Let @math be a symplectic manifold, @math a coisotropic submanifold, and @math a compact oriented (real) surface. I define a natural Maslov index for each continuous map @math that sends every connected component of @math to some isotropic leaf of @math . This index is real valued and generalizes the usual Lagrangian Maslov index. The idea is to use the linear holonomy of the isotropic foliation of @math to compensate for the loss of boundary data in the case codimension @math . The definition is based on the Salamon-Zehnder (mean) Maslov index of a path of linear symplectic automorphisms. I prove a lower bound on the number of leafwise fixed points of a Hamiltonian diffeomorphism, if @math is geometrically bounded and @math is closed, regular (i.e. "fibering"), and monotone. As an application, we obtain a presymplectic non-embedding result. I also prove a coisotropic version of the Audin conjecture.
For @math the map @math was introduced by R. Gaio and D. A. Salamon in @cite_8 . (More precisely, their definition relies on a choice of an @math -compatible almost complex structure @math on @math and a unitary trivialization of @math .)
@cite_8
2018165282 1886215707 2016553765 2073401105
A graph is planar if it can be embedded in the plane, or in the sphere, so that no two edges cross at an interior point. A planar graph together with a particular embedding is called a map. There is a rich theory of counting maps, started by Tutte in the 1960's. However, in this paper we are interested in counting graphs as combinatorial objects, regardless of how many nonequivalent topological embeddings they may have. As we are going to see, this makes the counting considerably more difficult. Wave maps are the simplest wave equations taking their values in a Riemannian manifold (M,g). Their Lagrangian is the same as for the scalar equation, the only difference being that lengths are measured with respect to the metric g. By Noether's theorem, symmetries of the Lagrangian imply conservation laws for wave maps, such as conservation of energy. In coordinates, wave maps are given by a system of semilinear wave equations. Over the past 20 years important methods have emerged which address the problem of local and global wellposedness of this system. Due to weak dispersive effects, wave maps defined on Minkowski spaces of low dimensions, such as @math , present particular technical difficulties. This class of wave maps has the additional important feature of being energy critical, which refers to the fact that the energy scales exactly like the equation. Around 2000 Daniel Tataru and Terence Tao, building on earlier work of Klainerman–Machedon, proved that smooth data of small energy lead to global smooth solutions for wave maps from 2+1 dimensions into target manifolds satisfying some natural conditions. In contrast, for large data, singularities may occur in finite time for @math as target. This monograph establishes that for @math as target the wave map evolution of any smooth data exists globally as a smooth function. While we restrict ourselves to the hyperbolic plane as target the implementation of the concentration-compactness method, the most challenging piece of this exposition, yields more detailed information on the solution. This monograph will be of interest to experts in nonlinear dispersive equations, in particular to those working on geometric evolution equations. A unitarily invariant projective framework is introduced to analyze the complexity of path-following methods for the eigenvalue problem. A condition number, and its relation to the distance to ill-posedness, is given. A Newton map appropriate for this context is defined, and a version of Smale’s ( )-theorem is proven. The main result of this paper bounds the complexity of path-following methods in terms of the length of the path in the condition metric. In this paper, we define the three-dimensional topological map, a model which represents both the topological and geometrical information of a three-dimensional labeled image. Since this model describes the image's topology in a minimal way, we can use it to define efficient image processing algorithms. The topological map is the last level of map hierarchy. Each level represents the region boundaries of the image and is defined from the previous level in the hierarchy, thus giving a simple constructive definition. This model is an extension of the similar model defined for 2D images. Progressive definition based on successive map levels allows us to extend this model to higher dimension. Moreover, with progressive definition, we can study each level separately. This simplifies the study of disconnection cases and the proofs of topological map properties. Finally, we provide an incremental extraction algorithm which extracts any map of the hierarchy in a single image scan. Moreover, we show that this algorithm is very efficient by giving the results of our experiments made on artificial images.