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arXiv · 2003.05595

Optimal regularity for the Pfaff system and isometric immersions in arbitrary dimensions

Siran Li
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Abstract

We prove the existence, uniqueness, and $W^{1,2}$-regularity for the solution to the Pfaff system with antisymmetric $L^2$-coefficient matrix in arbitrary dimensions. Hence, we establish the equivalence between the existence of $W^{2,2}$-isometric immersions and the weak solubility of the Gauss--Codazzi--Ricci equations on simply-connected domains. The regularity assumptions of these results are sharp. As an application, we deduce a weak compactness theorem for $W^{2,2}_{\rm loc}$-immersions.

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Siran Li. 2020-03-12. Optimal regularity for the Pfaff system and isometric immersions in arbitrary dimensions. https://arxiv.org/abs/2003.05595

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