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Given an arbitrary Hilbert manifold, can on find a complete Riemannian metric and a Morse function satisfying the Palais-Smale condition?

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up vote 6 down vote accepted

I recently learned that the answer to the question is YES, answered in the ETH preprint "H-cobordism for Hilbert Manifolds" by Dan Burghelea. I found the reference in the article "On the differential topology of Hilbert manifolds" of Eells and Elworthy.

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Well, I have been away from this kind of question for a long while, so please don't the following remarks as definitive in any way, but I am not aware of any counter-example and in the infinite dimensional case I also do not know any positive result. So, I think it is a nice question you are asking and if you can prove something in this direction it would be interesting and probably publishable.

A relatively minor point---I assume that you mean a differentiable Hilbert manifold, and you probably want to assume the manifold is separable. With those assumption I believe it is not hard to show that the manifold can be smoothly embedded as a closed submanifold of Hilbert space, so it gets a complete Riemannian metric, and it would be really nice if you could show that for some such embedding there was a ``height function'' (i.e., a continuous linear functional on the Hilbert space) that when restricted to the manifold satisfied Condition C. If I had to bet, I would guess this is so.

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And one can also play with the metric to make it satisfy the Palais-Smale condition, I guess. –  Pietro Majer Jul 26 '10 at 17:02
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