## Properly embedded minimal planar domains.(English)Zbl 1315.53008

It follows from the minimal surface equation that the plane, the catenoid and the helicoid are examples of properly embedded, minimal planar domains in $${\mathbb R}^3$$. And it is well known that those surfaces are of finite topology. A planar domain is a connected surface that embeds in the plane. Around 1860, Riemann discovered examples of properly embedded, minimal planar domains in $${\mathbb R}^3$$ with infinite topology. These examples, called the Riemann minimal examples by the authors, appear in a one-parameter family $${\mathcal R}_t, t \in (0, \infty)$$, and satisfy the property that, after a rotation, each $${\mathcal R}_t$$ intersects every horizontal plane in a circle or in a line. Moreover the $${\mathcal R}_t$$ have natural limits being a vertical catenoid as $$t \to 0$$ and a vertical helicoid as $$t \to \infty$$.
In this paper, the authors analyze the Riemann minimal examples and prove that the only connected properly embedded, minimal planar domains in $${\mathbb R}^3$$ with infinite topology are the Riemann minimal examples. From this result together with previously well-known facts, the authors complete the classification of properly embedded, minimal planar domains in $${\mathbb R}^3$$. Namely, they show that, up to scaling and rigid motion, any connected, properly embedded, minimal planar domain in $${\mathbb R}^3$$ is a plane, a helicoid, a catenoid or one of the Riemann minimal examples. In particular, for every such surface, there exists a foliation of $${\mathbb R}^3$$ by parallel planes, each of which intersects the surface transversely in a connected curve that is a circle or a line.

### MSC:

 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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### References:

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