Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The gap phenomenon for conformally related Einstein metrics

Jan Gregorovič, Josef Šilhan

Abstract

We determine the submaximal dimensions of the spaces of almost Einstein scales and normal conformal Killing fields for connected conformal manifolds. The results depend on the signature and dimension $n$ of the conformally nonflat conformal manifold. In the Riemannian case, these two dimensions are at most $n-3$ and $\frac{(n-4)(n-3)}{2}$, respectively. In the Lorentzian case, these two dimensions are at most $n-2$ and $\frac{(n-3)(n-2)}{2}$, respectively. In the remaining signatures, these two dimensions are at most $n-1$ and $\frac{(n-2)(n-1)}{2}$, respectively. This upper bound is sharp and to realize examples of submaximal dimensions, we first provide them directly in dimension 4. In higher dimensions, we construct the submaximal examples as the (warped) product of the (pseudo)-Euclidean base of dimension $n-4$ with one of the 4-dimensional submaximal examples.

The gap phenomenon for conformally related Einstein metrics

Abstract

We determine the submaximal dimensions of the spaces of almost Einstein scales and normal conformal Killing fields for connected conformal manifolds. The results depend on the signature and dimension of the conformally nonflat conformal manifold. In the Riemannian case, these two dimensions are at most and , respectively. In the Lorentzian case, these two dimensions are at most and , respectively. In the remaining signatures, these two dimensions are at most and , respectively. This upper bound is sharp and to realize examples of submaximal dimensions, we first provide them directly in dimension 4. In higher dimensions, we construct the submaximal examples as the (warped) product of the (pseudo)-Euclidean base of dimension with one of the 4-dimensional submaximal examples.
Paper Structure (11 sections, 11 theorems, 53 equations)

This paper contains 11 sections, 11 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Conformal geometry
  3. Notation and conventions
  4. Tractor calculus and invariant operators
  5. Elementary observation about submaximal dimensions $d_{aE}$ and $d_{ncK}$
  6. The upper bound on submaximal dimensions
  7. Realization of submaximal examples as warped products
  8. Submaximal examples in dimension 4
  9. Warped product
  10. Submaximal examples in a general dimension
  11. Final comments and open questions

Key Result

Theorem 1.1

Assume the conformal manifold $(M,[g])$ of dimension $n \geq 3$ is not locally conformally flat. Then dimensions $d_{aE}$ of the space almost Einstein scales and $d_{ncK}$ of of the space normal conformal Killing fields satisfy the following: Moreover, all upper bounds are sharp, i.e., there exist conformal classes where these inequalities are equalities.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • proof
  • Proposition 4.2
  • ...and 11 more