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A Note on Free Boundary Hypersurfaces in Space Forms Balls

In this article, we establish a relationship between geometric quantities of a hypersurface restricted to its boundary, and the geometric quantities of its boundary as a hypersurface of the boundary of the ball. As a first application, we prove that the quantity of umbilical points of a free boundary surface in the unit ball counted with multiplicities depend only on its topology; moreover, we obtain as consequences that free boundary surfaces are annuli if, and only if, they have no umbilical points, and a new proof of the Nitsche Theorem. Secondly, we prove two geometric integral inequalities for free boundary hypersurfaces, and use them to relate some geometric aspects of the hypersurface with topological aspects of its boundary in the three-dimensional case, and to give a new point of view to the Catenoid Conjecture.