The problem of finding metrics with constant Q-curvature in a prescribed conformal class is an important fourth-order cousin of the Yamabe problem. In this talk, I will explain how certain variational bifurcation techniques used to prove non-uniqueness of solutions to the Yamabe problem also yield non-uniqueness results for the constant Q-curvature problem. However, special emphasis will be given to the differences between multiplicity phenomena in these two variational problems. This is based on joint work with P. Piccione and Y. Sire.
Variational Methods in Geomatry
A Riemannian manifold is called Besse, if all of its geodesics are periodic. The goal of this talk is to study the energy functional on the free loop space of a Besse manifold. In particular, we show that this is a perfect Morse-Bott function for the rational, relative, S1-equivariant cohomology of the free loop space. We will show how this result is crucial in proving a conjecture of Berger for spheres of dimension at least 4, although it might be useful for proving the conjecture in full generality.
In the 80s Pitts-Rubinstein conjectured that certain kinds of Heegaard surfaces in three-manifolds can be isotoped to index 1 minimal surfaces. I'll describe in detail a proof of their conjecture and some applications. This is joint work with Liokumovich and Song.
We consider the classical problem of prescribing the scalar curvature of a manifold via conformal deformation of the metric, dating back to works by Kazdan and Warner. This problem is mainly understood in low dimensions, where blow-ups of solutions are proven to be "isolated simple". We find natural conditions to guarantee this also in arbitrary dimensions, when the prescribed curvatures are Morse functions. As a consequence, we improve some pinching conditions in the literature and derive existence
results of new type. This is joint work with M. Mayer.