6533b854fe1ef96bd12af543
RESEARCH PRODUCT
A note on Sobolev isometric immersions below W2,2 regularity
Zhuomin LiuJan Malýsubject
Hessian matrixPure mathematicsIsometric exercise01 natural sciencessymbols.namesake0103 physical sciencesGaussian curvatureImmersion (mathematics)Almost everywhereisometric immersions0101 mathematicsMathematics010102 general mathematicsMathematical analysista111Hessian determinantSobolev spaceComputational Theory and MathematicsBounded functionsymbolsGravitational singularityMathematics::Differential Geometry010307 mathematical physicsGeometry and Topologydegenerate Monge–Ampère equationAnalysisdescription
Abstract This paper aims to investigate the Hessian of second order Sobolev isometric immersions below the natural W 2 , 2 setting. We show that the Hessian of each coordinate function of a W 2 , 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 W 2 , p , p 2 , isometric immersion from a bounded domain of R 2 into R 3 that has multiple singularities.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2017-06-01 | Differential Geometry and its Applications |