Embedded minimal surfaces and total curvature of curves in a manifold

Let Mⁿ 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 + κ²inf_p∈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 M³ with total curvature ≤ 4π + κ²inf_p∈M Area(p×Γ) is unknotted. In the hemisphere S₊ⁿ, we prove the embeddedness of any minimal surface whose boundary curve has total curvature ≤ 4π - sup_p∈S+n Area(p×Γ).