### 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 G{cyrillic}⊂ℝ^{3} be an analytic plane Jordan curve which is symmetric with respect to a plane π{variant} (π{variant}⊅G{cyrillic}). Let S 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 whole S is proved to be symmetric with respect to π{variant}. 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.

Original language | English |
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Pages (from-to) | 311-325 |

Number of pages | 15 |

Journal | Manuscripta Mathematica |

Volume | 87 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1 1995 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

**A generalization of Steiner symmetrization for immersed surfaces and its applications.** / Koiso, Miyuki.

Research output: Contribution to journal › Article

*Manuscripta Mathematica*, vol. 87, no. 1, pp. 311-325. https://doi.org/10.1007/BF02570477

}

TY - JOUR

T1 - A generalization of Steiner symmetrization for immersed surfaces and its applications

AU - Koiso, Miyuki

PY - 1995/12/1

Y1 - 1995/12/1

N2 - 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 G{cyrillic}⊂ℝ3 be an analytic plane Jordan curve which is symmetric with respect to a plane π{variant} (π{variant}⊅G{cyrillic}). Let S 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 whole S is proved to be symmetric with respect to π{variant}. 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.

AB - 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 G{cyrillic}⊂ℝ3 be an analytic plane Jordan curve which is symmetric with respect to a plane π{variant} (π{variant}⊅G{cyrillic}). Let S 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 whole S is proved to be symmetric with respect to π{variant}. 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.

UR - http://www.scopus.com/inward/record.url?scp=51249164762&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249164762&partnerID=8YFLogxK

U2 - 10.1007/BF02570477

DO - 10.1007/BF02570477

M3 - Article

AN - SCOPUS:51249164762

VL - 87

SP - 311

EP - 325

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 1

ER -