Inthisarticle,weprovideasufficientandnecessaryconditiononparabolic isometries of positive translation lengths on complete visibility CAT(0) spaces. One of the consequences is that each parabolic isometry of a complete simply connected visibility manifold of nonpositive sectional curvature has zero translation length. Applications on the geometry of open negatively curved manifolds will also be discussed.

We define a hierarchy of Hamiltonian PDEs associated to an arbitrary tau-function in the semi-simple orbit of the Givental group action on genus expansions of Frobenius manifolds. We prove that the equations, the Hamiltonians, and the bracket are weightedhomogeneous polynomials in the derivatives of the dependent variables with respect to the space variable. In the particular case of a conformal (homogeneous) Frobenius structure, our hierarchy coincides with the Dubrovin–Zhang hierarchy that is canonically associated to the underlying Frobenius structure. Therefore, our approach allows to prove the polynomiality of the equations, Hamiltonians, and one of the Poisson brackets of these hierarchies, as conjectured by Dubrovin and Zhang.

This note discusses the recent work of Huang-Schoen-Wang [13] on specifying total conserved quantities of a vacuum initial data set.

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We study the action of the group of contact diffeomorphisms on CR deformations of compact three dimensional CR manifolds. Using anisotropic function spaces and an anisotropic structure on the space of contact diffeomorphisms, we establish the existence of local transverse slices to the action of the contact diffeomorphism group in the neighborhood of a fixed embeddable strongly pseudoconvex CR structure.