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 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 case addressed by Lührmann-Oh-Shahshahani, proving catenoid stability in dimensions shares additional difficulties with its dimensional analog, namely the slower spatial decay of the catenoid and slower temporal decay of waves. To overcome these difficulties for the 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 dimensions, our novelty is to introduce an appropriate commutator vector field to derive a new hierarchy with higher -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 dimensions.