Embeddedness of proper minimal submanifolds in homogeneous spaces

Abstract

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

Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space

“…In [21], the first author introduced the Möbius volume of a compact submanifold of S n 1 D @ 1 H n as follows. According to the definition of Li and Yau [17], the Möbius volume of is the same as the .n 1/-conformal volume of the inclusion of into S n 1 .…”