On the Palais-Smale condition for the prescribed Ricci curvature functional and the existence of saddle points

Abstract: Given a metric $T$, we want to solve the equation $Ric(g)=cT$ for a metric $g$ (and a constant $c$). It is well known that they are critical points of the scalar curvature $Scal$ under the constraint $\operatorname{tr}_gT=1$. We study this problem in the case of homogeneous spaces $G/H$. For the corresponding problem for Einstein metrics it was shown that $Scal$ satisfies the Palais-Smale condition, which gives rise to a large class of Einstein metrics which are saddle points of the functional. We will discuss this condition in our case and will see that Palais-Smale is not satisfied, and how one can nevertheless use a mountain pass type argument to produce saddle points. This is joint work with Artem Pulemotov.

differential geometry

Audience: researchers in the topic

Workshop on compact homogeneous Einstein manifolds

Series comments: The workshop aims at bringing together mathematicians that have contributed and are contributing to the study of Einstein metrics on the challenging class of all compact homogeneous spaces.