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Index-Energy Estimates for Yang–Mills Connections and Einstein Metrics

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Abstract

We prove a conformally invariant estimate for the index of Schrödinger operators acting on vector bundles over four-manifolds, related to the classical Cwikel–Lieb–Rozenblum estimate. Applied to Yang–Mills connections we obtain a bound for the index in terms of its energy which is conformally invariant, and captures the sharp growth rate. Furthermore we derive an index estimate for Einstein metrics in terms of the topology and the Einstein–Hilbert energy. Lastly we derive conformally invariant estimates for the Betti numbers of an oriented four-manifold with positive scalar curvature.

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Notes

  1. Note that in [Hui85], the norm of Weyl is the one induced by the metric on covariant 4-tensors, while we are using the norm of Weyl viewed as a section of \(\text{ End }(\Lambda ^2)\).

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Acknowledgements

The authors thank Elliott Lieb, Francesco Lin, Zhiqin Lu, and Richard Schoen for informative discussions.

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Authors and Affiliations

  1. Department of Mathematics, 255 Hurley Bldg, University of Notre Dame, Notre Dame, IN, 46556, USA

    Matthew J. Gursky

  2. Department of Mathematics, Princeton University, Princeton, NJ, 08540, USA

    Casey Lynn Kelleher

  3. Department of Mathematics, Rowland Hall, University of California, Irvine, Irvine, 92697, USA

    Jeffrey Streets

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  1. Matthew J. Gursky
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  2. Casey Lynn Kelleher
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  3. Jeffrey Streets
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Correspondence to Casey Lynn Kelleher.

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Communicated by P. Chrusciel

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M.J. Gursky is supported by NSF Grant DMS-1811034. C.L. Kelleher is supported by a National Science Foundation Postdoctoral Research Fellowship. J. Streets is supported by NSF Grant DMS-1454854.

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Gursky, M.J., Kelleher, C.L. & Streets, J. Index-Energy Estimates for Yang–Mills Connections and Einstein Metrics. Commun. Math. Phys. 376, 117–143 (2020). https://doi.org/10.1007/s00220-019-03627-w

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  • DOI: https://doi.org/10.1007/s00220-019-03627-w

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