Duke Mathematical Journal

Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space

Roland Donninger, Joachim Krieger, Jérémie Szeftel, and Willie Wong

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Abstract

We study timelike hypersurfaces with vanishing mean curvature in the (3+1)-dimensional Minkowski space, which are the hyperbolic counterparts to minimal embeddings of Riemannian manifolds. The catenoid is a stationary solution of the associated Cauchy problem. This solution is linearly unstable, and we show that this instability is the only obstruction to the global nonlinear stability of the catenoid. More precisely, we prove in a certain symmetry class the existence, in the neighborhood of the catenoid initial data, of a codimension one Lipschitz manifold transverse to the unstable mode consisting of initial data whose solutions exist globally in time and converge asymptotically to the catenoid.

Article information

Source
Duke Math. J., Volume 165, Number 4 (2016), 723-791.

Dates
Received: 20 December 2013
Revised: 28 April 2015
First available in Project Euclid: 20 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1453298455

Digital Object Identifier
doi:10.1215/00127094-3167383

Mathematical Reviews number (MathSciNet)
MR3474816

Zentralblatt MATH identifier
1353.35052

Subjects
Primary: 35L72: Quasilinear second-order hyperbolic equations
Secondary: 35B40: Asymptotic behavior of solutions 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35B35: Stability 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

Keywords
vanishing mean curvature flow extremal surfaces quasilinear wave equations nonlinear stability center manifold

Citation

Donninger, Roland; Krieger, Joachim; Szeftel, Jérémie; Wong, Willie. Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space. Duke Math. J. 165 (2016), no. 4, 723--791. doi:10.1215/00127094-3167383. https://projecteuclid.org/euclid.dmj/1453298455


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