It is shown that certain natural maps between the ideal, Gromov, and end boundaries of a complete CAT(0) space can fail to be either injective or surjective. Additionally the natural map from the Gromov boundary to the end boundary of a complete CAT(−1) space can fail to be either injective or surjective.

This paper wishes to foster communication between mathematicians and physicists working in mirror symmetry and orbifold Gromov-Witten theory. We provide a reader friendly review of the physics computation in [ABK06] that predicts Gromov-Witten invariants of [C/Z3] in arbitrary genus, and of the mathematical framework for expressing these invariants as Hodge integrals. Using geometric properties...

We make an estimation of the value of the Gromov norm of the Cartesian product of two surfaces. Our method uses a connection between these norms and the minimal size of triangulations of the products of two polygons. This allows us to prove that the Gromov norm of this product is between 32 and 52 when both factors have genus 2. The case of arbitrary genera is easy to deduce form this one.

Quantum Lefschetz theorem by Coates and Givental [4] gives a relationship between the genus 0 Gromov-Witten theory of X and the twisted theory by a line bundle L on X. We prove the convergence of the twisted theory under the assumption that the genus 0 theory for original X converges. As a byproduct, we prove the semisimplicity and the Virasoro conjecture for the Gromov-Witten theories of (not ...

We study sequences of conformal deformations a smooth closed Riemannian manifold dimension $n$, assuming uniform volume bounds and $L^{n/2}$ on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such underlying metric spaces are pre-compact Gromov-Hausdorff topology. Our is based use $A_\infty$-weights from harmonic analysis, provides geometric cont...

Abstract We establish the equivalence of Gromov ellipticity and subellipticity in algebraic category.

In this paper, we extend the concept of scaled Gromov hyperbolic graph, originally developed for the Thin Triangle Condition (TTC), to the computationally simplified, but less intuitive, Four-Point Condition (FPC). The original motivation was that, for a large but finite network graph to enjoy some of the typical properties to be expected in negatively curved Riemannian manifolds, the delta mea...

In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes what has until now been an ad hoc collection of results by many authors. In addition to providing much greater generality than any prior work, our results also give new insight into the nature of the connection between Dio...