Upper bounds for the critical values of homology classes of loops (Q6564486)

Let \(M\) be a compact differentiable manifold endowed with either a Riemannian or non-reversible Finsler metric. Let \(X\) denote either the free loop space \(\Lambda M\), the pathspace \(\Omega_{pq}M\), or the based loop space \(\Omega_pM\). Define the square root energy functional \[F : X \rightarrow \mathbb{R}\] by \N\[\NF(\gamma) = \sqrt{ \int_0^1 ||\gamma^\prime(t)||^2\thinspace dt}\N\]\Nwhere \(||\gamma^\prime(t)||\) is the norm of the velocity vector of \(\gamma \in X\) at \(t\). For \(a \in \mathbb{R}\), set \N\[\NX^{\leq a} = \{ \gamma \in X: F(\gamma) \leq a\}.\N\]\NGiven a nontrivial homology class \(h \in H_j(X,X^{\leq b})\), the critical value \(\mathrm{cr}_X(h)\) is defined to be the infimum of the \(a\) such that \(h\) lies in the image of \(H_j(X^{\leq a}, X^{\leq b})\) in \(H_j(X, X^{\leq b})\).\N\NIn this paper the author obtains an upper bound for \(\mathrm{cr}_X(h)\). When there is a positive lower bound on the sectional curvature or on the Ricci curvature of \(M\), the upper bound on \(\mathrm{cr}_X(h)\) is expressed in terms of the lower bounds on the sectional or Ricci curvature of \(M\). This leads to an upper bound on the length of the shortest periodic geodesic in \(M\) in terms of a positive lower bound on the Ricci curvature. This result improves previous bounds on the length of the shortest periodic geodesic.