### 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.

Original language | English (US) |
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Pages (from-to) | 117-143 |

Number of pages | 27 |

Journal | Communications In Mathematical Physics |

Volume | 376 |

Issue number | 1 |

DOIs | |

State | Published - May 1 2020 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Gursky, M. J., Kelleher, C. L., & Streets, J. (2020). Index-Energy Estimates for Yang–Mills Connections and Einstein Metrics.

*Communications In Mathematical Physics*,*376*(1), 117-143. https://doi.org/10.1007/s00220-019-03627-w