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We study the problem of stability of the catenoid, which is an asymptoticallyflat rotationally symmetric minimal surface in Euclidean space, viewed as astationary solution to the hyperbolic vanishing mean curvature equation inMinkowski space. The latter is a quasilinear wave equation that constitutes thehyperbolic counterpart of the minimal surface equation in Euclidean space. Ourmain result is the nonlinear asymptotic stability, modulo suitable translationand boost (i.e., modulation), of the $n$-dimensional catenoid with respect to acodimension one set of initial data perturbations without any symmetryassumptions, for $n \geq 5$. The modulation and the codimension one restrictionon the data are necessary and optimal in view of the kernel and the uniquesimple eigenvalue, respectively, of the stability operator of the catenoid.  In a broader context, this paper fits in the long tradition of studies ofsoliton stability problems. From this viewpoint, our aim here is to tackle somenew issues that arise due to the quasilinear nature of the underlyinghyperbolic equation. Ideas introduced in this paper include a new profileconstruction and modulation analysis to track the evolution of the translationand boost parameters of the stationary solution, a new scheme for provingintegrated local energy decay for the perturbation in the quasilinear andmodulation-theoretic context, and an adaptation of the vectorfield method inthe presence of dynamic translations and boosts of the stationary solution.

Stability of the Catenoid for the Hyperbolic Vanishing Mean Curvature Equation Outside Symmetry