The geometry of C1,α flat isometric immersions

We show that any isometric immersion of a flat plane domain into R³ is developable provided it enjoys the little Hölder regularity c1,2/3 . In particular, isometric immersions of local C1,α regularity with α >2/3 belong to this class. The proof is based on the existence of a weak notion of second fundamental form for such immersions, the analysis of the Gauss–Codazzi–Mainardi equations in this weak setting, and a parallel result on the very weak solutions to the degenerate Monge–Ampère equation analysed in [M. Lewicka and M. R. Pakzad. Anal. PDE 10 (2017), 695–727.].