Length minimizing property, Conley-Zehnder index and $C^1$-perturbations of Hamiltonian functions

The main purpose of this paper is to study the length minimizing property of Hamiltonian paths on closed symplectic manifolds $(M,\omega)$ such that there are no spherical homology class $A \in H_2(M)$ with $$\omega(A) > 0 \quad \text{and} \quad -n \leq c_1(A) < 0,$$ which we call {\it very strongly semi-positive}. We introduce the notion of {\it positively $\mu$-undertwisted} Hamiltonian paths and prove that any positively undertwisted quasi-autonomous Hamiltonian path is length minimizing in its homotopy class as long as it has a fixed maximum and a fixed minimum point that are generically under-twisted. This class of Hamiltonian can have non-constant large periodic orbits. The proof uses the chain level Floer theory, spectral invariants of Hamiltonian diffeomorphisms and the argument involving the thick and thin decomposition of Floer's moduli space of perturbed Cauchy-Riemann equation. And then based on this theorem and some closedness of length minimizing property, we relate the Minimality Conjecture on the very strongly semi-positive symplectic manifolds to a $C^1$-perturbation problem of Hamiltonian functions on general symplectic manifolds, which we also formulate here.