Calculus of Variations and Geometric Measure Theory

C. De Lellis - M. P. Reza

The geometry of flat $C^{1,\alpha}$ isometric immersions

created by delellis on 01 Dec 2020



Inserted: 1 dec 2020
Last Updated: 1 dec 2020

Year: 2020


We show that any isometric immersion of a flat plane domain into $\R^3$ is developable provided it enjoys $C^{1,\alpha}_{loc}$ H\"older regularity for some $\alpha>2/3$. The proof is based on showing 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 Monge-Amp\`ere equation which appeared in \cite{lepamonge}.