Displacement energy of unit disk cotangent bundles

We give an upper bound of a Hamiltonian displacement energy of a unit disk cotangent bundle $D^*M$ in a cotangent bundle $T^*M$, when the base manifold $M$ is an open Riemannian manifold. Our main result is that the displacement energy is not greater than $C r(M)$, where $r(M)$ is the inner radius of $M$, and $C$ is a dimensional constant. As an immediate application, we study symplectic embedding problems of unit disk cotangent bundles. Moreover, combined with results in symplectic geometry, our main result shows the existence of short periodic billiard trajectories and short geodesic loops.Comment: Title slightly changed. Close to the version published online in Math Zei

An exact sequence for contact- and symplectic homology

A symplectic manifold $W$ with contact type boundary $M = \partial W$ induces a linearization of the contact homology of $M$ with corresponding linearized contact homology $HC(M)$. We establish a Gysin-type exact sequence in which the symplectic homology $SH(W)$ of $W$ maps to $HC(M)$, which in turn maps to $HC(M)$, by a map of degree -2, which then maps to $SH(W)$. Furthermore, we give a description of the degree -2 map in terms of rational holomorphic curves with constrained asymptotic markers, in the symplectization of $M$.Comment: Final version. Changes for v2: Proof of main theorem supplemented with detailed discussion of continuation maps. Description of degree -2 map rewritten with emphasis on asymptotic markers. Sec. 5.2 rewritten with emphasis on 0-dim. moduli spaces. Transversality discussion reorganized for clarity (now Remark 9). Various other minor modification

Symplectic cohomology and q-intersection numbers

Given a symplectic cohomology class of degree 1, we define the notion of an equivariant Lagrangian submanifold. The Floer cohomology of equivariant Lagrangian submanifolds has a natural endomorphism, which induces a grading by generalized eigenspaces. Taking Euler characteristics with respect to the induced grading yields a deformation of the intersection number. Dehn twists act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz fibrations give fully computable examples. A key step in computations is to impose the "dilation" condition stipulating that the BV operator applied to the symplectic cohomology class gives the identity. Equivariant Lagrangians mirror equivariant objects of the derived category of coherent sheaves.Comment: 32 pages, 9 figures, expanded introduction, added details of example 7.5, added discussion of sign

The Minimal Length of a Lagrangian Cobordism between Legendrians

To investigate the rigidity and flexibility of Lagrangian cobordisms between Legendrian submanifolds, we investigate the minimal length of such a cobordism, which is a $1$-dimensional measurement of the non-cylindrical portion of the cobordism. Our primary tool is a set of real-valued capacities for a Legendrian submanifold, which are derived from a filtered version of Legendrian Contact Homology. Relationships between capacities of Legendrians at the ends of a Lagrangian cobordism yield lower bounds on the length of the cobordism. We apply the capacities to Lagrangian cobordisms realizing vertical dilations (which may be arbitrarily short) and contractions (whose lengths are bounded below). We also study the interaction between length and the linking of multiple cobordisms as well as the lengths of cobordisms derived from non-trivial loops of Legendrian isotopies.Comment: 33 pages, 9 figures. v2: Minor corrections in response to referee comments. More general statement in Proposition 3.3 and some reorganization at the end of Section

Localization for Involutions in Floer Cohomology

We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two applications to symplectic Khovanov cohomology are included.National Science Foundation (U.S.) (grant DMS-0405516)National Science Foundation (U.S.) (grant DMS-065260)European Research Council (grant ERC-2007-StG-205349

Revisiting the Local Scaling Hypothesis in Stably Stratified Atmospheric Boundary Layer Turbulence: an Integration of Field and Laboratory Measurements with Large-eddy Simulations

The `local scaling' hypothesis, first introduced by Nieuwstadt two decades ago, describes the turbulence structure of stable boundary layers in a very succinct way and is an integral part of numerous local closure-based numerical weather prediction models. However, the validity of this hypothesis under very stable conditions is a subject of on-going debate. In this work, we attempt to address this controversial issue by performing extensive analyses of turbulence data from several field campaigns, wind-tunnel experiments and large-eddy simulations. Wide range of stabilities, diverse field conditions and a comprehensive set of turbulence statistics make this study distinct