classical and modern differential geometry. The best known application is their parametrization of both geodesics and minimal surfaces in Riemannian manifolds. Other important examples are the holomorphic maps between Kähler manifolds, generalizing the classical holomorphic maps between complex vector spaces.
For further information on harmonic maps interested readers should check the following sources:
HM^{2}I-2000, Workshop on Harmonic Maps and Minimal Immersions, 1 - 5 February 2000 (LISBON/Portugal)
Harmonic Morphisms 28 May to 1 June 2001 (LUMINY/France) - Photograph
John C. Wood's 60th Birthday: A Harmonic Map Fest, 7 - 10 September 2009 (CAGLIARI/Italy)
50 Years of Harmonic maps, Differential Geometry Workshop 15 - 16 May 2014 (LUND/Sweden) - Photograph
Differential Geometry Workshop: Harmonic maps, biharmonic maps, harmonic morphisms and related topics, 10 - 12 June 2015 (CAGLIARI/Italy) - Photograph
The case when the manifold N is a surface, i.e. 2-dimensional, is of particular interest. Then harmonic morphisms (M,g) --> (N^2,h) have many nice geometric properties. Every regular fibre of such a map is a minimal submanifold of (M,g) of codimension 2. This means that harmonic morphisms are useful tools for the construction of such submanifolds. Interesting examples are holomorphic maps from Kähler manifolds to Riemann surfaces.
Most of my recent work is devoted to the existence theory of complex-valued harmonic morphisms from Riemannian homogeneous spaces of various types, such as symmetric spaces and semisimple, solvable and nilpotent Lie groups.
The Bibliography of Harmonic Morphisms
The Atlas of Harmonic Morphisms
The Bibliography on Foliations