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Free boundary minimal surfaces in the unit ball with low cohomogeneity


Authors: Brian Freidin, Mamikon Gulian and Peter McGrath
Journal: Proc. Amer. Math. Soc. 145 (2017), 1671-1683
MSC (2010): Primary 49Q05
DOI: https://doi.org/10.1090/proc/13424
Published electronically: November 21, 2016
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Abstract: We study free boundary minimal surfaces in the unit ball of low cohomogeneity. For each pair of positive integers $ (m,n)$ such that $ m, n >1$ and $ m+n\geq 8$, we construct a free boundary minimal surface $ \Sigma _{m, n} \subset B^{m+n}$(1) invariant under $ O(m)\times O(n)$. When $ m+n<8$, an instability of the resulting equation allows us to find an infinite family $ \{\Sigma _{m,n, k}\}_{k\in \mathbb{N}}$ of such surfaces. In particular, $ \{\Sigma _{2, 2, k}\}_{k\in \mathbb{N}}$ is a family of solid tori which converges to the cone over the Clifford torus as $ k$ goes to infinity. These examples indicate that a smooth compactness theorem for free boundary minimal surfaces due to Fraser and Li does not generally extend to higher dimensions.

For each $ n\geq 3$, we prove there is a unique nonplanar $ SO(n)$-invariant free boundary minimal surface (a ``catenoid'') $ \Sigma _n \subset B^n(1)$. These surfaces generalize the ``critical catenoid'' in $ B^3(1)$ studied by Fraser and Schoen.


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Additional Information

Brian Freidin
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912-0001

Mamikon Gulian
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912-0001

Peter McGrath
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912-0001

DOI: https://doi.org/10.1090/proc/13424
Received by editor(s): January 25, 2016
Received by editor(s) in revised form: June 12, 2016
Published electronically: November 21, 2016
Communicated by: Lei Ni
Article copyright: © Copyright 2016 American Mathematical Society