Entire downward translating solitons to the mean curvature flow in Minkowski space
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- by Joel Spruck and Ling Xiao PDF
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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 $\Sigma =\{(x, u(x)) | x\in \mathbb {R}^n\}$ 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.References
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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