Vanishing theorems for the cohomology groups of free boundary submanifolds

Abstract

In this paper, we prove that there exists a universal constant C, depending only on positive integers n ≥ 3 and p ≤ n − 1, such that if M n is a compact free boundary submanifold of dimension n immersed in the Euclidean unit ball B n+k whose size of the traceless second fundamental form is less than C, then the pth cohomology group of M n vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball B 2+k .

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).…”

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.…”

“…where we used the assumption (7) in the above inequality. We note that f ′′ < 0 if c > 0 by definition of the function f .…”

“…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.…”

“…generalized Theorem 1.2 to free boundary minimal surfaces of higher codimension in the n-dimensional Eucldiean unit ball B n . Cavalcante-Mendes-Vitório [7] obtained a topological gap result for compact free boundary submanifolds in the n-dimensional Euclidean unit ball B n which are not necessarily minimal. In [3], Barbosa-Cavalcante-Pereira proved an analogue of Theorem 1.2 for free boundary constant mean curvature surfaces in the 3-dimensional Euclidean unit ball B 3 .…”