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Two theorems on the intersections of horospheres in asymptotically harmonic spaces

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Abstract

We use Busemann functions to construct volume preserving mappings in an asymptotically harmonic manifold. If the asymptotically harmonic manifold satisfies the visibility condition, we construct mappings which preserve distances in some directions. We also prove that some integrals on the intersection of horospheres are independent of the differences between the values of the corresponding Busemann functions and we establish an upper bound of the volume of the intersection of two horospheres which is independent of the difference between values of corresponding Busemann functions.

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Acknowledgements

We thank the anonymous referee for the careful reading of the paper and the very helpful comments.

Funding

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1A2C1083957).

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  1. Department of Mathematics, Sungkyunkwan University, Suwon, 16419, Korea

    Sinhwi Kim & JeongHyeong Park

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  1. Sinhwi Kim
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  2. JeongHyeong Park
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Correspondence to JeongHyeong Park.

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Kim, S., Park, J. Two theorems on the intersections of horospheres in asymptotically harmonic spaces. Rev Mat Complut 37, 527–549 (2024). https://doi.org/10.1007/s13163-023-00459-0

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