Quasiconvexity in the Riemannian setting

Abstract: We introduce a notion of quasiconvexity for continuous functions $f$ defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold $(M,g)$ and $\mathbb{R}^m$, naturally generalizing the classical Euclidean definition. We prove that this condition characterizes the sequential lower semicontinuity of the associated integral functional [ F(u, Ω) = \int_Ω f(du) \, dμ] with respect to the weak$^*$ topology of $W^{1,\infty}(Ω, \mathbb{R}^m)$, for every bounded open subset $Ω\subseteq M$.