Differentiability of Mather's $\beta$-function vs Ma\~n\'e's conjecture

TL;DR

This paper investigates the differentiability properties of Mather's beta-function in relation to Mañé's conjecture, establishing a connection between the function's smoothness and the density of Legendre transforms of rational homology classes.

Contribution

It proves that strong differentiability conditions on Mather's beta-function imply the density of Legendre transforms of rational homology classes, advancing understanding of Mañé's conjecture.

Findings

Differentiability of Mather's beta-function implies density of Legendre transforms

Establishes a link between smoothness properties and homological density

Provides a step towards proving Mañé's conjecture

Abstract

We prove that if a time-periodic Tonelli Lagrangian on a closed manifold $M$ satisfies a strong version of the Differentiability Problem for Mather's $\beta$-function, then the Legendre transforms of rational homology classes are dense in the first cohomology of $M$, which is a first step towards Ma\~n\'e's conjecture.