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In this paper we provide a pinching condition for the characterization of thetotally geodesic disk and the rotational annulus among minimal surfaces withfree boundary in geodesic balls of three-dimensional hyperbolic space andhemisphere. The pinching condition involves the length of the secondfundamental form, the support function of the surface, and a natural potentialfunction in hyperbolic space and hemisphere.

A gap theorem for free boundary minimal surfaces in geodesic balls of hyperbolic space and hemisphere