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

초록

For a soap-film-like surface Sigma spanning a nonclosed curve Gamma in R-n, it is proved that 4 pi Area(Sigma) <= Length(Gamma)(2). In the upper hemisphere S-+(n) we use a mixed area M-p(Sigma), which was introduced by Choe and Gulliver [9] to prove an isoperimetric inequality 4 pi M-p(Sigma) <= Length(Gamma)(2) for a soap-film-like surface Sigma with nonclosed boundary partial derivative Sigma. In a complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant K, a soapfilm-like surface Sigma with an embedded, connected, and nonclosed boundary curve Gamma satisfies 4 pi Area(Sigma) - KArea(Sigma)(2) <= Length(Gamma)(2). Moreover, for a soap-film-like surface Sigma with nonclosed boundary partial derivative 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 Area(Sigma) - KArea(Sigma)(2) <= Length(partial derivative Sigma)(2). Finally, we prove that a soap-film-like surface Sigma subset of R-n satisfies 2 root 2 pi Area(Sigma) <= Length(partial derivative Sigma)(2).

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