Isoperimetric inequalities for soap-film-like surfaces spanning nonclosed curves

For a soap-film-like surface $\Sigma$ spanning a nonclosed curve $\Gamma$ in $\mathbf{R}^n$, it is proved that\[4\pi {\rm Area} (\Sigma) \leq {\rm Length} (\Gamma)^2.\]In the upper hemisphere $\mathbf{S}^n_+$ we use a mixed area $M_p (\Sigma)$, which was introduced by Choe and Gulliver \cite{CG} to prove an isoperimetric inequality\[4\pi M_p (\Sigma) \leq {\rm Length} (\Gamma)^2\]for a soap-film-like surface $\Sigma$ with nonclosed boundary $\partial \Sigma$. In a complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $K$, a soap-film-like surface $\Sigma$ with an embedded, connected, and nonclosed boundary curve $\Gamma$ satisfies\[4\pi {\rm Area} (\Sigma)- K {\rm Area} (\Sigma)^2 \leq {\rm Length} (\Gamma)^2.\]Moreover, for a soap-film-like surface $\Sigma$ with nonclosed boundary $\partial \Sigma$ in a complete simply connected Riemannian manifold $M$ with sectional curvature bounded above by a constant $K$, we obtain a weak isoperimetric inequality\[2\pi {\rm Area} (\Sigma) - K {\rm Area} (\Sigma)^2 \leq {\rm Length} (\partial \Sigma)^2.\]Finally, we prove that a soap-film-like surface $\Sigma \subset \mathbf{R}^n$ satisfies\[2\sqrt{2}\pi {\rm Area} (\Sigma) \leq {\rm Length} (\partial \Sigma)^2.\]

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Published 29 October 2014