Abstract
We show that for compact Riemannian manifolds of dimension at least 3 with nonempty boundary, we can modify the manifold by performing surgeries of codimension 2 or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimensions 3 and higher. Our result generalizes the 1-dimensional surgery in Fraser and Schoen (Adv Math 348:146–162, 2019) to higher dimensional surgeries and to higher eigenvalues. It is proved in Fraser and Schoen (Adv Math 348:146–162, 2019) that the unit ball does not maximize the first nonzero normalized Steklov eigenvalue among contractible domains in \(\mathbb {R}^n\), for \(n \ge 3\). We show that this is also true for higher Steklov eigenvalues. Using similar ideas, we show that in \(\mathbb {R}^n\), for \(n\ge 3\), the j-th normalized Steklov eigenvalue is not maximized in the limit by a sequence of contractible domains degenerating to the disjoint union of j unit balls, in contrast to the case in dimension 2 (Girouard and Polterovich in Funct Anal Appl 44:106–117, 2010).
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Acknowledgements
The author would like to thank his advisors Jingyi Chen and Ailana Fraser for helpful discussions.
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This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.
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Hong, H. Higher Dimensional Surgery and Steklov Eigenvalues. J Geom Anal 31, 11931–11951 (2021). https://doi.org/10.1007/s12220-021-00706-0
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DOI: https://doi.org/10.1007/s12220-021-00706-0