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Let † be a k-dimensional complete proper minimal submanifold in the Poincaré ball model B n of hyperbolic geometry. If we consider † as a subset of the unit ball B n in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold † and the ideal boundary @ 1 †, say Vol R . †/ and Vol R .@ 1 †/, respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if Vol R .@ 1 †/ Vol R .S k 1 /, then † satisfies the classical isoperimetric inequality. By proving the monotonicity theorem for such †, we further obtain a sharp lower bound for the Euclidean volume Vol R . †/, which is an extension of Fraser-Schoen and Brendle's recent results to hyperbolic space. Moreover we introduce the Möbius volume of † in B n to prove an isoperimetric inequality via the Möbius volume for †.

Let C be a strictly convex domain in a 3-dimensional Riemannian manifold with sectional curvature bounded above by a constant and let Σ be a constant mean curvature surface with free boundary in C. We provide a pinching condition on the length of the traceless second fundamental form on Σ which guarantees that the surface is homeomorphic to either a disk or an annulus. Furthermore, under the same pinching condition, we prove that if C is a geodesic ball of 3-dimensional space forms, then Σ is either a spherical cap or a Delaunay surface.

Abstract. We prove the three embeddedness results as follows. (i) Let Γ 2m+1 be a piecewise geodesic Jordan curve with 2m + 1 vertices in R n , where m is an integer ≥ 2. Then the total curvature of Γ 2m+1 < 2mπ. In particular, the total curvature of Γ 5 < 4π and thus any minimal surface Σ ⊂ R n bounded by Γ 5 is embedded. Let Γ 5 be a piecewise geodesic Jordan curve with 5 vertices in H n . Then any minimal surface Σ ⊂ H n bounded by Γ 5 is embedded. If Γ 5 is in a geodesic ball of radius π 4 in S n + , then Σ ⊂ S n + is also embedded. As a consequence, Γ 5 is an unknot in R 3 , H 3 and S 3 + .(ii) Let Σ be an m-dimensional proper minimal submanifold in H n with the ideal boundary ∂∞Σ = Γ in the infinite sphere S n−1 = ∂∞H n . If the Möbius volume of Γ Vol(Γ) < 2Vol(S m−1 ), then Σ is embedded. If Vol(Γ) = 2Vol(S m−1 ), then Σ is embedded unless it is a cone. (iii) Let Σ be a proper minimal surface in H 2 × R. If Σ is vertically regular at infinity and has two ends, then Σ is embedded.Mathematics Subject Classification(2010) : 53A10, 49Q05