Properly embedded minimal planar domains with infinite topology are Riemann minimal examples

2009/09/12 by William H. Meeks, Meeks, William H., Joaquín Pérez +1
Mathematics · #49Q05 #53A10 #53C42 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.0909.2326

openalex publication_date 2009/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

These notes outline recent developments in classical minimal surface theory that are essential in classifying the properly embedded minimal planar domains M in R³ with infinite topology (equivalently, with an infinite number of ends). This final classification result by Meeks, Perez, and Ros states that such an M must be congruent to a homothetic scaling of one of the classical examples found by Riemann in 1860. These examples \cal R_s, 0, are singly-periodic and intersect each horizontal plane in R³ in a circle or a line parallel to the x-axis. Earlier work by Collin, Lopez and Ros and Meeks and Rosenberg demonstrate that the plane, the catenoid and the helicoid are the only properly embedded minimal surfaces of genus zero with finite topology (equivalently, with a finite number of ends). Since the surfaces \cal R_s converge to a catenoid as s tends to 0 and to a helicoid as s tends to infinity, then the moduli space \cal M of all properly embedded, non-planar, minimal planar domains in R³ is homeomorphic to the closed unit interval [0,1].

Citations

• Limit leaves of a CMC lamination are stable

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