A remark on a curvature gap for minimal surfaces in the ball

Abstract

We extend to higher codimension earlier characterization of the equatorial disk and the critical catenoid by a pinching condition on the length of their second fundamental form among free boundary minimal surfaces in the three dimensional Euclidean ball due to L. Ambrozio and I. Nunes.

Uniqueness Results for Free Boundary Minimal Hypersurfaces in Conformally Euclidean Balls and Annular Domains

“…For example, the Conjecture 1 can be viewed as a free-boundary's version from the Lawson's conjecture about closed minimal surfaces in the sphere. Also, the classical gap results for minimal submanifolds immersed in spheres by Cherndo Carmo-Kobayashi have been evoked in order to establish similar results in the free-boundary's context, where, now, the ambient space is an Euclidean ball (see [1], [3], [5] and references therein). In this work, we also intend to approach gap results where the ambient space is a rotational ellipsoid (see Theorem 8) or a ball (see Theorem 9).…”

Pinching results for minimal submanifolds and a characterization of spherical caps

“…The authors in [4] raise the question if the above theorem can be generalized to higher dimension and co-dimension. In [6] we extended their result to 2-dimensional surfaces of any codimension in B n . In this work we answer their question at the level of topology and show some local second order rigidity properties, see Corollary 3.8 below.…”

Free boundary constant mean curvature surfaces in a strictly convex three-manifold

“…It turned out that any properly embedded complete minimal surface satisfying the same geometric condition must be either the plane or the catenoid, which was due to Meeks-Pérez-Ros [32]. Recently, there have been many interesting extensions [3,4,7,28] of the work by Ambrozio-Nunes. We extend the previous results into free boundary cmc-H surfaces Σ inside a strictly convex three-manifold under a similar pinching condition in terms of the distance function.…”

“…A number of characterizations of the critical catenoid have been obtained by numerous geometers. For example, refer to [10,40,44] for the Morse index estimate, [2,4,7,18,19,24,28,31] for uniqueness results of the critical catenoid, and [5] for a variational characterization in terms of 2-dimensional Hausdorff measure. We should mention that examples of free boundary minimal surfaces in B 3 are very rich.…”