The 4th DIFFERENTIAL GEOMETRY DAY
Wednesday the 14th of May 2008
Martin Svensson (Odense)
Harmonic morphisms from solvable Lie groups
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,
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.
Mattias Dahl (Stockholm)
The Yamabe invariant
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.
Gudlaugur Thorbergsson (Köln)
On the Funk transform on compact symmetric spaces
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.
Vicente Cortés (Hamburg)
Aspherical Kähler manifolds with solvable fundamental group
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.