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Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space. (English) Zbl 1353.35052

Summary: 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.

MSC:

35B35 Stability in context of PDEs
35L72 Second-order quasilinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58J45 Hyperbolic equations on manifolds
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