Locally Lipschitz graph property for lines

Cui, Xiaojun

doi:10.1090/S0002-9939-2015-12593-2

Locally Lipschitz graph property for lines
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Proc. Amer. Math. Soc. 143 (2015), 4423-4431 Request permission

Abstract:

On a non-compact, smooth, connected, boundaryless, complete Riemannian manifold $(M,g)$, some ideal boundary elements could be defined by rays (or equivalently, by Busemann functions). From the viewpoint of Aubry-Mather theory, these boundary elements could be regarded as an analogue to the static classes of Aubry sets, and thus lines should be thought of as the counterpart of the semi-static curves connecting different static classes. In Aubry-Mather theory, a core property is the Lipschitz graph property for Aubry sets and for some kind of semi-static curves. In this note, we prove such a result for a set of lines which connect the same pair of boundary elements. We also discuss an initial relation with ends (in the sense of Freudenthal).

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