2010/07/19 by Yasha Savelyev, Savelyev, Yasha
Mathematics · #53D35 #53D40 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1007.3213
openalex publication_date 2010/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce here a natural functional associated to any b ∈ QH_* (M, ω): spectral length functional, on the space of "generalized paths" in \text Ham(M, ω), closely related to both the Hofer length functional and spectral invariants and establish some of its properties. This functional is smooth on its domain of definition, and moreover the nature of extremals of this functional suggests that it may be variationally complete, in the sense that any suitably generic element of \widetilde\text Ham(M, ω) is connected to id by a generalized path minimizing spectral length. Rather strong evidence is given for this when M=S 2, where we show that all the Lalonde-McDuff Hamiltonian symplectomorphisms are joined to id by such a path. We also prove that the associated norm on \text Ham(M, ω) is non-degenerate and bounded from below by the the spectral norm. If the spectral length functional is variationally complete the associated norm reduces to the spectral norm.