existence of Morse functions satisfying the Palais-Smale condition - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T01:34:32Z http://mathoverflow.net/feeds/question/33401 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33401/existence-of-morse-functions-satisfying-the-palais-smale-condition existence of Morse functions satisfying the Palais-Smale condition Orbicular 2010-07-26T14:22:18Z 2010-11-05T23:49:27Z <p>Given an arbitrary Hilbert manifold, can on find a complete Riemannian metric and a Morse function satisfying the Palais-Smale condition?</p> http://mathoverflow.net/questions/33401/existence-of-morse-functions-satisfying-the-palais-smale-condition/33415#33415 Answer by Dick Palais for existence of Morse functions satisfying the Palais-Smale condition Dick Palais 2010-07-26T16:32:06Z 2010-07-26T16:32:06Z <p>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. </p> <p>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.</p> http://mathoverflow.net/questions/33401/existence-of-morse-functions-satisfying-the-palais-smale-condition/45007#45007 Answer by Orbicular for existence of Morse functions satisfying the Palais-Smale condition Orbicular 2010-11-05T23:49:27Z 2010-11-05T23:49:27Z <p>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.</p>