Codimension one stability of the catenoid for the hyperbolic vanishing mean curvature equation

Sohrab Shahshahani (UMass Amherst)

May 16. 2024, 10:00 — 11:00

The catenoid, which is a minimal surface, can be viewed as a stationary solution of the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in Euclidean space. Our main result is the nonlinear asymptotic stability, modulo suitable translation and Lorentz boost (i.e., modulation), of the n-dimensional catenoid with respect to a codimension one set of initial data perturbations without any symmetry assumptions, for n ≥ 5 and 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. The scheme of the proof for n = 3 and n ≥ 5 differ in a number of important points, which are ultimately related to the slow decay of the catenoid to a hyperplane in lower dimensions. In a broader context, this work fits in the long tradition of studies of soliton stability problems. From this viewpoint, our aim is to tackle some new issues that arise due to the quasilinear nature of the underlying hyperbolic equation. This talk is based on joint works with Jonas Luhrmann and Sung-Jin Oh.

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Nonlinear Waves and Relativity (Thematic Programme)
Roland Donninger (U of Vienna)
David Fajman (U of Vienna)
Birgit Schörkhuber (U of Innsbruck)