Year: 2024
Author: Siran Li
Communications in Mathematical Analysis and Applications, Vol. 3 (2024), Iss. 4 : pp. 532–557
Abstract
We survey recent developments on the analysis of Gauss-Codazzi-Ricci equations, the first-order PDE system arising from the classical problem of isometric immersions in differential geometry, especially in the regime of low Sobolev regularity. Such equations are not purely elliptic, parabolic, or hyperbolic in general, hence calling for analytical tools for PDEs of mixed types. We discuss various recent contributions – in line with the pioneering works [Chen et al., Proc. Amer. Math. Soc. 138 (2010), Commun. Math. Phys. 294 (2010)] – on the weak continuity of Gauss-Codazzi-Ricci equations, the weak stability of isometric immersions, and the fundamental theorem of submanifold theory with low regularity. Two mixed-type PDE techniques are emphasised throughout these developments: the method of compensated compactness and the theory of Coulomb-Uhlenbeck gauges.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cmaa.2024-0023
Communications in Mathematical Analysis and Applications, Vol. 3 (2024), Iss. 4 : pp. 532–557
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 26
Keywords: Isometric immersions nonlinear elasticity Gauss-Codazzi-Ricci equations compensated compactness gauge theory.