On the structure of complete simply connected embedded minimal surfaces (Q1312627)

This paper deals with the following problem (Osserman): Are there complete simply connected embedded minimal surfaces in \(\mathbb{R}^ 3\) other than the helicoid and the plane? The author establishes some geometric- analytic properties of this class of surfaces by deforming them into surfaces with self-intersections. In the first part he studies the case in which the deformation of a hyperbolic surface by its associates preserves the embedded condition at any time. Here, he obtains three alternatives. Two of them ((1) \(\inf K=- \infty\); and (2) the immersion is not proper) are of clear geometric-topological character. The third one is more analytic although it is reminiscent of the helicoid. In the second part, he studies the time evolution of the self- intersections of the above deformations (including both hyperbolic and parabolic cases) for a finite time. Both parts are summarised in the main theorem of the paper, where five alternatives for a complete simply connected embedded minimal surfaces are given: the three mentioned above and two more which are satisfied by the plane and helicoid respectively. As a consequence of the Weierstrass representation of minimal surfaces in \(\mathbb{R}^ 3\), the main tools used are those of complex variables.