Related papers: Hofer distance on Lagrangian links inside the disc

A Hopf algebra is co-Frobenius when it has a nonzero integral. It is proved that the composition length of the indecomposable injective comodules over a co-Frobenius Hopf algebra is bounded. As a consequence, the coradical filtration of a…

The maximum principle for hyperbolic inversive distance circle packings on polyhedral surfaces is established,which unifies and generalizes existing maximum principles for various types of circle packings in the literature.As an application…

Let $\Sigma$ be a compact surface equipped with an area form. There is an long standing open question by Katok, which, in particular, asks whether every entropy-zero Hamiltonian diffeomorphism of a surface lies in the $C^0$-closure of the…

Sakovich--Sormani introduced several notions of distance between certain classes of Lorentzian manifolds. These distances use the Hausdorff and Gromov-Hausdorff distances and therefore extend naturally to a broader class of spaces. Here we…

We find lower bounds on the number of intersection points between two relatively exact Hamiltonian isotopic Lagrangians. The bounds are given in terms of the cuplength of the Lagrangian in various multiplicative generalised cohomology…

For a class of Riemannian manifolds that include products of arbitrary compact manifolds with manifolds of nonpositive sectional curvature on the one hand, or with certain positive-curvature examples such as spheres of dimension at least 3…

We prove the following three results in Hamiltonian dynamics. 1. The Weinstein conjecture holds true for every displaceable hypersurface of contact type. 2. Every magnetic flow on a closed Riemannian manifold has contractible closed orbits…

Let D be a non-displaceable disk in an annulus A. Suppose that g is a compactly supported Hamiltonian which preserves D with translation number n. We show that Hofer's norm |g| is bounded from below by cn for a certain constant c. We also…

We argue that an infinite circumference limit can be obtained in 2-dimensional conformal field theory by adopting $L_0-(L_1+L_{-1})/2$ as a Hamiltonian instead of $L_0$. The theory obtained has a circumference of infinite length and hence…

We prove the Conley conjecture for cotangent bundles of oriented, closed manifolds, and Hamiltonians which are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits. For the conservative systems,…

For a given embedded Lagrangian in the complement of a complex hypersurface we show existence of a holomorphic disc in the complement having boundary on that Lagrangian.

In this article we prove that for a smooth fiberwise convex Hamiltonian, the asymptotic Hofer distance from the identity gives a strict upper bound to the value at 0 of Mather's $\beta$ function, thus providing a negative answer to a…

Let U be the open unit disc in C and let B be the open unit ball in C^2. We prove that every discrete subset of B is contained in the range f(U) of a complete, proper holomorphic embedding f:U-->B. Here the completeness of f means that for…

Chekanov showed that the Hofer norm on the Hamiltonian diffeomorphism group of a geometrically bounded symplectic manifold induces a nondegenerate metric on the orbit of any compact Lagrangian submanifold under the group. In this paper we…

We prove that the Gromov-Hausdorff distance from the circle with its geodesic metric to any simply connected geodesic space is never smaller than $\frac{\pi}{4}$. We also prove that this bound is tight through the construction of a simply…

Let $\Sigma$ be a compact surface of genus $g \geq 1$ equipped with an area form. We construct eggbeater Hamiltonian diffeomorphisms which lie arbitrarily far in the Hofer metric from the set of autonomous Hamiltonians. This result is…

We show that the space of Lagrangians which are Hamiltonian isotopic to the Clifford torus in a complex projective space or in the four-dimensional quadric, taken with Chekanov's Lagrangian Hofer metric, contains a quasi-isometric copy of…

We give three necessary and sufficient conditions so that a parabolic holomorphic semigroup $(\phi_t)$ in the unit disc is of finite shift. One is in terms of the asymptotic behavior of speeds of convergence, the second one is related to…

We prove that there is a universal measure on the unit circle such that any probability measure on the unit disk is the limit distribution of some subsequence of the corresponding orthogonal polynomials. This follows from an extension of a…

We introduce a persistence-like pseudo-distance on Tamarkin's category and prove that the distance between an object and its Hamiltonian deformation is at most the Hofer norm of the Hamiltonian function. Using the distance, we show a…