Stability of the 3-dimensional catenoid for the hyperbolic vanishing mean curvature equation

TL;DR

This paper proves the asymptotic stability of the 3-dimensional catenoid as a solution to the hyperbolic vanishing mean curvature equation in Minkowski space, addressing unique challenges due to slower decay rates and introducing new analytical techniques.

Contribution

The authors establish the stability of the 3D catenoid with novel methods for handling slow decay and modulation, extending stability analysis to this more complex setting.

Findings

Proved Morawetz estimates for the linearized operator with slow decay.

Developed a new approach for Price's law bounds on a moving catenoid.

Constructed a refined profile capturing wave-catenoid interactions.

Abstract

We prove that the $3$-dimensional catenoid is asymptotically stable as a solution to the hyperbolic vanishing mean curvature equation in Minkowski space, modulo suitable translation and boost (i.e., modulation) and with respect to a codimension one set of initial data perturbations. The modulation and the codimension one restriction on the initial data are necessary (and optimal) in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. The $3$-dimensional problem is more challenging than the higher (specifically, $5$ and higher) dimensional case addressed in the previous work of the authors with J.~L\"uhrmann, due to slower temporal decay of waves and slower spatial decay of the catenoid. To overcome these issues, we introduce several innovations, such as a proof of Morawetz- (or local-energy-decay-) estimates for the linearized operator with slowly decaying kernel elements based on the Darboux transform, a new method to obtain Price's-law-type bounds for waves on a moving catenoid, as well as a refined profile construction designed to capture a crucial cancellation in the wave-catenoid interaction. In conjunction with our previous work on the higher dimensional case, this paper outlines a systematic approach for studying other soliton stability problems for $(3+1)$-dimensional quasilinear wave equations.