  Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we
develop here a connection between Morse theory for the (positive) Hofer length
functional $L: \Omega \text {Ham}(M, \omega) \to \mathbb{R}$, with
Gromov-Witten/Floer theory, for monotone symplectic manifolds $ (M, \omega) $.
This gives some immediate restrictions on the topology of the group of
Hamiltonian symplectomorphisms (possibly relative to the Hofer length
functional), and a criterion for non-existence of certain higher index
geodesics for the Hofer length functional. The argument is based on a certain
automatic transversality phenomenon which uses Hofer geometry to conclude
transversality and may be useful in other contexts. Strangely the monotone
assumption seems essential for this argument, as abstract perturbations
necessary for the virtual moduli cycle, decouple us from underlying Hofer
geometry, causing automatic transversality to break.
