Duke Mathematical Journal
- Duke Math. J.
- Volume 165, Number 4 (2016), 723-791.
Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space
We study timelike hypersurfaces with vanishing mean curvature in the -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.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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]
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