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.

His 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 any two Kähler manifolds, generalizing the classical holomorphic maps between complex vector spaces.

For further information on harmonic maps interested readers should check the following sources:

Here you'll find THE HARMONIC MAPS BIBLIOGRAPHY a comprehensive list of articles on harmonic maps and here a key-word search on it.

Here you'll find HM2I-2000, Workshop on Harmonic Maps and Minimal Immersions, Caparide (Lisbon), Portugal · February 1-5, 2000


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.

Here you'll find THE BIBLIOGRAPHY OF HARMONIC MORPHISMS, a comprehensive list of articles on harmonic morphisms.

Here you'll find THE ATLAS OF HARMONIC MORPHISMS, a guide to the theory of harmonic morphisms.

Here you'll find a Bibliography on foliations.

The first international conference primarily devoted to harmonic morphisms was held 7 July - 11 July 1997 at the Université de Bretagne Occidentale, Brest, France.

The second international conference primarily devoted to harmonic morphisms was held at the Centre International de Rencontres Mathématiques in Luminy, France from 28 May to 1 June 2001. Here is a PHOTOGRAPH of the participants.