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Isoparametric foliations and critical sets of eigenfunctions

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Abstract

Jakobson and Nadirashvili (J Differ Geom 53(1):177–182, 1999) constructed a sequence of eigenfunctions on \(T^2\) with a bounded number of critical points, answering in the negative the question raised by Yau (Problem section, seminar on differential geometry. Princeton University Press, Princeton, 1982) which asks that whether the number of the critical points of eigenfunctions for the Laplacian increases with the corresponding eigenvalues. The present paper finds three interesting eigenfunctions on the minimal isoparametric hypersurface \(M^n\) in \(S^{n+1}(1)\). The corresponding eigenvalues are n, 2n and 3n, while their critical sets consist of 8 points, a submanifold (infinite many points) and 8 points, respectively. On one of its focal submanifolds, a similar phenomenon occurs.

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Notes

  1. Added in proof. It was recently proved by Enciso and Peralta-Salas that on a compact manifold, there is a Riemannian metric such that the first nontrivial eigenfunction can have as many non-degenerate critical points as one wishes (bigger in particular than the Morse number of the manifold). Moreover, any other metric \(C^{\infty }\) close to it carries the same property(cf. [6]).

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Authors and Affiliations

  1. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875, China

    Zizhou Tang & Wenjiao Yan

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  1. Zizhou Tang
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  2. Wenjiao Yan
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Correspondence to Wenjiao Yan.

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The project is partially supported by the NSFC (Nos. 11331002, 11301027) and SRFDP (No. 20130003120008).

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Tang, Z., Yan, W. Isoparametric foliations and critical sets of eigenfunctions. Math. Z. 286, 1217–1226 (2017). https://doi.org/10.1007/s00209-016-1798-3

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