A note on Sobolev isometric immersions below W-2,W-2 regularity

Abstrakt

This paper aims to investigate the Hessian of second order Sobolev isometric immersions below the natural W-2,W-2 setting. We show that the Hessian of each coordinate function of a W-2,W-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 W-2,W-P, p < 2, isometric immersion from a bounded domain of R-2 into R-3 that has multiple singularities.

Klíčová slova

• Isometric immersions
• Degenerate Monge Ampere equation
• Hessian determinant