Speaker
Dr
Alexander Haupt
(University of Hamburg)
Description
The classification of homogeneous compact Einstein manifolds in dimension six is an open problem.
We consider the remaining open case, namely left-invariant Einstein metrics $g$ on $G = \mathrm{SU}(2) \times \mathrm{SU}(2) = S^3 \times S^3$.
Einstein metrics are critical points of the total scalar curvature functional for fixed volume.
The scalar curvature $S$ of a left-invariant metric $g$ is constant and can be expressed as a rational function in the parameters determining the metric.
The critical points of $S$, subject to the volume constraint, are given by the zero locus of a system of polynomials in the parameters.
In general, however, the determination of the zero locus is out of reach with current technology. Instead, we consider the case where $g$ is, in addition to being left-invariant, assumed to be invariant under a non-trivial finite group $\Gamma$ of isometries.
When $\Gamma\not\cong \mathbb{Z}_2$ we prove that the Einstein metrics on $G$ are given by (up to homothety) either the standard metric or the nearly K\"ahler metric, based on representation-theoretic arguments and computer algebra.
For the remaining case $\Gamma\cong \mathbb{Z}_2$ we present partial results.
Primary author
Dr
Alexander Haupt
(University of Hamburg)
Co-authors
Mr
David Lindemann
(University of Hamburg)
Dr
Florin Belgun
(Humboldt-Universität zu Berlin)
Prof.
Vicente Cortés
(Department of Mathematics, University of Hamburg)
