Locally Lipschitz graph property for lines

TL;DR

This paper proves a locally Lipschitz graph property for lines connecting the same boundary elements in the context of Mather theory on Riemannian manifolds, extending known properties of Aubry sets and semi-static curves.

Contribution

It establishes a new locally Lipschitz graph property for a specific set of lines in Mather theory, linking boundary elements on Riemannian manifolds.

Findings

Proves Lipschitz graph property for lines connecting boundary elements

Extends properties of Aubry sets to new class of semi-static curves

Provides mathematical framework for analyzing boundary connections

Abstract

On a non-compact, smooth, connected, boundaryless, complete Riemannian manifold $(M,g)$, one can define its ideal boundary by rays (or equivalently, Busemann functions). From the viewpoint of Mather theory, boundary elements could be regarded as the static classes of Aubry sets, and thus lines should be think as the semi-statics curves connecting different static classes. In Mather theory, one core property is Lipschitz graph property for Aubry sets and for some kind of semi-static curves. In this article, we prove a such kind of result for a set of lines which connect the same pair of boundary elements.