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Topological properties of minimal surfaces
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Hall, Peter (1983) Topological properties of minimal surfaces. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b1464808~S1
Abstract
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.
| Item Type: | Thesis (PhD) | ||||
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| Subjects: | Q Science > QA Mathematics | ||||
| Library of Congress Subject Headings (LCSH): | Topology, Minimal surfaces, Algebraic topology, Boundary value problems | ||||
| Official Date: | June 1983 | ||||
| Dates: |
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| Institution: | University of Warwick | ||||
| Theses Department: | Mathematics Institute | ||||
| Thesis Type: | PhD | ||||
| Publication Status: | Unpublished | ||||
| Supervisor(s)/Advisor: | Epstein, D. B. A. | ||||
| Sponsors: | Science Research Council (Great Britain) | ||||
| Extent: | 93 leaves | ||||
| Language: | eng |
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