Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New quasi-Einstein metrics on a two-sphere

Alex Colling, Maciej Dunajski, Hari Kunduri, James Lucietti

Abstract

We construct all axi-symmetric non-gradient $m$-quasi-Einstein structures on a two-sphere. This includes the spatial cross-section of the extreme Kerr black hole horizon corresponding to $m=2$, as well as a family of new regular metrics with $m\neq 2$ given in terms of hypergeometric functions. We also show that in the case $m=-1$ with vanishing cosmological constant the only orientable compact solution in dimension two is the flat torus, which proves that there are no compact surfaces with a metrisable affine connection with skew Ricci tensor.

New quasi-Einstein metrics on a two-sphere

Abstract

We construct all axi-symmetric non-gradient -quasi-Einstein structures on a two-sphere. This includes the spatial cross-section of the extreme Kerr black hole horizon corresponding to , as well as a family of new regular metrics with given in terms of hypergeometric functions. We also show that in the case with vanishing cosmological constant the only orientable compact solution in dimension two is the flat torus, which proves that there are no compact surfaces with a metrisable affine connection with skew Ricci tensor.
Paper Structure (13 sections, 14 theorems, 65 equations)

This paper contains 13 sections, 14 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. General properties of the quasi-Einstein equations on surfaces
  3. Topology
  4. Prolongation
  5. Analyticity
  6. Kähler potential
  7. Symmetry reduction of the quasi-Einstein equation
  8. Inheritance of symmetry
  9. General axi-symmetric solution
  10. Extension to a two-sphere
  11. Hypergeometric function
  12. The parity of $B$
  13. Projective metrizability and quasi-Einstein with $m=-1$

Key Result

Theorem 1.1

Let $(g, X)$ be a solution to the $m$-quasi-Einstein equation (QEE) on a two-dimensional connected surface $M$, with $\mathrm{d} X^\flat$ not identically zero, and a $U(1)$ isometric action.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • ...and 20 more