GAP RESULTS FOR FREE BOUNDARY CMC SURFACES IN THE EUCLIDEAN THREE-BALL EZEQUIEL BARBOSA, MARCOS P. CAVALCANTE, AND EDNO PEREIRA Abstract. In this note, we prove that if a free boundary constant mean cur- vature surface Σ in an Euclidean 3-ball satisfies a pinching condition on the length of traceless second fundamental tensor, then either Σ is a totally um- bilical disk or an annulus of revolution. The pinching is sharp since there are portions of some Delaunay surfaces inside the unit Euclidean 3-ball which are free boundary and satisfy the pinching condition. 1. Introduction Let W 3 be a Riemannian 3-manifold with smooth boundary and let Σ be a compact orientable surface immersed in W such that int(Σ) ⊂ int(W ), and ∂ Σ ⊂ ∂W . We say that Σ is a free boundary CMC surface if its mean curvature H is constant and the boundary ∂ Σ intersects ∂W orthogonally. If follows from the first variation formula that free boundary CMC surfaces are critical points of the area functional for volume preserving variations of Σ whose boundaries are free to move in ∂W (see [14] for details). In the last decades, free boundary CMC surfaces have been investigated by many authors, and there is a crescent interest in this theme, particularly when H = 0, i.e., free boundary minimal surfaces. The most important case to be considered is when W 3 is the unit ball B 3 in the Euclidean space. In this case, the most simple examples are the equatorial flat disk and a portion of the catenoid (called the critical catenoid ) if H = 0, and spherical caps and some portions of Delaunay surfaces in the case H is constant and nonzero. Many other examples of free boundary minimal surfaces in the unit ball had been recently constructed by using the desingularization method or the gluing method (see [8, 7, 11, 9]), and it is expected these methods can also be used to construct other examples of free boundary CMC surfaces with high genus and many boundary components. These notes are motivated by the following geometrical characterization of the disk and the catenoid discovered by Ambrozio and Nunes. Theorem 1.1 (Ambrozio-Nunes, [2]). Let Σ be a compact free boundary minimal surface in B 3 . Assume that for all points x in Σ, (1.1) |A| 2 (x) 〈x, N (x)〉 2 ≤ 2, where N (x) denotes a unit normal vector at the point x ∈ Σ and A denotes the second fundamental form of Σ. Then, i) either |A| 2 (x) 〈x, N 〉 2 ≡ 0 and Σ is a flat equatorial disk, Date : February 12, 2022. 2010 Mathematics Subject Classification. 53A10, 49Q10, 35P15. Key words and phrases. Constant mean curvature surfaces, free boundary surfaces. 1 arXiv:1908.09952v1 [math.DG] 26 Aug 2019