Mathematics, Faculty of Science

Sigmundur Gudmundsson

Research Interests

 

Sigmundur Gudmundsson is an Associate Professor mainly interested in differential geometry and analysis on manifolds, especially the study of harmonic maps and harmonic morphisms between Riemannian manifolds. This field is a facinating focal point for differential geometry, analysis, topology, algebra and even probability theory.

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

A Riemannian metric g on a smooth manifold M gives rise to the notion of a real-valued harmonic function (M,g) --> R. This generalizes the classical situation when the manifold is a flat Euclidean space. One can generalize further to harmonic maps (M,g) --> (N,h) between any two Riemannian manifolds. Harmonic maps are solutions to an elliptic system of partial differential equations, which in general is non-linear.

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:

Harmonic maps, Harmonic Morphisms and Related Topics 7 July - 11 July 1997 (BREST/France)

HM2I-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

Harmonic Morphisms

Harmonic morphisms are maps (M,g) --> (N,h) between Riemannian manifolds which pull back locally defined real-valued harmonic functions on (N,h) to harmonic functions on (M,g). They form a special class of harmonic maps, namely those that are horizontally conformal. This means that harmonic morphisms are solutions to an over-determined, non-linear system of partial differential equations.

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