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.
Volume growth, curvature decay, and critical metrics
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abstract
We make some improvements to our previous results. First, we prove a version of our volume growth theorem which does not require any assumption on the first Betti number. Second, we show that our local regularity theorem only requires a lower volume growth assumption, not a full Sobolev constant bound. These results allow us to weaken the assumptions of our previous volume growth and convergence theorems.
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math.DG 1years
2019 1verdicts
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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.