The round metric on the Poincaré dodecahedral space violates spectral selection for minimal Morse functions because its first eigenspace contains non-minimal ones, but nearby metrics restore the property by splitting the eigenvalue to a simple minimal Morse eigenfunction with six critical points.
Kato,Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer, Berlin
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Spectral Selection and Minimal Morse Structures on the Poincar\'e Dodecahedral Space
The round metric on the Poincaré dodecahedral space violates spectral selection for minimal Morse functions because its first eigenspace contains non-minimal ones, but nearby metrics restore the property by splitting the eigenvalue to a simple minimal Morse eigenfunction with six critical points.