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) - (v) +c, where c and 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) =cL in the barrier sense. This analysis enables us to prove that each weak KAM solution u is constant if and only if is a harmonic 1-form. Furthermore, we explore several applications to the Mather quotient and Ma\~n\'e's Lagrangian.