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 $L(x,v):TM\to \mathbb{R}$ defined by $L(x,v):=\frac{1}{2}{g}_x(v,v)-\omega (v)+c$, where $c\in \mathbb{R}$ and $\omega$ 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 $\omega$ is a harmonic 1-form. Furthermore, we explore several applications to the Mather quotient and Mañé’s Lagrangian.