On any closed symplectic manifold, we construct a path-connected neighborhood of the identity in the Hamiltonian diﬀeomorphism group with the property that each Hamiltonian diﬀeomorphism in this neighborhood admits a Hofer and spectral length minimizing path to the identity. This neighborhood is open in the $C^1$-topology. The construction utilizes a continuation argument and chain level result in the Floer theory of Lagrangian intersections.
"Length minimizing paths in the Hamiltonian diffeomorphism group." J. Symplectic Geom. 6 (2) 159 - 187, June 2008.