## Free boundary minimal surfaces in the unit ball with low cohomogeneity

HTML articles powered by AMS MathViewer

- by Brian Freidin, Mamikon Gulian and Peter McGrath PDF
- Proc. Amer. Math. Soc.
**145**(2017), 1671-1683 Request permission

## 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.

## References

- Hilário Alencar,
*Minimal hypersurfaces of $\textbf {R}^{2m}$ invariant by $\textrm {SO}(m)\times \textrm {SO}(m)$*, Trans. Amer. Math. Soc.**337**(1993), no. 1, 129–141. MR**1091229**, DOI 10.1090/S0002-9947-1993-1091229-1 - Hilário Alencar, Abdênago Barros, Oscar Palmas, J. Guadalupe Reyes, and Walcy Santos,
*$\textrm {O}(m)\times \textrm {O}(n)$-invariant minimal hypersurfaces in $\Bbb R^{m+n}$*, Ann. Global Anal. Geom.**27**(2005), no. 2, 179–199. MR**2131912**, DOI 10.1007/s10455-005-2572-7 - Personal communication, H. Alencar and O. Palmas.
- Simon Brendle,
*A sharp bound for the area of minimal surfaces in the unit ball*, Geom. Funct. Anal.**22**(2012), no. 3, 621–626. MR**2972603**, DOI 10.1007/s00039-012-0167-6 - E. Bombieri, E. De Giorgi, and E. Giusti,
*Minimal cones and the Bernstein problem*, Invent. Math.**7**(1969), 243–268. MR**250205**, DOI 10.1007/BF01404309 - Ailana Fraser and Martin Man-chun Li,
*Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary*, J. Differential Geom.**96**(2014), no. 2, 183–200. MR**3178438** - Ailana Fraser and Richard Schoen,
*The first Steklov eigenvalue, conformal geometry, and minimal surfaces*, Adv. Math.**226**(2011), no. 5, 4011–4030. MR**2770439**, DOI 10.1016/j.aim.2010.11.007 - Ailana Fraser and Richard Schoen,
*Sharp eigenvalue bounds and minimal surfaces in the ball*, Invent. Math.**203**(2016), no. 3, 823–890. MR**3461367**, DOI 10.1007/s00222-015-0604-x - Ailana Fraser and Richard Schoen,
*Uniqueness theorems for free boundary minimal disks in space forms*, Int. Math. Res. Not. IMRN**17**(2015), 8268–8274. MR**3404014**, DOI 10.1093/imrn/rnu192 - Wu-yi Hsiang and H. Blaine Lawson Jr.,
*Minimal submanifolds of low cohomogeneity*, J. Differential Geometry**5**(1971), 1–38. MR**298593** - Wu-Yi Hsiang,
*Minimal cones and the spherical Bernstein problem. I*, Ann. of Math. (2)**118**(1983), no. 1, 61–73. MR**707161**, DOI 10.2307/2006954 - Wu-yi Hsiang,
*Generalized rotational hypersurfaces of constant mean curvature in the Euclidean spaces. I*, J. Differential Geometry**17**(1982), no. 2, 337–356. MR**664499** - Wu-Yi Hsiang,
*On the construction of infinitely many congruence classes of imbedded closed minimal hypersurfaces in $S^n(1)$ for all $n\geq 3$*, Duke Math. J.**55**(1987), no. 2, 361–367. MR**894586**, DOI 10.1215/S0012-7094-87-05520-7 - T. Ilmanen,
*Lectures on Mean Curvature Flow and Related Equations*, https://people.math.ethz.ch/~ilmanen/papers/notes.pdf - Saadia Fakhi and Frank Pacard,
*Existence result for minimal hypersurfaces with a prescribed finite number of planar ends*, Manuscripta Math.**103**(2000), no. 4, 465–512. MR**1811769**, DOI 10.1007/PL00005863 - A. Folha, F. Pacard, and T. Zolotareva,
*Free boundary minimal surfaces in the unit 3-ball*, http://arxiv.org/abs/1502.06812 - Johannes C. C. Nitsche,
*Stationary partitioning of convex bodies*, Arch. Rational Mech. Anal.**89**(1985), no. 1, 1–19. MR**784101**, DOI 10.1007/BF00281743 - R. Osserman (ed.),
*Geometry. V*, Encyclopaedia of Mathematical Sciences, vol. 90, Springer-Verlag, Berlin, 1997. Minimal surfaces. MR**1490037**, DOI 10.1007/978-3-662-03484-2 - Richard Schoen and Karen Uhlenbeck,
*Regularity of minimizing harmonic maps into the sphere*, Invent. Math.**78**(1984), no. 1, 89–100. MR**762354**, DOI 10.1007/BF01388715 - James Simons,
*Minimal varieties in riemannian manifolds*, Ann. of Math. (2)**88**(1968), 62–105. MR**233295**, DOI 10.2307/1970556 - Rabah Souam,
*On stability of stationary hypersurfaces for the partitioning problem for balls in space forms*, Math. Z.**224**(1997), no. 2, 195–208. MR**1431192**, DOI 10.1007/PL00004289

## 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
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 1671-1683 - MSC (2010): Primary 49Q05
- DOI: https://doi.org/10.1090/proc/13424
- MathSciNet review: 3601558