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

  • Matthew J. Gursky
  • Casey Lynn KelleherEmail author
  • Jeffrey Streets


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.



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


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Matthew J. Gursky
    • 1
  • Casey Lynn Kelleher
    • 2
    Email author
  • Jeffrey Streets
    • 3
  1. 1.Department of Mathematics, 255 Hurley BldgUniversity of Notre DameNotre DameUSA
  2. 2.Department of MathematicsPrinceton UniversityPrincetonUSA
  3. 3.Department of Mathematics, Rowland HallUniversity of California, IrvineIrvineUSA

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