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This article investigates the complex symplectic geometry of the deformation space of complex projective structures on a closed oriented surface of genus at least 2. The cotangent symplectic structure given by the Schwarzian parametrization…

Let $\OO$ be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $(X,\omega).$ We define a functional $\CC:\OO \to \R$ for each differential form $\beta$ of middle…

This paper constructs a Hodge theory of noncompact topologically tame manifolds $M$. The main result is an isomorphism between the de Rham cohomology with compact supports of $M$ and the kernel of the Hodge--Witten--Bismut Laplacian…

Let $f$ be a harmonic map from a Riemann surface to a Riemannian $n$-manifold. We prove that if there is a holomorphic diffeomorphism $h$ between open subsets of the surface such that $f\circ h = f$, then $f$ factors through a holomorphic…

We explicitly construct a symplectomorphism that relates magnetic twists to the invariant hyperk\"ahler structure of the tangent bundle of a Hermitian symmetric space. This symplectomorphism reveals foliations by (pseudo-) holomorphic…

The group Hameo (M,\omega) of Hamiltonian homeomorphisms of a connected symplectic manifold (M,\omega) was defined and studied in [7] and further in [6]. In these papers, the authors consistently used the L^{(1,\infty)}-Hofer norm (and not…

Mishchenko-Oliveira proved the piecewise smooth cohomology and Lie algebroid cohomology of a Lie algebroid on a combinatorial compact manifold are isomorphic. In this paper, we describe an application of that result locally trivial Lie…

We show that a symplectic isotopy that is a $C^0$ limit of Hamiltonian isotopies is itself Hamiltonian, if the corresponding sequence of generating Hamiltonians converge in $L^{(1, \infty)}$ topology.

Denote by $\DC(M)_0$ the identity component of the group of compactly supported $C^\infty$ diffeomorphisms of a connected $C^\infty$ manifold $M$, and by $\HR$ the group of the homeomorphisms of $\R$. We show that if $M$ is a closed…

The discrete Laplacian on Euclidean triangulated surfaces is a well-established notion. We introduce discrete Laplacians on spherical and hyperbolic triangulated surfaces. On the one hand, our definitions are close to the Euclidean one in…

We investigate nicely embedded H--holomorphic maps into stable Hamiltonian three--manifolds. In particular we prove that such maps locally foliate and satisfy a no--first--intersection property. Using the compactness results of…

We introduce a class of volume-contracting surface diffeomorphisms whose dynamics is intermediate between one-dimensional dynamics and general surface dynamics. For that type of systems one can associate to the dynamics a reduced…

Finite-sheeted covering mappings onto compact connected groups are studied. It is shown that a finite-sheeted covering mapping from a connected Hausdorff topological space onto a compact connected abelian group G must be a homeomorphism…

It is well known that a foliation F of a smooth manifold M gives rise to a rich cohomological theory, its characteristic (i.e., leafwise) cohomology. Characteristic cohomologies of F may be interpreted, to some extent, as functions on the…

We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications,…

We use functions of a bicomplex variable to unify the existing constructions of harmonic morphisms from a 3-dimensional Euclidean or pseudo-Euclidean space to a Riemannian or Lorentzian surface. This is done by using the notion of…

We set out some general criteria to prove the K-property, refining the assumptions used in arXiv:1906.09315 for the flow case, and introducing the analogous discrete-time result. We also introduce one-sided $\lambda$-decompositions, as well…

This work is motivated by the problem of finding locally compact group topologies for piecewise full groups (a.k.a.~ topological full groups). We determine that any piecewise full group that is locally compact in the compact-open topology…

For any connected component $H_0$ of the space of real meromorphic functions we build a compactification $N(H_0)$ of the space $H_0$. Then we express the Euler characteristics of the spaces $H_0$ and $N(H_0)$ in terms of topological…

We study some analytic properties of distance decreasing self-maps onto the complement of a smooth curve $\Sigma$ in $S^n$. For $n>4$ and $n\equiv 0 \mod 4$, let $\Sigma$ be an embedded circle in $S^n$ and let $g$ be a complete Riemannian…