Stability of the catenoid for the hyperbolic vanishing mean curvature equation in 4 spatial dimensions

TL;DR

This paper proves the asymptotic stability of the catenoid as a stationary solution to the hyperbolic vanishing mean curvature equation in four-dimensional Minkowski space, overcoming unique challenges posed by slower decay rates.

Contribution

It introduces a novel commutator vector field technique to establish decay estimates for the 4D case, extending stability results to this critical dimension without symmetry assumptions.

Findings

Established stability of the catenoid in 4D HVMC equation.

Developed a hierarchy of estimates with higher r^p-weights for decay.

Applicable methods for other wave equations in even dimensions.

Abstract

We establish the asymptotic stability of the catenoid, as a nonflat stationary solution to the hyperbolic vanishing mean curvature (HVMC) equation in Minkowski space $\mathbb{R}^{1 + (n + 1)}$ for $n = 4$. Our main result is under a ``codimension-$1$'' assumption on initial perturbation, modulo suitable translation and boost (i.e. modulation), without any symmetry assumptions. In comparison to the $n \geq 5$ case addressed by L\"{u}hrmann-Oh-Shahshahani arxiv:2212.05620, proving catenoid stability in $4$ dimensions shares additional difficulties with its $3$ dimensional analog, namely the slower spatial decay of the catenoid and slower temporal decay of waves. To overcome these difficulties in the $n = 3$ case, the strong Huygens principle, as well as a miraculous cancellation in the source term, plays an important role in arxiv:2409.05968 to obtain strong late time tails. In $n = 4$ dimensions, without these special structural advantages, our novelty is to introduce an appropriate commutator vector field to derive a new hierarchy of estimates with higher $r^p$-weights so that an improved pointwise decay can be established. We expect this to be applicable for proving improved late time tails of other quasilinear wave equations in even dimensions or wave equations with inverse square potential.