Einstein Metrics, Harmonic Forms, and Symplectic Four-Manifolds

Abstract

If $M$ is the underlying smooth oriented $4$-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics $h$ on $M$ such that $W^{+}(\omega ,\omega )>0$, where $W^{+}$ is the self-dual Weyl curvature of $h$, and $\omega$ is a non-trivial self-dual harmonic $2$-form on $(M,h)$. While this open region in the space of Riemannian metrics contains all the known Einstein metrics on $M$, we show that it contains no others. Consequently, it contributes exactly one connected component to the moduli space of Einstein metrics on $M$.

in Annals of Global Analysis and Geometry (2015)

References in corpus (1)

• On Conformally Kaehler, Einstein Manifolds

Cited by in corpus (2)

• Cosmology: macro and micro
• Berger curvature decomposition, Weitzenböck formula, and canonical metrics on four-manifolds