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The Vanishing of the Fundamental Gap of Convex Domains in \(\mathbb {H}^n\)

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Abstract

For the Laplace operator with Dirichlet boundary conditions on convex domains in \(\mathbb H^n\), \(n\ge 2\), we prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small for domains of any diameter.

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Notes

  1. It is the case that \(\lambda _2 = \lambda _2(\Omega _{\sqrt{\mu }, L})\) for \(\mu \) large. Lemma 5.1 and Proposition 5.2 show that for \(\eta \in (0, L)\) and \(\mu > \mu _2\), we have \( \lambda _1^{4\mu } \ge (\cos L)^2 4 \mu > (\cos \eta )^2 \mu \ge \lambda _2^{\mu }, \) where \(\lambda _1^{4\mu }\) is the first eigenvalue of \( h_{\varphi \varphi }+\lambda (\sec \varphi )^{2} h=4\mu h\).

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Acknowledgements

This research originated at the workshop “Women in Geometry 2” at the Casa Matemática Oaxaca (CMO) from June 23 to June 28, 2019. We would like to thank CMO-BIRS for creating the opportunity to start work on this problem through their support of the workshop. Julie Clutterbuck thanks Ben Andrews for a useful discussion about the existence of a double-peaked eigenfunction.

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Authors and Affiliations

  1. Department of Mathematics, University of Tennessee, 1403 Circle Dr, Knoxville, TN,  37916, USA

    Theodora Bourni

  2. School of Mathematics, Monash University, 9 Rainforest Walk, Melbourne, VIC,  3800, Australia

    Julie Clutterbuck

  3. Department of Mathematics, Iowa State University, 411 Morrill Rd, Ames, IA,  50011, USA

    Xuan Hien Nguyen

  4. Department of Mathematics and Statistics, Concordia University, 1455 Blvd. de Maisonneuve Ouest, Montreal, QC,  H3G 1M8, Canada

    Alina Stancu

  5. Department of Mathematics, UC Santa Barbara, Santa Barbara, CA,  93106, USA

    Guofang Wei

  6. Valentina-Mira Wheeler, Institute for Mathematics and its Applications, University of Wollongong, Northfields Avenue, Wollongong, NSW,  2522, Australia

    Valentina-Mira Wheeler

Authors
  1. Theodora Bourni
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  2. Julie Clutterbuck
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  3. Xuan Hien Nguyen
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  4. Alina Stancu
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  5. Guofang Wei
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  6. Valentina-Mira Wheeler
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Corresponding author

Correspondence to Julie Clutterbuck.

Additional information

Communicated by Claude-Alain Pillet.

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The research of Julie Clutterbuck was supported by Grant FT1301013 of the Australian Research Council. The research of Xuan Hien Nguyen was supported by Grant 579756 of the Simons Foundation. The research of Alina Stancu was supported by NSERC Discovery Grant RGPIN 327635. The research of Guofang Wei was supported by NSF Grant DMS 1811558. The research of Valentina-Mira Wheeler was supported by Grants DP180100431 and DE190100379 of the Australian Research Council.

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Bourni, T., Clutterbuck, J., Nguyen, X.H. et al. The Vanishing of the Fundamental Gap of Convex Domains in \(\mathbb {H}^n\). Ann. Henri Poincaré 23, 595–614 (2022). https://doi.org/10.1007/s00023-021-01096-3

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