The HADES seminar on Tuesday, November 5th, will be at 2:00pm in Room 740.

Speaker: Ning Tang

Abstract: The hyperbolic vanishing mean curvature (HVMC) equation in Minkowski space is a quasilinear wave equation that serves as the hyperbolic counterpart of the 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.