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

The hyperbolic vanishing mean curvature (HVMC) equation in Minkowski space is a quasilinear wave equation that serves as the hyperbolic counterpart ofthe minimal surface equation in Euclidean space. This talk will concern the modulated nonlinear asymptotic stability of the $1+4$ dimensional hyperbolic catenoid, viewed as a stationary solution to the HVMC equation. This stability result is under a ``codimension-one'' 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 Lührmann-Oh-Shahshahani, proving catenoid stability in $n = 4$ 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 for the $n = 3$ case, the strong Huygens principle, as well as a miracle cancellation in the source term, plays an important role in the work of Oh-Shahshahani to obtain strong late time tails. Without these special structural advantages in $n = 4$ dimensions, our novelty is to introduce an appropriate commutator vector field to derive a new hierarchy with higher $r^p$-weights so that an improved pointwise decay can be established. 

In this talk, I will first give an outline of the proof and then focus on a model case to illustrate our approach to achieving an improved decay in $4$ dimensions.