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Let $L$ be a closed simply-connected smooth manifold with a Riemannian metric $g$. We say $g$ is index bounded if the energy functional (which is assumed to be Morse/Morse-Bott) $$ E: C^k(L,g) \rightarrow \mathbb{R} $$ is a bounded function for each $k$. Here $C^k(L,g)$ is the set of closed geodesics with Morse index $k$.

Question 1: When does $L$ admit an index bounded metric? (Examples I know are spheres.)

Question 2: If $L$ admits an index bounded metric, then under what kind of perturbations the index bounded property will be preserved?

My motivation: Morse theory of the energy functional computes the homology of the loop space of $L$. Suppose we know that on the homology level it is finite dimensional in each degree, then when is there a chain level analogue? Question 2 maybe related to the homotopy theory between different minimal model DGA's for the loop space, which I am not very familiar with. There are also parallel questions in the contact/symplectic setting, replacing closed geodesics by Reeb orbits.

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  • $\begingroup$ Do you have an example of a metric that is not index bounded? $\endgroup$
    – RBega2
    Jul 8, 2020 at 19:56
  • $\begingroup$ @RBega2 Not yet. I tried for some time to show any simply-connected manifold admits such a metric, but there is not much progress. $\endgroup$
    – Yuhan
    Jul 8, 2020 at 20:23

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