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A generalization of Steiner symmetrization for immersed surfaces and its applications

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Abstract

We generalize the classical Steiner symmetrization to surfaces with self-intersections. Then we apply the generalized Steiner symmetrization to several isoperimetric problems. For example, let Г⊂ℝ3 be an analytic plane Jordan curve which is symmetric with respect to a plane ϖ (ϖ⊅Г). LetS be a compact immersed surface bounded by Λ which has the smallest area among all compact surfaces bounded by Λ with a fixed volume. In this situation, under some additional assumptions, the wholeS is proved to be symmetric with respect to ϖ. When Λ is a round circle,S is proved to be a spherical cap or the flat disk bounded by Λ without any additional assumptions.

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Authors and Affiliations

  1. Department of Mathematics, Osaka University, 560, Toyonaka, Osaka, Japan

    Miyuki Koiso

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  1. Miyuki Koiso
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Dedicated to Professor Hideki Ozeki on his 60th birthday

This article was processed by the author using the Springer-Verlag TEX P Jourlg macro package 1991.

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Koiso, M. A generalization of Steiner symmetrization for immersed surfaces and its applications. Manuscripta Math 87, 311–325 (1995). https://doi.org/10.1007/BF02570477

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  • DOI: https://doi.org/10.1007/BF02570477

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