Fractional Sobolev isometric immersions of planar domains

TL;DR

This paper extends the understanding of $C^1$ regularity and developability of isometric immersions of flat domains into three-dimensional space, focusing on fractional Sobolev regularity regimes.

Contribution

It generalizes known results on Sobolev and H"older regimes by analyzing fractional Sobolev regularity and the behavior of the distributional Jacobian determinant in this context.

Findings

Established $C^1$ regularity for fractional Sobolev isometric immersions.

Analyzed the weak Codazzi-Mainardi equations for these immersions.

Showed the Jacobian determinant operator's behavior under scalar multiplication.

Abstract

We discuss $C^1$ regularity and developability of isometric immersions of flat domains into $\mathbb R^3$ enjoying a local fractional Sobolev $W^{1+s, \frac2s}$ regularity for $2/3 \le s< 1 $, generalizing the known results on Sobolev and H\"older regimes. Ingredients of the proof include analysis of the weak Codazzi-Mainardi equations of the isometric immersions and study of $W^{2,\frac2s}$ planar deformations with symmetric Jacobian derivative and vanishing distributional Jacobian determinant. On the way, we also show that the distributional Jacobian determinant, conceived as an operator defined on the Jacobian matrix, behaves like determinant of gradient matrices under products by scalar functions.