The gap phenomenon for conformally related Einstein metrics (Q6634559)

This paper studies how many Einstein metrics can there be in a given conformal class \([g]\). The main result that the authors prove is the following:\N\NTheorem. Assume the conformal manifold \((M, [g]\)) of dimension \(n\geq3\) is not locally conformally flat. Then the dimensions \(d_{aE}\) of the space of almost Einstein scales and \(d_{ncK}\) of the space of normal conformal Killing fields satisfy the following:\N\begin{itemize}\N\item[1.] if \(g\) has definite signature, then \(d_{aE} \leq n-3\) and \(d_{ncK} \leq \frac{(n-3)(n-4)}{2}\)\N\item[2.] if \(g\) has Lorentzian signature, then \(d_{aE} \leq n-2\) and \(d_{ncK} \leq \frac{(n-3)(n-2)}{2}\)\N\item[3.] if \(g\) has signature \((p,q)\) with \(2\leq p \leq q\), then \(d_{aE} \leq n-1\) and \(d_{ncK} \leq \frac{(n-2)(n-1)}{2}\)\N\end{itemize}\NMoreover, all upper bounds are sharp, that is, there exist conformal classes where these inequalities are equalities.