Abstract
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.
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