Compact half-conformally flat manifolds of negative type with bounded L2 energy, small scalar curvature, and non-collapsing have bounded Betti numbers; related singularity models are 2-ended and asymptotically Kähler, with decay rates O(r^{-4}) or better for certain self-dual forms on ALE ends.
Heras The kirchhoff gauge
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A Gap Theorem for Half-Conformally Flat Manifolds
Compact half-conformally flat manifolds of negative type with bounded L2 energy, small scalar curvature, and non-collapsing have bounded Betti numbers; related singularity models are 2-ended and asymptotically Kähler, with decay rates O(r^{-4}) or better for certain self-dual forms on ALE ends.