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Flowing maps to minimal surfaces

Rupflin, M., & Topping, P. M. (2016). Flowing maps to minimal surfaces. American Journal of Mathematics, 138(4), 1095-1115. doi:10.1353/ajm.2016.0035.

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1205.6298 (Preprint), 249KB
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1205.6298
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Creators:
Rupflin, Melanie1, Author
Topping, Peter M., Author
Affiliations:
1Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_24012

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Free keywords: Mathematics, Differential Geometry, math.DG,Mathematics, Analysis of PDEs, math.AP,
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 and Struwe etc. In the genus 1 case, we show that our flow is exactly equivalent to that considered by Ding-Li-Lui. In general, we recover the result of Schoen-Yau and Sacks-Uhlenbeck 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 $\pi_2=0$.

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Dates: 2012-05-2920132016
Publication Status: Published in print
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Identifiers: arXiv: 1205.6298
URI: http://arxiv.org/abs/1205.6298
DOI: 10.1353/ajm.2016.0035
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Title: American Journal of Mathematics
Source Genre: Journal
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Pages: - Volume / Issue: 138 (4) Sequence Number: - Start / End Page: 1095 - 1115 Identifier: -