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Non-existence of cohomogeneity one Einstein metrics of two summands

Hanci Chi

TL;DR

The work addresses the existence problem for cohomogeneity one Einstein metrics on compact double disk bundles where the principal orbit splits into two inequivalent irreducible summands. It casts the cohomogeneity one Einstein equations as a phase-space dynamical system and develops a barrier-based method, introducing the $P$-barrier and an $\omega$-barrier together with a detailed algebraic sign analysis to obstruct trajectories. The main result provides a new nonexistence obstruction for invariant cohomogeneity one Einstein metrics when the coupling parameter $A$ lies in a prescribed interval bounded below by $\Psi_{d_1,d_2}$, extending Böhm-type results to settings where the principal orbit is Einstein but global metrics fail to exist. The findings sharpen understanding of Einstein metrics on double disk bundles and offer a systematic barrier framework potentially applicable to other cohomogeneity-one problems.

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.

Non-existence of cohomogeneity one Einstein metrics of two summands

TL;DR

The work addresses the existence problem for cohomogeneity one Einstein metrics on compact double disk bundles where the principal orbit splits into two inequivalent irreducible summands. It casts the cohomogeneity one Einstein equations as a phase-space dynamical system and develops a barrier-based method, introducing the -barrier and an -barrier together with a detailed algebraic sign analysis to obstruct trajectories. The main result provides a new nonexistence obstruction for invariant cohomogeneity one Einstein metrics when the coupling parameter lies in a prescribed interval bounded below by , extending Böhm-type results to settings where the principal orbit is Einstein but global metrics fail to exist. The findings sharpen understanding of Einstein metrics on double disk bundles and offer a systematic barrier framework potentially applicable to other cohomogeneity-one problems.

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.
Paper Structure (8 sections, 16 theorems, 73 equations, 1 table)

This paper contains 8 sections, 16 theorems, 73 equations, 1 table.

Table of Contents

  1. Introduction
  2. Böhm's non-existence theorem
  3. Barrier of non-existence: Part I
  4. Barrier of non-existence: Part II
  5. Appendix
  6. Formula of $P_Y$, $P_X$, $Q_Y$ and $Q_X$
  7. Formula of $\omega_i$
  8. Properties of some elementary functions

Key Result

Theorem 1.1

Define If $\mathsf{G}/\mathsf{K}$ is a principal orbit in eqn_homogeneous fibration with $d_2\geq d_1\geq 2$, $(d_1,d_2)\notin\{(2,2), (2,3), (2,4)\}$, and $A\in\left[\Psi_{d_1,d_2},\frac{1}{n+d_1}\frac{d_2(d_2-1)^2}{4(d_1-1)}\right)$, there does not exist any $\mathsf{G}$-invariant cohomogeneity one Einst

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 4.3
  • ...and 22 more