Caltech/UCLA/USC Joint Analysis Seminar

The catenoid is one of the simplest examples of a minimal hypersurface, next to the hyperplane. In this talk, we will view the catenoid as a stationary solution to the hyperbolic vanishing mean curvature flow, which is the hyperbolic analog of the (elliptic) minimal hypersurface equation, and study its nonlinear stability under no symmetry assumptions. The main result, which is a recent joint work with Jonas Luhrmann and Sohrab Shahshahani, is that with respect to a "codimension one" set of initial data perturbations of the n-dimensional catenoid, the corresponding flow asymptotes to an adequate translation and Lorentz boost of the catenoid for n greater than or equal to 5. Note that the codimension one condition is necessary and sharp in view of the fact that the catenoid is an index 1 minimal hypersurface.

Among the key challenges of the present problem compared to the more classical stability problems for nontrivial stationary solutions are: (1) the quasilinearity of the equation, (2) the slow (polynomial) decay of the catenoid at infinity, and (3) the lack of symmetry assumptions. To address these challenges, we introduce several new ideas, such as a geometric construction of modulated profiles, smoothing of modulation parameters, and a robust framework for proving decay for the radiation.