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

Abstract

We study time-like 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 co-dimension 1 Lipschitz manifold transverse to the unstable mode consisting of initial data whose solutions exist globally in time and converge asymptotically to the catenoid.

59 pages, 2 figures ; v3 fixed some minor typos and problems with cross references within the document, submitted; v4 added some additional references and a minor clarification in Appendix A; v5 incorporated comments from the referees

References in corpus (2)

• On smoothness of timelike maximal cylinders in three dimensional vacuum spacetimes
• Regular hyperbolicity, dominant energy condition and causality for Lagrangian theory of maps

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