## The 4th DIFFERENTIAL GEOMETRY DAY

## at LUND

## Wednesday the 14th of May 2008

## ABSTRACTS

Martin Svensson(Odense)Harmonic morphisms are maps between Riemannian manifolds which preserve Laplace's equation. They are examples of harmonic maps, satisfying some additional conformality conditions. While giving the maps interesting geometrical properties, these conditions amount to an over-determined, non-linear system of partial differential equations. As a result, harmonic morphisms are in general difficult to find. However, we know many examples of harmonic morphisms, and in some special situations, even classifications. In this talk I will discuss the open question of the existence problem of harmonic morphisms from symmetric spaces, and how the study of solvable groups has helped us to get closer to a solution. In this context, I will also discuss some particular examples where no harmonic morphisms can exist, not even locally.Harmonic morphisms from solvable Lie groupsMattias Dahl(Stockholm)On a compact manifold $M$ the Einstein-Hilbert functional associates to a Riemannian metric $g$ the total scalar curvature $\int_M \scal^g dv^g$. The Yamabe invariant of $M$ is computed by first taking the infimum of this quantity over metrics of unit volume in a conformal class, followed by the supremum over all conformal classes on $M$. The resulting invariant is a good candidate for a quantitative measure of "how much" positive scalar curvature a given manifold can have. In this talk I will discuss what is known about the Yamabe invariant, and try to explain some new results which control the Yamabe invariant when surgery is performed on the manifold.The Yamabe invariantGudlaugur Thorbergsson(Köln)Paul Funk showed in 1913 that an even function $f$ on the two-sphere is determined by its integrals $\hat f(\xi)$ over the great circles on the sphere. Recently, Helgason proposed a generalization of the transform $f\to \hat f$ to compact symmetric spaces. We will discuss the injectivity of the transform and a support theorem that is valid for compact symmetric spaces which are not spheres. The results are joint work with Sebastian Klein and Laszlo Verhoczki.On the Funk transform on compact symmetric spacesVicente Cortés(Hamburg)I will survey recent developments which led to the solution of the Benson-Gordon conjecture on Kähler quotients of completely solvable Lie groups and to the classification of compact aspherical Kähler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. The talk is based on arXiv:math/0601616.Aspherical Kähler manifolds with solvable fundamental group