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Locally Lipschitz graph property for lines


Author: Xiaojun Cui
Journal: Proc. Amer. Math. Soc. 143 (2015), 4423-4431
MSC (2010): Primary 53C22, 54D35
DOI: https://doi.org/10.1090/S0002-9939-2015-12593-2
Published electronically: April 2, 2015
MathSciNet review: 3373941
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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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Additional Information

Xiaojun Cui
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, People’s Republic of China.
Email: xcui@nju.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2015-12593-2
Keywords: Busemann function, Barrier function, Aubry-Mather theory, Lipschitz graph
Received by editor(s): February 25, 2014
Received by editor(s) in revised form: June 18, 2014, and July 20, 2014
Published electronically: April 2, 2015
Additional Notes: The author was supported by the National Natural Science Foundation of China (Grant 11271181), the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) and the Fundamental Research Funds for the Central Universities.
Communicated by: Guofang Wei
Article copyright: © Copyright 2015 American Mathematical Society