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 $-\kappa^2$. It is proved that every branched minimal surface in $M$ bounded by a smooth Jordan curve $\Gamma$ with total curvature $\leq4\pi+\kappa^2\inf_{p\in M}\Area(p \smash \Gamma)$ is embedded. $p \smash \Gamma$ denotes the geodesic cone over $\Gamma$ with vertex $p$. It follows that a Jordan curve $\Gamma$ in $M^3$ with total curvature $\leq4\pi+\kappa^2\inf_{p\in M}\Area(p \smash \Gamma)$ is unknotted. In the hemisphere $\sn,$ we prove the embeddedness of any minimal surface whose boundary curve has total curvature $\leq 4\pi - \sup_{p\in \sn}$ $\Area(p \smash \Gamma)$.

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Published 1 January 2003