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Flowing maps to minimal surfaces
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Rupflin, Melanie and Topping, Peter (2016) Flowing maps to minimal surfaces. American Journal of Mathematics, 138 (4). pp. 1095-1115. doi:10.1353/ajm.2016.0035 ISSN 0002-9327.
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Official URL: http://dx.doi.org/10.1353/ajm.2016.0035
Abstract
We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal surfaces. In the genus $0$ case, our flow is just the harmonic map flow, and it tries to find branched minimal 2-spheres as in Sacks-Uhlenbeck (1981) and Struwe (1985), etc. In the genus 1 case, we show that our flow is exactly equivalent to that considered by Ding-Li-Liu (2006). In general, we recover the result of Schoen-Yau (1979) and Sacks-Uhlenbeck (1982) that an incompressible map from a surface can be adjusted to a branched minimal immersion with the same action on $\pi_1$, and this minimal immersion will be homotopic to the original map in the case that $\pi_2=0$.
| Item Type: | Journal Article | ||||||
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| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| Library of Congress Subject Headings (LCSH): | Harmonic maps, Riemannian manifolds, Teichmüller spaces | ||||||
| Journal or Publication Title: | American Journal of Mathematics | ||||||
| Publisher: | The Johns Hopkins University Press | ||||||
| ISSN: | 0002-9327 | ||||||
| Official Date: | August 2016 | ||||||
| Dates: |
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| Volume: | 138 | ||||||
| Number: | 4 | ||||||
| Page Range: | pp. 1095-1115 | ||||||
| DOI: | 10.1353/ajm.2016.0035 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Date of first compliant deposit: | 3 January 2018 | ||||||
| Date of first compliant Open Access: | 3 January 2018 | ||||||
| RIOXX Funder/Project Grant: |
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