The 7th DIFFERENTIAL GEOMETRY DAY

at LUND

Wednesday the 11th of May 2011


ABSTRACTS

Steen Markvorsen (Technical University of Denmark) Manifolds on fire Marina Ville (University of Tours) Knots and minimal surfaces in $4$-manifolds A complex curve $C$ in a complex surface $X$ has only isolated singularities; if we intersect a small sphere in $X$ around one of the singular points of $C$ we get a knot or a link. Such links, called {\it algebraic links} have been very much studied and are now well understood. Minimal surfaces in $\mathbb{R}^4$ can have codimension $1$ singularities; but when a singularity is isolated, it defines a knot or a link as in the complex case. Very little is known about these {\it minimal links}; I will give some examples of minimal knots and describe questions concerning them. More generally it is surprising to see how little attention has been given to the interface between minimal surfaces in $4$-manifolds and knot theory. I will try and interest the audience on this interface by suggesting possible problems. Chris Wood (University of York) Harmonic vector fields on space forms "The application of harmonic map theory to vector fields suffered an early blow when it was observed that any harmonic section of the tangent bundle of a compact manifold is necessarily parallel (Ishihara, Nouhaud, W). This problem was partially overcome by restricting the energy functional to unit vector fields (Wiegmink, Vanhecke, Gil-Medrano, W, et al). However, this is not possible on manifolds of non-zero Euler characteristic. In this talk I will show how a fairly tightly prescribed perturbation of the background geometry of the tangent bundle, away from the Sasaki metric, allows us to formulate a general theory of harmonicity for vector fields, which to a large extent preserves the existing theory of harmonic unit fields. As a criterion for optimality, it produces some interesting results when applied to Killing fields on space forms." Andrew Swann (University of Århus) Moment map geometry for three-forms