We are gald to announce the upcoming talk given by Sigmundur Gudmundsson in the Oberseminar "Geometrie und Visualisierung" (Geometry and Visualization).

Title
Harmonic Morphisms and Minimal Foliations on Low-Dimensional Lie Groups - Computer Algebra in Action

Affiliation
Mathematical Institute of Lund University, Lund, Sweden

Date, Time and Place
31.03.2022, 3 pm, Arnimallee 6 - Room 108/109

Abstract

Harmonic morphisms are maps F: (M^m,g) → (N^n,h) between Riemannian manifolds which preserve harmonic functions on the manifolds involved. In the late 1970s they were characterised, independently by B. Fuglede and T. Ishihara, as those harmonic maps which are horizontally conformal. This means that they are solutions to an over-determined non-linear system of partial differential equations. For this reason their existence theory becomes non-trivial. From the geometric point of view, the case when the codomain is a surface (n=2) is particularly interesting. Then each regular fibre of F: (M,g) → (N^2,h) is a minimal submanifold of (M,g) of codimension 2. This means that harmonic morphisms can be seen as a useful tool for constructing such interesting geometric objects.

In the first part of this talk, we will describe connections with classical complex analysis and a certain duality to the theory of minimal surfaces in Euclidean 3-space. We will then produce multi-dimensional conformal and minimal foliations on low-dimensional Lie groups and show how these induce local complex-valued solutions to our overdetermined non-linear problem.

Biography

Name: Sigmundur Gudmundsson (b. 1960)
University Studies: Reykjavik, Bonn and Leeds
Postdoc: Reykjavik and Copenhagen
Employment: University of Lund (since 1994)