Embedded minimal ends of finite type (Q2701660)

The authors investigate complete embedded minimal surfaces (CEMS) in \({\mathbb R}^3\) with infinite total curvature and finite type, and they give a partial answer to an old conjecture that the only nonplanar simply connected CEMS in \({\mathbb R}^3\) is the helicoid.NEWLINENEWLINENEWLINEA complete minimal surface \(M\) in \({\mathbb R}^3\) is said to be of finite type if it has the conformal type of a compact Riemann surface with a finite number of points removed and the differentials \(dg/g\), \(\eta\) extend meromorphically to the punctures, where \((g, \eta)\) is the Weierstrass data of \(M\) given by the stereographically projected Gauss map \(g\) and the height differential \(\eta\). NEWLINENEWLINENEWLINEThe authors prove that the end of a CEMS with infinite total curvature and finite type has an explicit Weierstrass representation which depends only on a holomorphic function that vanishes at the puncture. Conversely, it is proved that any choice of such a holomorphic function gives rise to a properly embedded minimal end provided it solves the corresponding period problem. Moreover, if the flux along the boundary vanishes, then the end proves to be \(C^0\)-asymptotic to a Helicoid. By applying these results, the authors prove that any one-ended CEMS with infinite total curvature and finite type is \(C^0\)-asymptotic to a Helicoid, and that the only simply connected CEMS of finite type are the plane and the Helicoid. NEWLINENEWLINENEWLINEThe proof is achieved especially by very precise analyses of curves on a minimal end of finite type. In the whole study, it is essential that the end is represented in terms of good Weierstrass data.NEWLINENEWLINENEWLINEThe paper also serves as an erratum for a former work of the third author [Ann. Global Anal. Geom. 12, No. 4, 341-355 (1994; Zbl 0873.53002)] which tackles the conjecture mentioned above. This conjecture also has been studied by several other authors and partial results have been obtained, but it is still open.