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

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

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    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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Correspondence to Julie Clutterbuck.

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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.

Communicated by Claude-Alain Pillet.

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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 (2021). https://doi.org/10.1007/s00023-021-01096-3

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