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We give here some extensions of Gromov’s and Polterovich’s theorems on k –area of CP~n, particularly in the symplectic and Hamiltonian context. Our main methods involve Gromov–Witten theory, and some connections with Bott peri...
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We give here some extensions of Gromov’s and Polterovich’s theorems on k –area of CP~n, particularly in the symplectic and Hamiltonian context. Our main methods involve Gromov–Witten theory, and some connections with Bott periodicity and the theory of loop groups. The argument is closely connected with the study of jumping curves in CP~n, and as an upshot we prove a new symplectic-geometric theorem on these jumping curves.
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We give a construction of the moduli space of stable maps to the classifying stack Bu_r of a cyclic group by a sequence of rth root constructions on M_(0,n). We prove a closed formula for the total Chern class of u_r-eigenspaces o...
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We give a construction of the moduli space of stable maps to the classifying stack Bu_r of a cyclic group by a sequence of rth root constructions on M_(0,n). We prove a closed formula for the total Chern class of u_r-eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus-zero Gromov{Witten theory of stacks of the form [C~N=u_r]. We deduce linear recursions for genus-zero Gromov-Witten invariants.
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We find necessary and sufficient conditions under which the complex coordinates on a flag manifold of a classical group described in Alekseevsky and Perelomov (Funct. Anal. Appl. 20(3):171-182, 1986) are Bochner coordinates.
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We find necessary and sufficient conditions under which the complex coordinates on a flag manifold of a classical group described in Alekseevsky and Perelomov (Funct. Anal. Appl. 20(3):171-182, 1986) are Bochner coordinates.
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If X is a geodesic metric space and x 1,x 2,x 3 ∈ X, a geodesic triangle T = {x 1,x ...
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If X is a geodesic metric space and x 1,x 2,x 3 ∈ X, a geodesic triangle T = {x 1,x 2,x 3} is the union of the three geodesics [x 1 x 2], [x 2 x 3] and [x 3 x 1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e., \({\delta}(X)=\inf\{{\delta}\ge 0: \, X \, \text{is $\delta$-hyperbolic}\}. \) In this paper we study the hyperbolicity of median graphs and we also obtain some results about general hyperbolic graphs. In particular, we prove that a median graph is hyperbolic if and only if its bigons are thin.
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We prove that the moduli space of gauge equivalence classes of symplectic vortices with uniformly bounded energy in a compact Hamiltonian manifold admits a Gromov compactification by polystable vortices. This extends results of Mu...
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We prove that the moduli space of gauge equivalence classes of symplectic vortices with uniformly bounded energy in a compact Hamiltonian manifold admits a Gromov compactification by polystable vortices. This extends results of Mundet i Riera for circle actions to the case of arbitrary compact Lie groups. Our argument relies on an a priori estimate for vortices that allows us to apply techniques used by McDuff and Salamon in their proof of Gromov compactness for pseudoholomorphic curves. As an intermediate result we prove a removable singularity theorem for symplectic vortices.
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The classic 2π-Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3–manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cro...
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The classic 2π-Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3–manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the "2π-metric" and the hyperbolic metric. We make partial progress in the program, proving long-time existence, preservation of negative sectional curvature, curvature bounds and integral convergence to hyperbolic for the metrics under consideration.
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We recall that a Riemannian manifold has bounded geometry if its sectional curvature is bounded above and below and its injectivity radius is bounded away from zero. Two metric spaces are quasi-isometric if they contain bi-Lipschi...
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We recall that a Riemannian manifold has bounded geometry if its sectional curvature is bounded above and below and its injectivity radius is bounded away from zero. Two metric spaces are quasi-isometric if they contain bi-Lipschitz homeomor-phic nets. A piecewise smooth function u : X —> R. on a bounded degree graph is harmonic if it minimizes energy on finite subgraphs, or, equivalently, if its derivative is constant along each edge and its value at each vertex coincides with its average.
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In this article we investigate the Gromov hyperbolicity of Denjoy domains equipped with the hyperbolic or the quasihyperbolic metric. We first prove the existence of suitable families of quasigeodesics. The main result shows that ...
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In this article we investigate the Gromov hyperbolicity of Denjoy domains equipped with the hyperbolic or the quasihyperbolic metric. We first prove the existence of suitable families of quasigeodesics. The main result shows that a Denjoy domain is Gromov hyperbolic with respect to the hyperbolic metric if and only it is Gromov hyperbolic with respect to the quasihyperbolic metric. Using these tools we give a characterization in terms of Euclidean distances of when the domains are Gromov hyperbolic. We also give several concrete examples of families of domains satisfying the criteria of the theorems.
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We show continuity under equivariant Gromov-Hausdorff convergence of the critical exponent of discrete, non-elementary, torsion-free, quasiconvex-cocompact groups with uniformly bounded codiameter acting on uniformly Gromov-hyperb...
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We show continuity under equivariant Gromov-Hausdorff convergence of the critical exponent of discrete, non-elementary, torsion-free, quasiconvex-cocompact groups with uniformly bounded codiameter acting on uniformly Gromov-hyperbolic metric spaces.
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