Sharp asymptotics of the first eigenvalue on some degenerating surfaces
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- by Henrik Matthiesen and Anna Siffert PDF
- Trans. Amer. Math. Soc. 373 (2020), 5903-5936 Request permission
Abstract:
We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be used to strictly increase the first eigenvalue normalized by area if the initial surface has some symmetries. If these symmetries are not present we show that the first eigenvalue normalized by area strictly decreases for the same range of parameters. These results are the main motivation for the construction in [preprint, arXiv:1909.03105v2], where we show a monotonicity result for the normalized first eigenvalue without any symmetry assumptions.References
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Additional Information
- Henrik Matthiesen
- Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
- MR Author ID: 1157404
- Email: hmatthiesen@math.uchicago.edu
- Anna Siffert
- Affiliation: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
- MR Author ID: 1060420
- Email: siffert@mpim-bonn.mpg.de
- Received by editor(s): September 8, 2019
- Received by editor(s) in revised form: January 12, 2020, and January 21, 2020
- Published electronically: May 28, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 5903-5936
- MSC (2010): Primary 35P15, 49Q05, 49Q10, 58E11
- DOI: https://doi.org/10.1090/tran/8114
- MathSciNet review: 4127896