In 1950’s, Nash-Kuiper built up the C^1 isometric embedding for any surface into $\mathbb{R}^3$, this can be viewed as analysis side of isometric embedding. On the other hand, there is obstruction for the existence of $C^2$ isometric embedding of surface into $\mathbb{R}^3$ known since Hilbert, which reflects the geometry flavor of isometric embedding. What’s happening from $C^1$ to $C^2$ (from analysis to geometry)? We will present our partial progress along this direction. The talk will be accessible to audience with basic knowledge of analysis and differential geometry.
