A geometric approach to Mather quotient problem

TL;DR

This paper investigates the properties of weak KAM solutions on Riemannian manifolds with nonnegative Ricci curvature, establishing conditions under which these solutions are constant and exploring implications for the Mather quotient and M\'an\'e's Lagrangian.

Contribution

It provides a geometric estimate of the Laplacian of weak KAM solutions and characterizes when these solutions are constant based on harmonicity of the 1-form, with applications to Mather quotient theory.

Findings

Weak KAM solutions are constant iff the 1-form is harmonic.

Laplacian estimates for weak KAM solutions are derived.

Applications to Mather quotient and M\'an\'e's Lagrangian are demonstrated.

Abstract

Let $(M,g)$ be a closed, connected and orientable Riemannian manifold with nonnegative Ricci curvature. Consider a Lagrangian $L(x,v):TM\to\R$ defined by $L(x,v):=\frac 12g_x(v,v)-\omega(v)+c$, where $c\in\R$ 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 Ma\~n\'e's Lagrangian.