# Geometry, Gravity and Supersymmetry

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
45m
02.430 (Mainz Institute for Theoretical Physics, Johannes Gutenberg University)

### 02.430

#### Mainz Institute for Theoretical Physics, Johannes Gutenberg University

Staudingerweg 9 / 2nd floor, 55128 Mainz

### 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)

### Presentation Materials

 Slides
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