Free boundary minimal surfaces in the unit ball with low cohomogeneity

Abstract

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 ≥ 8, we construct a free boundary minimal surface Σm,n ⊂ B m+n (1) invariant under O(m) × O(n). When m + n < 8, an instability of the resulting equation allows us to find an infinite family {Σ m,n,k } k∈N of such surfaces. In particular, {Σ 2,2,k } k∈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 ≥ 3, we prove there is a unique nonplanar SO(n)-invariant free boundary minimal surface (a "catenoid") Σn ⊂ B n (1). These surfaces generalize the "critical catenoid" in B 3 (1) studied by Fraser and Schoen.

Index estimates for free boundary minimal hypersurfaces

“…Given (Ω n+1 , g) a smooth Riemannian manifold with boundary, we shall be concerned here with certain global properties of free boundary minimal hypersurfaces M n ⊂ Ω n+1 , namely hypersurfaces that are critical points of the area functional when the boundary ∂M is not fixed (like in Plateau's problem) but subject to the sole constraint ∂M ⊂ ∂Ω. Due to their self-evident geometric interest (which can be traced back at least to Courant [3]), these variational objects have been widely studied and a number of existence results have been obtained via surprisingly diverse methods (see, among others, [4,[9][10][11][17][18][19][20]28,30] and references therein). Free boundary minimal hypersurfaces also naturally arise in partitioning problems for convex bodies, in capillarity problems for fluids and, as has significantly emerged in recent years, in connection to extremal metrics for Steklov eigenvalues for manifolds with boundary (see primarily the works by Fraser-Schoen [7][8][9] and references therein).…”

Deformations of Free Boundary CMC Hypersurfaces

“…It is not clear whether all of these surfaces are (equivariantly) nondegenerate, but it is natural to expect that many of them are. Furthermore, in higher dimensions, if the new free boundary minimal hypersurfaces obtained by Freidin, Gulian, and McGrath [11] using equivariant methods are equivariantly nondegenerate, then it seems very likely that our deformation result recovers all the free boundary CMC hypersurfaces constructed by Cruz, Palmas, and Reyes [7] x i − z 2 , x i + z 2 , satisfies the following properties:…”

Uniqueness of stable capillary hypersurfaces in a ball

“…Recently one of inspiring work is a series of papers of Fraser-Schoen [22,23,24] about minimal hypersurfaces with free boundary in a ball and the first Steklov eigenvalue. See also [11,15,63,35,20,25,3]. Our research on the stability on CMC hypersurfaces are motivated by these results.…”

“…25) γ := n|x| 2 + n cos θ x, ν − 1 2 (|x| 2 − 1)H x, ν may have no definite sign. In order to handle this problem, we introduce the following functionΦ = 1 2 (|x| 2 − 1)H − n( x, ν + cos θ).Using(3.15) and (3.17), one can check that Φ satisfies ∆Φ = (n|h| 2 − H 2 ) x, ν .…”