A rigidity theorem for Einstein $4$-manifolds with semi-definite sectional curvature, and its consequences
Luca F. Di Cerbo
TL;DR
This work establishes a sharp pointwise bound $|s|/sqrt(6) >= |W^+|+|W^-|$ for oriented 4D Einstein manifolds with semi-definite sectional curvature and completely classifies closed saturating examples via the cosmological constant, revealing universal covers that are either S^2 x S^2, R^4, or H^2_R x H^2_R. It develops the curvature analysis needed for this rigidity, proving a rigidity theorem that forces local symmetry in the saturating non-positive case and yields a global product structure on the universal cover with tau(M)=0. The paper then derives geography-type consequences for non-positively curved Einstein and Kaehler-Einstein 4-manifolds, including a sharp Kaehler-Einstein Gromov-Luck type inequality and implications for aspherical surfaces of general type. Collectively, it ties curvature pinching to global topology and complex-analytic structures, and highlights several open questions about the existence and shape of aspherical complex surfaces with constrained invariants.
Abstract
Any oriented $4$-dimensional Einstein metric with semi-definite sectional curvature satisfies the pointwise inequality \[ \frac{|s|}{\sqrt{6}}\geq|W^+|+|W^-|, \] where $s$, $W^+$ and $W^-$ are respectively the scalar curvature, the self-dual and anti-self-dual Weyl curvatures. We give a complete characterization of closed $4$-dimensional Einstein metrics with semi-definite sectional curvature saturating this pointwise inequality. We then present further consequences of this circle of ideas, in particular to the study of the geography of non-positively curved closed Einstein and Kaehler-Einstein $4$-manifolds. In the Kaehler-Einstein case, we obtain a sharp Gromov-Lueck type inequality.