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

作者：Liu Zhuomin; Maly Jan^*

来源：Differential Geometry and its Applications, 2017, 52: 1-10.

DOI：10.1016/j.difgeo.2017.02.009

• Isometric immersions
• Degenerate Monge Ampere equation
• Hessian determinant

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.

• 出版日期2017-6