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Uniqueness of stable minimal surfaces with partially free boundaries. (English) Zbl 0831.53006

We consider parametric minimal surfaces \(X : B \to \mathbb{R}^3\) of disc- type which are stationary in a boundary configuration \(\langle T, S\rangle\) consisting of a support surface \(S \subset \mathbb{R}^3\) and a Jordan arc \(\Gamma\) with endpoints on \(S\). We introduce the notion of “freely stable minimal surfaces”, whose second variation of the area functional is nonnegative with respect to displacements of \(X\) keeping the arc \(\Gamma\) fixed and remaining with the support surface \(S\). If \(\Gamma\) is a graph above the \(x,y\)-plane \(E\) with a convex projection curve \(\underline {\Gamma} \subset E\) and \(S\) is a cylindrical surface with a generating curve \(\sum_0 \subset E\) satisfying a certain oscillation condition, we prove: Each freely stable, parametric minimal surface in this configuration \(\langle \Gamma, S\rangle\) is necessarily a graph above \(E\), and its height function solves a mixed boundary value problem for the minimal surface equation. Consequently, these configurations \(\langle \Gamma, S\rangle\) only bound one freely stable, parametric minimal surface.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q05 Minimal surfaces and optimization
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