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arXiv · 2503.09570

A rigidity theorem for Einstein $4$-manifolds with sectional curvature of a fixed sign, and its consequences

Luca F. Di Cerbo
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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.

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Luca F. Di Cerbo. 2025-03-12. A rigidity theorem for Einstein $4$-manifolds with sectional curvature of a fixed sign, and its consequences. https://arxiv.org/abs/2503.09570

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