We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact n-dimensional Riemannian manifold which has nonnegative Ricci curvature and strictly convex boundary. When n " 3, this implies an apriori curvature estimate for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. An important consequence of the estimate is a smooth compactness theorem for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed.

In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces satisfying a uniform area bound, which generalizes the celebrated Schoen-Simon-Yau interior curvature estimates [16] up to the free boundary. Our curvature estimates imply a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors [13]. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For 3-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without a-prioi area bound. This generalizes Schoen's interior curvature estimates [17] to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi in [5].

For any smooth Riemannian metric on an (n + 1)-dimensional compact manifold with boundary (M, ∂M ) where 3 ≤ (n + 1) ≤ 7, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates to prove that for almost every (in the C ∞ Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M . If ∂M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.

Abstract. In this paper, we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact Riemannian 3-manifold M with boundary ∂M . These minimal surfaces are either disjoint from ∂M or meet ∂M orthogonally. The main feature of our result is that there is no assumptions on the curvature of M or convexity of ∂M . We prove the boundary regularity of the minimal surfaces at their free boundaries. Furthermore, we define a topological invariant, the filling genus, for compact 3-manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold M . Our proof employs a variant of the min-max construction used by Colding and De Lellis on closed embedded minimal surfaces, which were first developed by Almgren and Pitts.

Abstract. Let M be a compact n-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary ∂M . Assume that the mean curvature H of the boundary ∂M satisfies H ≥ (n − 1)k > 0 for some positive constant k. In this paper, we prove that the distance function d to the boundary ∂M is bounded from above by

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disc. The surfaces are constructed by singular perturbation methods and have three boundary components. They are the free boundary analogue of the Costa–Hoffman–Meeks surfaces and the surfaces constructed by Kapouleas by desingularizing coaxial catenoids and planes. It is plausible that the minimal surfaces we constructed here are the same as the ones obtained recently by Ketover by using the min-max method.

No abstract

For any smooth Riemannian metric on an $$(n+1)$$ ( n + 1 ) -dimensional compact manifold with boundary $$(M,\partial M)$$ ( M , ∂ M ) where $$3\le (n+1)\le 7$$ 3 ≤ ( n + 1 ) ≤ 7 , we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min–max theory in the Almgren–Pitts setting. We apply our Morse index estimates to prove that for almost every (in the $$C^\infty $$ C ∞ Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M. If $$\partial M$$ ∂ M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.