Properly embedded minimal planar domains. (English) Zbl 1315.53008

Meeks, William H. III.; Pérez, Joaquín; Ros, Antonio

Ann. Math. (2) 181, No. 2, 473-546 (2015).

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.

Reviewer: Gabjin Yun (Yongin)

MSC:

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

Keywords:

curvature estimates; finite total curvature; index of stability; Jacobi function; KdV hierarchy; Korteweg-de Vries equation; limit tangent plane at infinity; minimal surface; Shiffman function; stability

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