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Entropy, stability and harmonic map flow


Authors: Jess Boling, Casey Kelleher and Jeffrey Streets
Journal: Trans. Amer. Math. Soc. 369 (2017), 5769-5808
MSC (2010): Primary 53C43, 53C44
DOI: https://doi.org/10.1090/tran/6949
Published electronically: April 24, 2017
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Abstract: Inspired by work of Colding-Minicozzi (2012) on mean curvature flow, Zhang (2012) introduced a notion of entropy stability for harmonic map flow. We build further upon this work in several directions. First we prove the equivalence of entropy stability with a more computationally tractable $ \mathcal {F}$-stability. Then, focusing on the case of spherical targets, we prove a general instability result for high-entropy solitons. Finally, we exploit results of Lin-Wang (2008) to observe long time existence and convergence results for maps into certain convex domains and how they relate to generic singularities of harmonic map flow.


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Additional Information

Jess Boling
Affiliation: Department of Mathematics, Rowland Hall, University of California, Irvine, California 92617
Email: jboling@uci.edu

Casey Kelleher
Affiliation: Department of Mathematics, Rowland Hall, University of California, Irvine, California 92617
Email: clkelleh@uci.edu

Jeffrey Streets
Affiliation: Department of Mathematics, Rowland Hall, University of California, Irvine, California 92617
Email: jstreets@uci.edu

DOI: https://doi.org/10.1090/tran/6949
Received by editor(s): July 3, 2015
Received by editor(s) in revised form: February 10, 2016
Published electronically: April 24, 2017
Additional Notes: The second author was supported by an NSF Graduate Research Fellowship DGE-1321846
The third author was supported by the NSF via DMS-1301864 and a Sloan Foundation Fellowship
Article copyright: © Copyright 2017 American Mathematical Society