Compensated Compactness in Banach Spaces and Weak Rigidity of Isometric Immersions of Manifolds

Abstract

We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div-curl lemma for tensor fields on Riemannian manifolds is obtained. Then we show how this intrinsic div-curl lemma can be employed to establish the global weak rigidity of the Gauss-Codazzi-Ricci equations, the Cartan formalism, and the corresponding isometric immersions of Riemannian submanifolds.

arXiv comment22 pages; Partial Differential Equations, Mathematical Physics and Stochastic Analysis, A Volume in Honor of Helge Holden's 60th Birthday, EMS Congress Reports, EMS, 2018