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arXiv · 1804.08201

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

Chung-Jun Tsai·Mu-Tao Wang
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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.

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Chung-Jun Tsai, Mu-Tao Wang. 2018-04-23. Global uniqueness of the minimal sphere in the Atiyah-Hitchin manifold. https://arxiv.org/abs/1804.08201

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