A geometric approach to Mather quotient problem

Let $(M,g)$ be a closed, connected and orientable Riemannian manifold with nonnegative Ricci curvature. Consider a Lagrangian L(x,v):TM→ defined by L(x,v):= 12gₓ(v,v)-ω(v)+c, where c∈ and ω is a closed 1-form. From the perspective of differential geometry, we estimate the Laplacian of the weak KAM solution u to the associated Hamilton-Jacobi equation $H(x,du)=c[L]$ in the barrier sense. This analysis enables us to prove that each weak KAM solution u is constant if and only if ω is a harmonic 1-form. Furthermore, we explore several applications to the Mather quotient and Ma\~n\'e's Lagrangian.