A note on Sobolev isometric immersions below W2,2 regularity

This paper aims to investigate the Hessian of second order Sobolev isometric immersions below the natural W2,2 setting. We show that the Hessian of each coordinate function of a W2,p, p<2, isometric immersion satisfies a low rank property in the almost everywhere sense, in particular, its Gaussian curvature vanishes almost everywhere. Meanwhile, we provide an example of a W2,p, p<2, isometric immersion from a bounded domain of R2 into R3 that has multiple singularities.