In a previous paper, new closed embedded smooth minimal surfaces in the round three‐sphere S3()1 were constructed, each resembling two parallel copies of the equatorial two‐sphere Snormaleq2 joined by small catenoidal bridges, with the catenoidal bridges concentrating along two parallel circles, or the equatorial circle and the poles. In this sequel, we generalize those constructions so that the catenoidal bridges can concentrate along an arbitrary number of parallel circles, with the further option to include bridges at the poles. The current constructions follow the linearized doubling (LD) methodology developed in the previous paper. The LD solutions constructed here can be modified readily for use to doubling constructions of rotationally symmetric minimal surfaces with asymmetric sides [15]. In particular, they allow us to develop doubling constructions for the catenoid in Euclidean three‐space, the critical catenoid in the unit ball, and the spherical shrinker of the mean curvature flow. Unlike in the previous paper, our constructions here allow for sequences of minimal surfaces where the catenoidal bridges tend to be “densely distributed,” that is, they do not miss any open set of Snormaleq2 in the limit. This in particular leads to interesting observations that seem to suggest that it may be impossible to construct embedded minimal surfaces with isolated singularities by concentrating infinitely many catenoidal necks at a point. © 2019 Wiley Periodicals, Inc.

We show that an embedded minimal annulus Σ 2 ⊂ B 3 which intersects ∂B 3 orthogonally and is invariant under reflection through the coordinate planes is the critical catenoid. The proof uses nodal domain arguments and a characterization, due to Fraser and Schoen, of the critical catenoid as the unique free boundary minimal annulus in B n with lowest Steklov eigenvalue equal to 1. We also give more general criteria which imply that a free boundary minimal surface in B 3 invariant under a group of reflections has lowest Steklov eigenvalue 1.

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.

We prove that the area of a free boundary minimal surface Σ 2 ⊂ B n , where B n is a geodesic ball contained in a round hemisphere S n + , is at least as big as that of a geodesic disk with the same radius as B n ; equality is attained only if Σ coincides with such a disk. More generally, we prove analogous results for a class of conformally euclidean ambient spaces. This follows work of Brendle and Fraser-Schoen in the euclidean setting.