Stability of the Catenoid for the Hyperbolic Vanishing Mean Curvature Equation in 4 Spatial Dimensions
Year: 2026
Type: article
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 "Equation missing" for $$n = 4$$ 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 $$n \ge 5$$ n ≥ 5 case addressed by Lührmann–Oh–Shahshahani [40], 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 $$n = 3$$ n = 3 case, the strong Huygens principle, as well as a miraculous cancellation in the source term, plays an important role in [50] to obtain strong late time tails. In $$n = 4$$ 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 $$r^p$$ 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.
Source: Annals of PDE
Author: Ning Tang
Institution: University of California, Berkeley
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Funder: National Science Foundation
OpenAlex ID: W4404407511