Non-existence of cohomogeneity one Einstein metrics of two summands

TL;DR

This paper proves that certain compact manifolds with specific symmetry properties cannot admit cohomogeneity one Einstein metrics, introducing a new obstruction method applicable even when principal orbits are homogeneous Einstein.

Contribution

It establishes a non-existence result for cohomogeneity one Einstein metrics on double disk bundle manifolds with split principal orbits, using a novel phase space barrier argument.

Findings

No cohomogeneity one Einstein metrics exist for the studied class of manifolds.

The phase space barrier method provides a new obstruction criterion.

Applicable even when principal orbits are homogeneous Einstein.

Abstract

We prove the non-existence of cohomogeneity one Einstein metrics on a class of compact manifolds arising as double disk bundles, whose principal orbits split into two inequivalent irreducible summands. The proof uses a phase space barrier argument that yields a new obstruction to the existence of closed cohomogeneity one Einstein metrics, even when the principal orbits admit homogeneous Einstein metrics.