The uniqueness of the helicoid

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The classical theory of minimal surfaces (2011, methodology): “Theorems 4.2 and 4.3 have been applied to obtain a number of results, among which we highlight three: Meeks and Rosenberg [136] proved that the helicoid and the plane are the unique properly embedded, simply connected minimal surfaces in R 3 (see Theorem 1.4); Meeks, Pérez and Ros used Colding-Mini”
Stable constant mean curvature surfaces (2008, methodology): “For the sake of completeness, we give the proof of this elementary fact in the case H 6= 0 (see [40] for the proof in the minimal case).”
Constant mean curvature surfaces in $M^2\times \mathbf{R}$ (2005, background): “A properly embedded minimal surface in R3 cannot be compact; if it is simply connected, it must be the plane or the helicoid; and if it is an annulus, it must be the catenoid [15, 3].”