Isometric Immersions, via PDE and Physical Approaches

We report our recent progress on the problem of isometric immersions of Riemannian/semi-Riemannian manifolds. The weak stability of $W^{2,p}$-isometric immersions; or, equivalently, the $L^p$-weak continuity of the Gauss--Codazzi--Ricci equations, will be discussed. This is done via the theory of compensated compactness as well as a smoothability result for Uhlenbeck gauges. We also present two new results on the existence of isometric immersions, in elliptic (the Weyl problem via $J$-holomorphic curves) and hyperbolic (fluid formulation and artificial viscosity) regimes, respectively. PDE theories and techniques shall be emphasised throughout the talk. (*Joint with Prof. Gui-Qiang Chen.)