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Mean curvature flow of entire graphs evolving away from the heat flow


Authors: Gregory Drugan and Xuan Hien Nguyen
Journal: Proc. Amer. Math. Soc. 145 (2017), 861-869
MSC (2010): Primary 53C44, 35K15; Secondary 35B40
DOI: https://doi.org/10.1090/proc/13238
Published electronically: September 15, 2016
MathSciNet review: 3577885
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Abstract: We present two initial graphs over the entire $ \mathbb{R}^n$, $ n \geq 2$ for which the mean curvature flow behaves differently from the heat flow. In the first example, the two flows stabilize at different heights. With our second example, the mean curvature flow oscillates indefinitely while the heat flow stabilizes. These results highlight the difference between dimensions $ n \geq 2$ and dimension $ n=1$, where Nara-Taniguchi proved that entire graphs in $ C^{2,\alpha }(\mathbb{R})$ evolving under curve shortening flow converge to solutions to the heat equation with the same initial data.


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Additional Information

Gregory Drugan
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: drugan@uoregon.edu

Xuan Hien Nguyen
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: xhnguyen@iastate.edu

DOI: https://doi.org/10.1090/proc/13238
Keywords: Mean curvature flow, heat flow, asymptotic behavior
Received by editor(s): December 14, 2015
Received by editor(s) in revised form: April 7, 2016
Published electronically: September 15, 2016
Communicated by: Lei Ni
Article copyright: © Copyright 2016 American Mathematical Society