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Entire downward translating solitons to the mean curvature flow in Minkowski space


Authors: Joel Spruck and Ling Xiao
Journal: Proc. Amer. Math. Soc. 144 (2016), 3517-3526
MSC (2010): Primary 53C44, 35J62
DOI: https://doi.org/10.1090/proc/12969
Published electronically: December 22, 2015
MathSciNet review: 3503719
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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)) \vert 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
Email: js@math.jhu.edu

Ling Xiao
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
Email: lx70@math.rutgers.edu

DOI: https://doi.org/10.1090/proc/12969
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
Article copyright: © Copyright 2015 American Mathematical Society

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