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Consider the Laplace-Beltrami operator on a compact manifold. Generically, Uhlenbeck has shown that eigenfunctions of the Laplace-Beltrami operator are Morse functions.

But there are some manifolds, such as the two-dimensional torus, where some eigenfunctions of the Laplace-Beltrami operator have non-isolated maxima.

For the only examples I know of, the eigenfunctions are separable, i.e. they can be written as a product of lower-dimensional functions. I heard somewhere that separable eigenfunctions are the only eigenfunctions whose critical points are not isolated, but I can't remember where to find it, or even where to look.

Question. Is this true, and does anyone know of a reference or how to prove it?

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    $\begingroup$ What is a 'lower-dimensional function'? $\endgroup$
    – Leo Moos
    Feb 19 at 10:51
  • $\begingroup$ I don't know. I suspect it has something to do with $\Delta$ being separable as well as the eigenfunction. I'm struggling to come up with a good definition which allows for the "standard" separation of variables for e.g. the "banded" spherical harmonics, but doesn't allow for eigenfunctions I wouldn't normally consider separable, such as $\frac{5}{13}\cos(x)+\frac{12}{13}\sin(y)$ on a torus. $\endgroup$
    – user7868
    Feb 22 at 2:56

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