Abstract

We report our recent work on a classical problem in differential geometry: isometric immersions of Riemannian manifolds. In particular, we shall explore the existence theory and geometric characterisations of isometric immersions with critical/supercritical Sobolev regularities, mainly via the theory of compensated compactness. Connections to other problems in mathematical physics, including nonlinear elasticity, harmonic maps, and Uhlenbeck's gauge theory, will also be emphasised. The talk is based on various joint work with Gui-Qiang Chen (Oxford), Marshall Slemrod (Wisconsin), Armin Schikorra (Pittsburgh), Mohammad Reza Pakzad (Toulon), and Xiangxiang Su (SJTU).