Some uniqueness and nonexistence theorems for embedded minimal surfaces (Q1318022)

We develop a method to study embedded minimal surfaces in the Euclidean space \(\mathbb{R}^ 3\) with vertical flux, which is based on the existence of a 1-parameter deformation for such a surface. As application we first give a new and simpler proof of the known fact -- see \textit{F. J. López} and the second author, J. Differ. Geom. 33, No. 1, 293-300 (1991; Zbl 0719.53004) -- that the plane and the catenoid are the only properly embedded minimal surfaces with finite total curvature and genus zero in \(\mathbb{R}^ 3\). We also prove that a properly embedded minimal surface of finite total curvature and genus zero whose boundary is a planar convex curve must be an annulus, and a similar result when the boundary consists in two planar convex curves lying in parallel planes. When we apply our method to singly periodic minimal surfaces, we obtain that the helicoid is the only properly embedded minimal surface in \(\mathbb{R}^ 3/T\), \(T\) being a non trivial vertical translation, with genus zero and a finite number of helicoidal type ends. We also demonstrate that there are no properly embedded minimal tori with a finite number of planar type ends in \(\mathbb{R}^ 3/S_ \theta\), where \(S_ \theta\) is a non trivial screw motion, \(0 < \theta < 2\pi\).