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Infinite-time incompleteness of noncompact Yamabe flow

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Abstract

We show the noninheritance of the completeness of the noncompact Yamabe flow. Our main theorem states the existence of a long time solution which is complete for each time and converges to an incomplete Riemannian metric. This shows the occurrence of the infinite-time incompleteness.

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Acknowledgements

The authors would like to thank Professor Marek Fila for his valuable comments on Theorem 2.2 and Professor Kin Ming Hui for informing them of the paper [21]. The authors also would like to thank the referee for the careful reading and the helpful comments to improve the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematical and Computing Science, Tokyo Institute of Technology, Meguro, Tokyo, 152-8552, Japan

    Jin Takahashi

  2. Department of Mathematics, Faculty of Pure and Applied Science, University of Tsukuba, Tsukuba, Ibaraki, 305-8571, Japan

    Hikaru Yamamoto

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  1. Jin Takahashi
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  2. Hikaru Yamamoto
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Correspondence to Jin Takahashi.

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Communicated by A. Neves.

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The first author was supported in part by JSPS KAKENHI Grant-in-Aid for Early-Career Scientists 19K14567. The second author was supported in part by JSPS KAKENHI Grant-in-Aid for Early-Career Scientists 18K13415.

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Takahashi, J., Yamamoto, H. Infinite-time incompleteness of noncompact Yamabe flow. Calc. Var. 61, 212 (2022). https://doi.org/10.1007/s00526-022-02331-3

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