The first APDE seminar on Monday, 2/6, will be given by our own Sung-Jin Oh in-person in Evans 732, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Mengxuan Yang (firstname.lastname@example.org).
Title: Codimension one stability of the catenoid under the hyperbolic vanishing mean curvature flow
Abstract: The catenoid is one of the simplest examples of minimal hypersurfaces 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 condition is necessary and sharp in view of the fact that the catenoid is an index 1 minimal hypersurface.
In a broader context, our result is a part of stability theory for solitary waves (i.e., stationary solutions) in the presence of modulation (i.e., symmetries, such as translation and Lorentz boosts, create nearby stationary solutions), which was mostly studied for semilinear PDEs in the past. Among the key challenges of the present problem compared to the more classical context are: (1) the quasilinearity of the equation, (2) slow (polynomial) decay of the catenoid at infinity, and (3) 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 part, which are hoped to be useful in the larger context of stability theory for solitary waves for quasilinear wave equations in the presence of modulation.