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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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

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

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

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Zentralblatt MATH identifier

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]

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


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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