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 definite signature, these two dimensions are at most and , respectively. In Lorentzian signature, 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.