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  1. ; ; (, International Mathematics Research Notices)
    Abstract In a previous paper [7], the first two authors classified complete Ricci-flat ALF Riemannian 4-manifolds that are toric and Hermitian, but non-Kähler. In this article, we consider general Ricci-flat metrics on these spaces that are close to a given such gravitational instanton with respect to a norm that imposes reasonable fall-off conditions at infinity. We prove that any such Ricci-flat perturbation is necessarily Hermitian and carries a bounded, non-trivial Killing vector field. With mild additional hypotheses, we are then able to show that the new Ricci-flat metric must actually be one of the known gravitational instantons classified in [7]. 
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  2. (, Pure and Applied Mathematics Quarterly)
    The infimum of the Weyl functional is shown to be surprisingly small on many compact 4-manifolds that admit positive- scalar-curvature metrics. Results are also proved that systematically compare the scalar and self-dual Weyl curvatures of certain almost-Kähler 4-manifolds. 
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  3. (, Symmetry, Integrability and Geometry: Methods and Applications)
    The Yamabe invariant is a diffeomorphism invariant of smooth compact manifolds that arises from the normalized Einstein-Hilbert functional. This article highlights the manner in which one compelling open problem regarding the Yamabe invariant appears to be closely tied to static potentials and the first eigenvalue of the Laplacian. 
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  4. ; (, Communications in Analysis and Geometry)
    Abstract. For compact complex surfaces (M^4,J) of Kaehler type, it was previously shown (LeBrun 1999) that the sign of the Yamabe invariant Y (M) only depends on the Kodaira dimension Kod(M,J). In this paper, we prove that this pattern in fact extends to all compact complex surfaces except those of class VII. In the process, we also reprove a result from (Albanese 2021) that explains why the exclusion of class VII is essential here. 
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