A generalization of Steiner symmetrization for immersed surfaces and its applications
The author generalizes the Steiner symmetrization to surfaces with self-intersections. Then he gives applications to the isoperimetric problem: consider an analytic Jordan curve \(C\) in a plane \(P\) of \(\mathbb{R}^3\), which is symmetric with respect to a plane \(P_1\) orthogonal to \(P\), and such that each plane orthogonal to \(P\) and \(P_1\) meets \(C\) in at most two points. If \(M\) is an immersed surface in one of the closed halfspaces determinated by \(P\) with boundary \(C\), which has smallest area among immersed surfaces bounded by \(C\) which enclose a given volume, then \(M\) is an embedding and is symmetric with respect to the plane \(P_1\).
scientific article; zbMATH DE number 784077