Skip to main content
We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate
arXiv logo
Cornell University Logo

Mathematics > Differential Geometry

arXiv:1804.08201 (math)
[Submitted on 23 Apr 2018]

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

Authors:Chung-Jun Tsai, Mu-Tao Wang
View PDF
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.
Comments: 12 pages
Subjects: Differential Geometry (math.DG)
Cite as: arXiv:1804.08201 [math.DG]
  (or arXiv:1804.08201v1 [math.DG] for this version)
  https://doi.org/10.48550/arXiv.1804.08201

Submission history

From: Chung-Jun Tsai [view email]
[v1] Mon, 23 Apr 2018 00:39:22 UTC (13 KB)
Full-text links:

Access Paper:

  • View PDF
  • TeX Source
  • Other Formats
view license
Current browse context:
math.DG
< prev   |   next >
new | recent | 2018-04
Change to browse by:
math

References & Citations

export BibTeX citation

Bookmark

BibSonomy logo Reddit logo

Bibliographic and Citation Tools

Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)

Code, Data and Media Associated with this Article

alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)

Demos

Replicate (What is Replicate?)
Hugging Face Spaces (What is Spaces?)
TXYZ.AI (What is TXYZ.AI?)

Recommenders and Search Tools

Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)