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

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.