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Nonuniqueness in the Plateau problem for surfaces of constant mean curvature. (English) Zbl 0603.49027
Let B denote the open unit disc in \({\mathbb{R}}^ 2\) with oriented boundary C. Let \(\gamma\) : \(C\to {\mathbb{R}}^ 3\) be an oriented closed Jordan curve in \({\mathbb{R}}^ 3\). For a given real constant H, the Plateau problem for surfaces of constant mean curvature H consists of determining a function \(x\in C^ 2(B;{\mathbb{R}}^ 3)\cap C^ 0(\bar B;{\mathbb{R}}^ 3)\) such that \(\Delta x=2Hx_ u\wedge x_ v\) in B, where \(x|_ C: C\to \gamma\) is an oriented monotone, continuous, parametrization of \(\gamma\), and \(| x_ u|^ 2-| x_ v|^ 2=0=x_ u\cdot x_ v\) in B.
Such a solution is called an H-surface spanning \(\gamma\). For sufficiently small \(| H|\), that is for \(| H| \sup | \gamma (\omega)| \leq 1\), existence of an H-surface is known from the work of other authors. Letting \(\gamma\) be a plane circular arc with \(0<| H| \sup | \gamma (\omega | <1\), one easily sees using portions of a sphere that for \(| H|\) small enough, but non- zero, there exist two geometrically distinct surfaces of (signed) mean curvature H spanning \(\gamma\). Motivated by this example, the author gives a sufficient condition for a curve under which at least two geometrically distinct H-surfaces will exist for any H in a punctured neighborhood of 0. This sufficient condition is satisfied by any sufficiently smooth simple curve lying in a plane.
To describe the method of proof, let \(D(x)=2^{-1}\int_{B}| \nabla x|^ 2d\omega\), \(V(x)=3^{-1}\int_{B}x_ u\wedge x_ v\bullet x d\omega\) denote the Dirichlet and volume integrals, respectively. The H- surfaces are extremals of the functional \(E_ H(x)=D(x)+2H V(x)\). In over-simplified terms, the proof proceeds by minimizing \(E_ H\) over a collection of functions x, which satisfy (*) D(x-h)\(\geq 1\), where h is the harmonic extension of \(x|_ C\). The condition (*) insures that the solution obtained is ”large” while the solution arising from the earlier existence result, mentioned above, is ”small”.
Reviewer: H.Parks

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