Abstract

We consider isometric immersions of 3-dimensional domains into $\mathbb{R}^3$ at low regularity regimes, in particular in the case of a flat domain. A notion of second fundamental form can be defined when only $1/2$-fractional derivatives of the Gauss map are well-controlled. Through an analysis of the Codazzi system which - in a sense - survives at $2/3$-fractional differentiability, we can pass to a degenerate Monge-Ampère equation and extract valuable geometric information about the rigidity of these isometric immersions. Time permitting, we will discuss a conjecture of Gromov in this regard and some connected problems in nonlinear analysis and geometric function theory.