Minimal surfaces with bounded curvature in Euclidean space (Q1599918)

The author states that this work is related to the problem of proving the uniqueness of the helicoid and plane among all simply-connected surfaces that are completely and minimally embedded in the three-dimensional Euclidean space \({\mathbf E}^3\). Since this goal seems distant, the author studies the set \(\mathcal M\) of surfaces which are completely and minimally embedded in \({\mathbf E}^3\) each of whose Gauss curvatures is bounded. One of the motivations to consider this class of surfaces is the following result proved by \textit{F. Xavier} [Geom. Funct. Anal. 11, 1344-1356 (2001; Zbl 1004.53007); a former result in this direction was given by \textit{L. Rodríguez} and \textit{H. Rosenberg}, J. Geom. Anal. 7, 329-342 (1997; Zbl 0936.53008)]: If \(M \in {\mathcal M}\) is transverse to one horizontal plane \(\Pi\) and its intersection with \(\Pi\) consists of a finite number of connected curves, then \(M\) is either a plane or a helicoid. The first main result of this paper is as follows: For each \(M \in {\mathcal M}\), there exists an embedded tubular neighborhood of constant radius which is at least equal to \((\sqrt{3}-1)/(\|K\|_{\infty}^{1/2})\), where \(K\) is the Gauss curvature of \(M\). By applying this result, uniform bounds for area and for spherical area are given: Especially, it is proved that the area growth of \(M \in {\mathcal M}\) is not more than cubic, and moreover, by using the co-area formula, the spherical area growth proves to be not more than linear. Finally, by using the uniform area bounds and results of \textit{H. Choi} and \textit{R. Schoen} [Invent. Math. 81, 387-394 (1985; Zbl 0577.53044)] and \textit{B. White} [Invent. Math. 88, 243-256 (1987; Zbl 0615.53044)], the following compactness theorem is proved: For any sequence of minimal surfaces in \({\mathcal M}\) with at least one accumulation point in \({\mathbf E}^3\), a sub-sequence converges in \({\mathcal M}\) on all compact sets of \({\mathbf E}^3\).