Some recent developments on isometric immersions via compensated compactness and gauge transforms

TL;DR

This paper surveys recent advances in analyzing the Gauss--Codazzi--Ricci equations for isometric immersions, focusing on low regularity regimes using compensated compactness and gauge transform techniques.

Contribution

It highlights new analytical methods for PDEs of mixed types applied to isometric immersion problems at low Sobolev regularity.

Findings

Weak continuity of Gauss--Codazzi--Ricci equations established

Weak stability of low-regularity isometric immersions demonstrated

Fundamental theorem of submanifold theory extended to low regularity

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 by G.-Q. Chen, M. Slemrod, and D. Wang [Proc. Amer. Math. Soc. (2010); Comm. Math. Phys. (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.