In this work, we investigate the existence of compact free-boundary minimal hypersurfaces immersed in several domains. Using an original integral identity for compact free-boundary minimal hypersurfaces that are immersed in a domain whose boundary is a regular level set, we study the case where this domain is a quadric or, more generally, a rotational domain. This existence study is done without topological restrictions. We also obtain a new gap theorem for free boundary hypersurfaces immersed in an Euclidean ball and in a rotational ellipsoid.

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In this note, we prove that if a free boundary constant mean curvature 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 umbilical 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.

In this paper we prove that a flat free-boundary minimal n-disk, n ≥ 3, in the unit Euclidean ball B n+1 is the unique compact free boundary minimal hypersurface in the unit Euclidean ball which the squared norm of the second fundamental form is less than either n 2 4 or4|x| 2 . Moreover, we prove analogous results for compact free boundary minimal hypersurfaces in annular domains with a conformally Euclidean metric.

In this paper, we prove gap results for constant mean curvature (CMC) surfaces. First, we find a natural inequality for CMC surfaces that imply convexity for distance function. We then show that if Σ is a complete, properly embedded CMC surface in the Euclidean space satisfying this inequality, then Σ is either a sphere or a right circular cylinder. Next, we show that if Σ is a free boundary CMC surface in the Euclidean 3-ball satisfying the same inequality, then either Σ is a totally umbilical disk or an annulus of revolution. These results complete the picture about gap theorems for CMC surfaces in the Euclidean 3-space. We also prove similar results in the hyperbolic space and in the upper hemisphere, and in higher dimensions.M S C 2 0 2 0 53A10, 49Q10 (primary)