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Busemann Functions and Barrier Functions

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Abstract

On a smooth, non-compact, complete, boundaryless, connected Riemannian manifold there are two kinds of functions: Busemann functions with respect to rays and barrier functions with respect to lines (if there exists at least one). In this paper we collect some known properties on Busemann functions and introduce some new fundamental properties on barrier functions. Based on these properties of barrier functions, we could define some relations on the set of lines and thus classify them. With the equivalence relation we introduced, we present a generalization of a rigidity conjecture.

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Acknowledgements

We would like to thank Professor G. Knieper for some remarks on Conjecture 7.1. We would like to thank two anonymous referees for advices and criticisms which improved the paper substantially.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, Jiangsu Province, People’s Republic of China

    Xiaojun Cui & Jian Cheng

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  1. Xiaojun Cui
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  2. Jian Cheng
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Correspondence to Xiaojun Cui.

Additional information

The first author is supported by the National Natural Science Foundation of China (Grants 11271181, 11571166, 11631006). Both authors are supported by the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) and the Fundamental Research Funds for the Central Universities.

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Cui, X., Cheng, J. Busemann Functions and Barrier Functions. Acta Appl Math 152, 93–110 (2017). https://doi.org/10.1007/s10440-017-0114-5

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  • DOI: https://doi.org/10.1007/s10440-017-0114-5

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