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Degenerating sequences of conformal classes and the conformal Steklov spectrum

Part of: Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  05 March 2021

Vladimir Medvedev
Vladimir Medvedev*
Affiliation:
Département de Mathématiques et de Statistique, Pavillon André-Aisenstadt, Université de Montréal, Montréal, QC H3C 3J7, Canada and Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow 117198, Russian Federation and Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva Street, Moscow 119048, Russian Federation
*
e-mail: medvedevv@dms.umontreal.ca
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Abstract

Let $\Sigma $ be a compact surface with boundary. For a given conformal class c on $\Sigma $ the functional $\sigma _k^*(\Sigma ,c)$ is defined as the supremum of the kth normalized Steklov eigenvalue over all metrics in c. We consider the behavior of this functional on the moduli space of conformal classes on $\Sigma $ . A precise formula for the limit of $\sigma _k^*(\Sigma ,c_n)$ when the sequence $\{c_n\}$ degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander–Nadirashvili invariants of closed manifolds defined as $\inf _{c}\sigma _k^*(\Sigma ,c)$ , where the infimum is taken over all conformal classes c on $\Sigma $ . We show that these quantities are equal to $2\pi k$ for any surface with boundary. As an application of our techniques we obtain new estimates on the kth normalized Steklov eigenvalue of a nonorientable surface in terms of its genus and the number of boundary components.

Keywords

Spectral geometrySteklov spectrumupper boundsconformal Steklov spectrumcobordisms

MSC classification

Secondary: 35P15: Estimation of eigenvalues, upper and lower bounds
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 4 , August 2022 , pp. 1093 - 1136
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

This work is supported by the Ministry of Science and Higher Education of the Russian Federation: agreement no. 075-03-2020-223/3 (FSSF-2020-0018).

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