Quasiconvexity in the Riemannian setting

We introduce a notion of quasiconvexity for integrands defined on the tangent bundle of a Riemannian manifold. We prove that this condition characterizes the sequential lower semicontinuity of the associated integral functionals with respect to the weak^* topology of W^{1,\infty}, generalizing the classical Euclidean results by Morrey and Acerbi--Fusco.
Moreover, we also extend the notions of polyconvexity and rank--one convexity to this context and establish the hierarchy between polyconvexity, quasiconvexity, and rank--one convexity, as in the Euclidean setting.
Joint work with Aurora Corbisiero e Chiara Leone.