Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of cohomogeneity one Einstein metrics

Hanci Chi

TL;DR

This work establishes a sufficient condition for the existence of cohomogeneity one Einstein metrics on double disk bundles of two summands type, under a bound on the structural parameter A associated with the principal orbit G/K. By formulating the Einstein equations as a dynamical system in a compact eta-coordinate and developing a global barrier framework, the authors prove that for all d_1,d_2 >= 2 there exists a chi_{d_1,d_2} such that if A in (0,chi_{d_1,d_2}), at least one complete Einstein metric exists on M. The analysis combines local stability near the initial cone with a global barrier argument to guarantee a heteroclinic connection between the endpoint critical points, yielding new compact cohomogeneity one Einstein manifolds beyond the A=0 (product) case. The results connect to Ricci-flat first-integral structures and provide explicit examples where the bounds recover known metrics on spaces like certain S^1-bundles over quaternionic projective spaces. This approach opens a path to broader existence results via quantitative bounds on the principal-orbit data.

Abstract

This paper derives a sufficient condition for the existence of cohomogeneity one Einstein metrics on double disk bundles of two summands type. The condition is an inequality that involves geometric data from the principal orbits.

Existence of cohomogeneity one Einstein metrics

TL;DR

This work establishes a sufficient condition for the existence of cohomogeneity one Einstein metrics on double disk bundles of two summands type, under a bound on the structural parameter A associated with the principal orbit G/K. By formulating the Einstein equations as a dynamical system in a compact eta-coordinate and developing a global barrier framework, the authors prove that for all d_1,d_2 >= 2 there exists a chi_{d_1,d_2} such that if A in (0,chi_{d_1,d_2}), at least one complete Einstein metric exists on M. The analysis combines local stability near the initial cone with a global barrier argument to guarantee a heteroclinic connection between the endpoint critical points, yielding new compact cohomogeneity one Einstein manifolds beyond the A=0 (product) case. The results connect to Ricci-flat first-integral structures and provide explicit examples where the bounds recover known metrics on spaces like certain S^1-bundles over quaternionic projective spaces. This approach opens a path to broader existence results via quantitative bounds on the principal-orbit data.

Abstract

This paper derives a sufficient condition for the existence of cohomogeneity one Einstein metrics on double disk bundles of two summands type. The condition is an inequality that involves geometric data from the principal orbits.
Paper Structure (7 sections, 24 theorems, 121 equations, 1 table)

This paper contains 7 sections, 24 theorems, 121 equations, 1 table.

Table of Contents

  1. Introduction
  2. The cohomogeneity one Einstein equation
  3. Local Analysis
  4. Global Analysis
  5. Ricci–flat metrics with first integrals
  6. Computations of Rational functions
  7. Examples

Key Result

Theorem 1.1

For any $(d_1,d_2)$ with $d_1,d_2\geq 2$, there exists a constant $\chi_{d_1,d_2}\in \left(0,\frac{d_2(d_2-1)^2}{d_1^2(d_1d_2-d_2+4)}\right]$ such that if $\mathsf{G}/\mathsf{K}$ is a principal orbit with the structural triple $(d_1,d_2,A)$ and $A\in (0,\chi_{d_1,d_2})$, there exists at least one co

Theorems & Definitions (56)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Proposition 4.1
  • ...and 46 more