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Persistence barcodes and Laplace eigenfunctions on surfaces

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Abstract

We obtain restrictions on the persistence barcodes of Laplace–Beltrami eigenfunctions and their linear combinations on compact surfaces with Riemannian metrics. Some applications to uniform approximation by linear combinations of Laplace eigenfunctions are also discussed.

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Notes

  1. These are sometimes refered to as pointwise finite dimensional persistence modules.

  2. Our definition is slightly different from the one in [36] since we do not assume that \(\int _{M} f~\sigma =0\) if M has no boundary. However, this assumption is not needed for any of the results of [36] which we use.

  3. Formally speaking, \(F_k\) should be a small perturbation of \(f_k+1\) in order to make it Morse, but we will ignore this detail for the sake of clarity.

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Acknowledgements

The authors are grateful to Lev Buhovsky for providing Example 1.4.12 that has lead to a reformulation of Conjectures 1.4.7 and 1.4.8 . The authors would like to thank Yossi Azar, Allan Pinkus and Justin Solomon for useful discussions, as well as Lev Buhovsky and Mikhail Sodin for helpful remarks on the early version of the paper. We also thank Yuliy Baryshnikov for bringing the reference [11] to our attention. Part of this research was accomplished while I.P. was supported by the Weston Visiting Professorship program at the Weizmann Institute of Science.

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Authors and Affiliations

  1. Département de Mathématiques et de Statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, QC, H3C 3J7, Canada

    Iosif Polterovich

  2. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel

    Leonid Polterovich & Vukašin Stojisavljević

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  1. Iosif Polterovich
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  2. Leonid Polterovich
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  3. Vukašin Stojisavljević
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Correspondence to Leonid Polterovich.

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I. Polterovich: Partially supported by NSERC, FRQNT and Canada Research Chairs program.

L. Polterovich, V. Stojisavljević: Partially supported by the European Research Council Advanced Grant 338809.

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Polterovich, I., Polterovich, L. & Stojisavljević, V. Persistence barcodes and Laplace eigenfunctions on surfaces. Geom Dedicata 201, 111–138 (2019). https://doi.org/10.1007/s10711-018-0383-9

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