# A Gap Theorem for Half-Conformally Flat Manifolds

## TL;DR

This paper proves a gap theorem for compact half-conformally flat manifolds with specific geometric bounds, showing all Betti numbers are bounded and analyzing the asymptotic behavior of certain solutions on ALE ends.

## Contribution

It establishes a new gap theorem for half-conformally flat manifolds under geometric constraints and characterizes the asymptotic behavior of bounded self-dual solutions on ALE ends.

## Key findings

- All Betti numbers are bounded under the given conditions.
- Singularity models can be 2-ended and asymptotically Kähler.
- Bounded self-dual solutions on ALE ends are either asymptotically Kähler or decay as O(r^{-4}).

## Abstract

We show that any compact half-conformally flat manifold of negative type, with bounded $L^2$ energy, sufficiently small scalar curvature, and a non-collapsing assumption, has all betti numbers bounded. We show that this result is optimal from an analytic perspective by demonstrating singularity models that are 2-ended, and are asymptotically K\"ahler on both ends. We show that bounded self-dual solutions of $d\omega=0$ on ALE manifold ends are either asymptotically K\"ahler, or they have a decay rate of $O(r^{-4})$ or better.