15 March 2016 Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space
Roland Donninger, Joachim Krieger, Jérémie Szeftel, Willie Wong
Duke Math. J. 165(4): 723-791 (15 March 2016). DOI: 10.1215/00127094-3167383


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.


Download Citation

Roland Donninger. Joachim Krieger. Jérémie Szeftel. Willie Wong. "Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space." Duke Math. J. 165 (4) 723 - 791, 15 March 2016. https://doi.org/10.1215/00127094-3167383


Received: 20 December 2013; Revised: 28 April 2015; Published: 15 March 2016
First available in Project Euclid: 20 January 2016

zbMATH: 1353.35052
MathSciNet: MR3474816
Digital Object Identifier: 10.1215/00127094-3167383

Primary: 35L72
Secondary: 35B30 , 35B35 , 35B40 , 53A10

Keywords: center manifold , extremal surfaces , nonlinear stability , Quasilinear wave equations , vanishing mean curvature flow

Rights: Copyright © 2016 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.165 • No. 4 • 15 March 2016
Back to Top