Rigidity and compactness of Einstein metrics (Q2771295)

The author derives a very general Weitzenböck type formula for the Lichnerowicz Laplacian acting on \(E\)-valued tensors on a Riemannian manifold \((M,g)\), where \(E\) is a vector bundle over \(M\) endowed with a metric and a compatible connection. Using this formula and the fact that the Einstein property of the metric \(g\) implies that its curvature tensor \(R^g\) considered as an \(E = \Lambda^2TM\)-valued 2-form is coclosed (\(d^*R^g =0\)), he proves the following rigidity theorems for a compact \(n\)-manifold \(M\). NEWLINENEWLINENEWLINETheorem 1. Let \(\lambda >0\) be given, there is a constant \(\varepsilon = \varepsilon (n,\lambda)>0 \) so that any Einstein metric on \(M\) with \(L^{n/2}\)-norm \(||R^g - \lambda id ||_{n/2} < \varepsilon \) has constant curvature. NEWLINENEWLINENEWLINETheorem 2. Let \(D>0\) is given, there is a constant \(\varepsilon = \varepsilon (n,D)>0 \) so that any Ricci flat metric \(g\) on \(M\) with the diameter \(<D\) and \(||R^g ||_{n/2}< \varepsilon \) is flat.NEWLINENEWLINENEWLINEOther deep results about compactness and degeneration phenomena for Einstein manifolds and, more generally, manifolds with bounded Ricci curvature, are also proved. For example,NEWLINENEWLINENEWLINETheorem 3. Given \(v>0, D>0, C \), there are only finitely many diffeomorphism classes of Einstein 4-manifolds with the volume \(>v\), the diameter \(<D\) and Euler characteristic \(\xi <C \). Moreover, a sequence of such manifolds always has a subsequence which converges to an orbifold. NEWLINENEWLINENEWLINETheorem 4. A sequence of 4-dimensional Einstein manifolds with Einstein constant 3 and volume \(> v\) has a subsequence which converges to an orbifold with only point singularities.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00021].