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

TL;DR

This paper develops a compensated compactness theorem in Banach spaces, motivated by weak rigidity problems in geometry, leading to new div-curl lemmas and results on the weak rigidity of isometric immersions of manifolds.

Contribution

It introduces a novel compensated compactness theorem in Banach spaces and applies it to establish weak rigidity results for isometric immersions of manifolds.

Findings

Established a compensated compactness theorem in Banach spaces.

Derived a geometrically intrinsic div-curl lemma for tensor fields.

Proved global weak rigidity of Gauss-Codazzi-Ricci equations and isometric immersions.

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.