A wedge product theorem of compensated compactness theory with critical exponents on Riemannian manifolds

TL;DR

This paper extends compensated compactness theorems to wedge products of differential forms on Riemannian manifolds, especially at critical regularity exponents, impacting geometric analysis and weak convergence theories.

Contribution

It generalizes the div-curl lemma to critical exponents on manifolds, advancing the understanding of weak convergence of differential forms in geometric contexts.

Findings

Proves a wedge product theorem for weakly convergent forms at critical exponents.

Extends compensated compactness results beyond H"older regularity regimes.

Discusses implications for weak continuity of geometric equations and extrinsic geometry.

Abstract

We formulate and prove compensated compactness theorems concerning the limiting behaviour of wedge products of weakly convergent differential forms on closed Riemannian manifolds \`{a} la Robbin--Rogers--Temple [Trans. Amer. Math. Soc. 303 (1987), 609--618]. The case of critical regularity exponents is considered, which generalises the div-curl lemma in Briane--Casado-D\'{i}az--Murat [J. Math. Pures Appl. 91 (2009), 476--494] for vectorfields, thus going beyond the regularity regime entailed by H\"{o}lder's inequality. Implications on the weak continuity of Gauss--Codazz--Ricci equations and $L^p$-extrinsic geometry of isometric immersions of Riemannian manifolds are discussed.