## 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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