# Index-energy estimates for Yang-Mills connections and Einstein metrics

## TL;DR

This paper establishes conformally invariant estimates for the index of Schrödinger operators, Yang-Mills connections, and Einstein metrics on four-manifolds, linking geometric analysis with topological invariants.

## Contribution

It introduces new conformally invariant bounds for the index of operators and metrics, connecting energy, topology, and geometric properties in four dimensions.

## Key findings

- Bound for the index of Yang-Mills connections in terms of energy
- Index estimate for Einstein metrics involving topology and energy
- Conformally invariant estimates for Betti numbers in four-manifolds

## Abstract

We prove a conformally invariant estimate for the index of Schr\"odinger 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.