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We prove a theorem which asserts that the Lie algebra of all holomorphic vector ﬁelds on a compact Kähler manifold with a perturbed extremal metric has the structure similar to the case of an unperturbed extremal Kähler metric proved by Calabi.
The purpose of this article is to view Penrose rhombus tilings from the perspective of symplectic geometry. We show that each thick rhombus in such a tiling can be naturally associated to a highly singular 4-dimensional compact symplectic space $M_R$, while each thin rhombus can be associated to another such space $M_r$; both spaces are invariant under the Hamiltonian action of a 2-dimensional quasitorus, and the images of the corresponding moment mappings give the rhombuses back. The spaces $M_R$ and $M_r$ are diﬀeomorphic but not symplectomorphic.
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.
The Lagrangian and Hamiltonian structures for an ideal gauge-charged ﬂuid are determined. Using a Kaluza–Klein point of view, the equations of motion are obtained by Lagrangian and Poisson reductions associated to the automorphism group of a principal bundle. As a consequence of the Lagrangian approach, a Kelvin–Noether theorem is obtained. The Hamiltonian formulation determines a non-canonical Poisson bracket associated to these equations.