24-28 April 2017
Mainz Institute for Theoretical Physics, Johannes Gutenberg University
Europe/Berlin timezone

Left-invariant Einstein metrics on $S^3 \times S^3$

26 Apr 2017, 15:45
02.430 (Mainz Institute for Theoretical Physics, Johannes Gutenberg University)


Mainz Institute for Theoretical Physics, Johannes Gutenberg University

Staudingerweg 9 / 2nd floor, 55128 Mainz


Dr Alexander Haupt (University of Hamburg)


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)


Mr David Lindemann (University of Hamburg) Dr Florin Belgun (Humboldt-Universität zu Berlin) Prof. Vicente Cortés (Department of Mathematics, University of Hamburg)

Presentation Materials

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