# Limits of Yang-Mills {\alpha}-connections

## TL;DR

This paper investigates the behavior of Yang-Mills { extalpha}-connections, demonstrating that for the SU(2) Hopf fibration over the four sphere, the SO(4) invariant ADHM instanton uniquely minimizes the { extalpha}-energy below a certain threshold for small { extalpha}.

## Contribution

It establishes the uniqueness of the SO(4) invariant ADHM instanton as the { extalpha}-critical point with minimal energy in a specific geometric setting for small { extalpha}.

## Key findings

- The SO(4) invariant ADHM instanton is the unique { extalpha}-critical point below a certain energy threshold.
- For sufficiently small { extalpha}, the ADHM instanton minimizes the { extalpha}-energy.
- The results connect { extalpha}-connections with classical Yang-Mills instantons in a geometric context.

## Abstract

In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps, we identify Yang-Mills connections obtained by approximations with respect to the Yang-Mills {\alpha}-energy. More specifically, we show that for the SU(2) Hopf fibration over the four sphere, for sufficiently small {\alpha} values the SO(4) invariant ADHM instanton is the unique {\alpha}-critical point which has Yang-Mills {\alpha}-energy lower than a specific threshold.