Global uniqueness of the minimal sphere in the Atiyah-Hitchin manifold

TL;DR

This paper proves the global uniqueness of the minimal 2-sphere in the Atiyah-Hitchin manifold, confirming a conjecture and demonstrating its strong stability, which has implications for the geometry of monopole moduli spaces.

Contribution

It establishes the global uniqueness of the minimal 2-sphere in the Atiyah-Hitchin manifold by proving its strong stability, confirming a longstanding conjecture.

Findings

The minimal 2-sphere satisfies the strong stability condition.

The minimal 2-sphere is globally unique among all closed minimal 2-surfaces.

Confirmed the conjecture by Micallef and Wolfson regarding uniqueness.

Abstract

In this note, we study submanifold geometry of the Atiyah-Hitchin manifold, the double cover of the $2$-monopole moduli space. When the manifold is naturally identified as the total space of a line bundle over $S^2$, the zero section is a distinguished minimal $2$-sphere of considerable interest. In particular, there has been a conjecture by Micallef and Wolfson [Math. Ann. 295 (1993), Remark on p.262] about the uniqueness of this minimal $2$-sphere among all closed minimal $2$-surfaces. We show that this minimal $2$-sphere satisfies the "strong stability condition" proposed in our earlier work [arXiv:1710.00433], and confirm the global uniqueness as a corollary.