## 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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