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

## TL;DR

This paper classifies certain six-dimensional homogeneous Einstein metrics on S^3 x S^3, showing that under specific symmetry conditions, only known standard or nearly Kähler metrics exist, with partial results for a special case.

## Contribution

It proves that for non-trivial isotropy groups other than Z_2, the only Einstein metrics are standard or nearly Kähler, advancing the classification of Einstein metrics on S^3 x S^3.

## Key findings

- Einstein metrics with non-trivial isotropy groups are either standard or nearly Kähler.
- Partial results are obtained for the case where the isotropy group is Z_2.
- The classification leverages representation theory and computer algebra methods.

## Abstract

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 apparently out of reach. Instead, we consider the case where the isotropy group $K$ of $g$ in the group of motions is non-trivial. When $K\not\cong \mathbb{Z}_2$ 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 $K\cong \mathbb{Z}_2$ we present partial results.