Related papers: Symplectic Applicability of Lagrangian Surfaces
We introduce a symplectic surgery in six dimensions which collapses Lagrangian three-spheres and replaces them by symplectic two-spheres. Under mirror symmetry it corresponds to an operation on complex 3-folds studied by Clemens, Friedman…
We prove that a very general cubic fourfold containing a plane can be embedded into a holomorphic symplectic eightfold as a Lagrangian submanifold. We construct the desired holomorphic symplectic eightfold as a moduli space of Bridgeland…
We survey the progress on the study of symplectic geometry past five decades. The survey focuses on the convexity properties of a moment map, the classification of symplectic actions, the symplectic embedding problems, and the theory of…
Long-term stability studies of nonlinear Hamiltonian systems require symplectic integration algorithms which are both fast and accurate. In this paper, we study a symplectic integration method wherein the symplectic map representing the…
We study the physical content of quadratic diff-invariant Lagrangians in arbitrary dimensions by using covariant symplectic techniques. This paper extends previous results in dimension four. We discuss the difference between the even and…
On general symplectic manifolds we describe a correspondence between symplectic transformations and their phase functions. On the quantum level, this is a correspondence between unitary operators and phase functions of the…
Surface charges and their algebra in interacting Lagrangian gauge field theories are investigated by using techniques from the variational calculus. In the case of exact solutions and symmetries, the surface charges are interpreted as a…
The graph of a real symplectic linear transformation is an R-Lagrangian subspace of a complex symplectic vector space. The restriction of the complex symplectic form is thus purely imaginary and may be expressed in terms of the generating…
We introduce the Euler-Lagrange cohomology to study the symplectic and multisymplectic structures and their preserving properties in finite and infinite dimensional Lagrangian systems respectively. We also explore their certain difference…
We present a method for explicit leapfrog integration of inseparable Hamiltonian systems by means of an extended phase space. A suitably defined new Hamiltonian on the extended phase space leads to equations of motion that can be…
We propose a symplectic structure for the phase space of a generic Lagrangian field theory expressed in the framework of $L_\infty$ algebras. The symplectic structure does not require explicit knowledge of the derivative content of the…
A material-based, i.e., Lagrangian, methodology for exact integration of flux by volume-preserving flows through a surface has been developed recently in [Karrasch, SIAM J. Appl. Math., 76 (2016), pp. 1178-1190]. In the present paper, we…
In this article, we study the singularities of Lagrangian immersions into Cartesian product of surfaces. After applying a Hamiltonian isotopy in the Weinstein tubular neighbourhood of the Lagrangian immersion, the singular points of the…
We review computations of joint invariants on a linear symplectic space, discuss variations for an extension of group and space and relate this to other equivalence problems and approaches, most importantly to differential invariants.
This article describes the use of symplectic cut-and-paste methods to compute Gromov-Witten invariants. Our focus is on recent advances extending these methods to Kahler surfaces with geometric genus p_g>0, for which the usual GW invariants…
We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.
The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with a useful Riemannian metric which is…
The Hamiltonian formalism is extremely elegant and convenient to mechanics problems. However, its application to the classical field theories is a difficult task. In fact, you can set one to one correspondence between the Lagrangian and…
We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures. In particular, we define local angle functions encoding…
The geometry of a Lagrangian mechanical system is determined by its associated evolution semispray. We uniquely determine this semispray using the symplectic structure and the energy of the Lagrange space and the external force field. We…