Topological properties of minimal surfaces

• Main Author: Hall, Peter
• Published: University of Warwick 1983
• Subjects: 510; QA Mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.354724

This thesis describes examples which answer two questions posed by Meeks about the topology of minimal surfaces. Question 1 [Meeksl, conjecture 5][Meeks2, Problem l]. Given a set Г of disjoint smooth Jordan curves on the standard 2-sphere S2, such that Г bounds two homeomorphic embedded compact connected minimal surfaces F and G in B³, is there an isotopy of B³ fixing Г and taking F to G? Meeks has shown that such surfaces always split B³ into two handlebodies; it then follows that such an isotopy exists if T consists of a single curve or if F and G are annuli [Meeks2, Theorem 2]. We give two examples where F and G are not isotopic in one example F and G are planar domains with three boundary components and in the other they have genus one and two boundary components. Question 2 [Meeksl, conjecture 2][Nitschel, §910(b)]. Can a Jordan curve on the boundary of a convex set in R³ bound a minimal disc that is not embedded? Meeks and Yau have proved that such a disc is embedded under the assumption that it solves the problem of least area for its boundary [MYI, Theorem 2j. We give an example that shows this assumption is necessary. Our examples can be described informally using the "bridge principle," a heuristic method for constructing minimal surfaces which was introduced by Courant [Courant, Lemma 3.3] and Levy [Livy, Chapter I, Section 6]. A method for making such examples rigorous was given by Meeks and Yau [MY2, Theorem 7], and we include an exposition of the results of theirs that we need.