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.

We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved and CAT(-1) metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this and other formulas in terms of curvature, we prove an analogue of the Eels-Sampson Bochner formula in this more general setting. In particular, we show that harmonic maps from spaces of non-negative Ricci curvature into non-positively curved spaces have subharmonic energy density. When the domain is compact the energy density is constant, and if the domain has a point of positive Ricci curvature every harmonic map into an NPC space must be constant.