# A generalization of Steiner symmetrization for immersed surfaces and its applications

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4 Citations (Scopus)

### 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}⊂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.

Original language English 311-325 15 Manuscripta Mathematica 87 1 https://doi.org/10.1007/BF02570477 Published - Dec 1 1995 Yes

### Fingerprint

Symmetrization
Isoperimetric Problem
Jordan Curve
Self-intersection
Plane Curve
Circle
Generalise
Generalization

### All Science Journal Classification (ASJC) codes

• Mathematics(all)

### Cite this

In: Manuscripta Mathematica, Vol. 87, No. 1, 01.12.1995, p. 311-325.

Research output: Contribution to journalArticle

title = "A generalization of Steiner symmetrization for immersed surfaces and its applications",
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}⊂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.",
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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.

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