Embedded Minimal Surfaces and Total Curvature of Curves in a Manifold

Let M^n be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant −κ2. It is proved that every branched minimal surface in M bounded by a smooth Jordan curve Γ with total curvature ≤ 4π + κ2 infp∈M Area(p×Γ) is embedded. p×Γ denotes the geodesic cone over Γ with vertex p. It follows that a Jordan curve Γ in M 3 with total curvature ≤ 4π + κ2 infp∈M Area(p×Γ) is unknotted. In the hemisphere Sn +, we prove the embeddedness of any minimal surface whose boundary curve has total curvature ≤ 4π − sup p∈S n + Area(p×Γ)