We report our recent work on a classical problem in geometric analysis: isometric immersions and/or embeddings of Riemannian and semi-Riemannian manifolds. The underlying PDE (partial differential equation) is the Gauss–Codazzi–Ricci equations. Existence of isometric immersions is studied under various curvature conditions, via elliptic and hyperbolic PDE techniques. Weak continuity of isometric immersions is investigated with the help of the theory of compensated compactness. Connections to other problems, including harmonic maps and Coloumb gauges, will also be discussed.
Our talk contains joint works with Gui-Qiang Chen and Marshall Slemrod.