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Spectral Geometry of the Steklov Problem on Orbifolds

Research output: Contribution to journalArticle

  • Asma Hassannezhad
  • Teresa Arias-Marco
  • Emily Dryden
  • Carolyn Gordon
  • Allie Ray
  • Elizabeth Stanhope
Original languageEnglish
Article numberrnx117
Pages (from-to)90-139
Number of pages50
JournalInternational Mathematics Research Notices
Volume2019
Issue number1
Early online date19 Jun 2017
DOIs
DateAccepted/In press - 10 May 2017
DateE-pub ahead of print - 19 Jun 2017
DatePublished (current) - Jan 2019

Abstract

We consider how the geometry and topology of a compact $n$-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich, and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in terms of only boundary data. As a consequence, we prove that the Steklov spectrum detects the presence and number of orbifold singularities on the boundary of an orbisurface and it detects the number each of smooth and singular boundary components. Moreover, we find that the Steklov spectrum also determines the lengths of the boundary components modulo an equivalence relation, and we show by examples that this result is the best possible. We construct various examples of Steklov isospectral Riemannian orbifolds which demonstrate that these two-dimensional results do not extend to higher dimensions.

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    Rights statement: This is the accepted author manuscript (AAM). The final published version (version of record) is available online via OUP at https://doi.org/10.1093/imrn/rnx117 . Please refer to any applicable terms of use of the publisher.

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