Stability of the catenoid for the hyperbolic vanishing mean curvature equation

Abstract: The catenoid, which is a minimal surface, can be viewed as a stationary solution of the hyperbolic vanishing mean curvature equation in Minkowski spacetime. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in Euclidean space. The main result discussed in this talk is the nonlinear asymptotic stability, modulo suitable translation, and Lorentz boost (i.e., modulation), of the n-dimensional catenoid for a codimension-one set of initial data perturbations without any symmetry assumptions, for n ≥ 3. The modulation and the codimension-one restriction on the data are necessary and optimal in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. 

In a broader context, these works fit in the long tradition of studies of soliton stability problems. From this viewpoint, they outline a systematic approach for studying soliton stability problems for quasilinear wave equations. This talk is based on joint works with Jonas Lührmann and Sohrab Shahshahani, as well as an independent work of Ning Tang.