Global well-posedness for radial extremal hypersurface equation in $(1+3)$-dimensional Minkowski space-time in critical Sobolev space

In this article, we prove the global well-posedness in the critical Sobolev space $H_{rad}^2\left({\mathbb{R}}^2\right)\times H_{rad}^1\left({\mathbb{R}}^2\right)$ for the radial time-like extremal hypersurface equation in $(1+3)$- dimensional Minkowski space-time. This is achieved by deriving a new div-curl type lemma and combined it with energy and ``momentum" balance law to get some space-time estimates of the nonlinearity.