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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Entire downward translating solitons to the mean curvature flow in Minkowski space
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by Joel Spruck and Ling Xiao PDF
Proc. Amer. Math. Soc. 144 (2016), 3517-3526 Request permission

Abstract:

In this paper, we study entire translating solutions $u(x)$ to a mean curvature flow equation in Minkowski space. We show that if $\Sigma =\{(x, u(x)) | x\in \mathbb {R}^n\}$ is a strictly spacelike hypersurface, then $\Sigma$ reduces to a strictly convex rank $k$ soliton in $\mathbb {R}^{k,1}$ (after splitting off trivial factors) whose “blowdown” converges to a multiple $\lambda \in (0,1)$ of a positively homogeneous degree one convex function in $\mathbb {R}^k$. We also show that there is nonuniqueness as the rotationally symmetric solution may be perturbed to a solution by an arbitrary smooth order one perturbation.
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Additional Information
  • Joel Spruck
  • Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
  • MR Author ID: 165780
  • Email: js@math.jhu.edu
  • Ling Xiao
  • Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
  • Email: lx70@math.rutgers.edu
  • Received by editor(s): May 9, 2015
  • Received by editor(s) in revised form: September 9, 2015
  • Published electronically: December 22, 2015
  • Additional Notes: The research of the first author was partially supported by the NSF
  • Communicated by: Lei Ni
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 3517-3526
  • MSC (2010): Primary 53C44, 35J62
  • DOI: https://doi.org/10.1090/proc/12969
  • MathSciNet review: 3503719