arXiv subjects

Ronen Brilleslijper

Publications and source records attributed to Ronen Brilleslijper.

5 recordsLinked to original sources

arXiv2025

Floer sections in multisymplectic geometry

In symplectic geometry, Floer theory is the most important tool to prove the existence of time-periodic solutions in Hamiltonian mechanics. The core observation is that the $L^2$-gradient lines of the symplectic action functional are pseudo-holomorphic curves, enabling the use of elliptic PDE methods. Multisymplectic geometry is the geometric framework underlying Hamiltonian field theory, where the time line is replaced by higher-dimensional manifolds. In the case of two dimensions and using complex structures, we introduce a novel multisymplectic framework that is fit for the generalization of the elliptic methods from symplectic geometry. Besides proving a Darboux theorem, we show that the $L^2$-gradient lines of our multisymplectic action functional are now pseudo-Fueter curves defined using a compatible almost hyperkähler structure.

math.SG↗

arXiv2025

Generalizing symplectic topology from 1 to 2 dimensions

In symplectic topology one uses elliptic methods to prove rigidity results about symplectic manifolds and solutions of Hamiltonian equations on them, where the most basic example is given by geodesics on Riemannian manifolds. Harmonic maps from surfaces are the natural 2-dimensional generalizations of geodesics. In this paper, we give the corresponding generalization of symplectic manifolds and Hamiltonian equations, leading to a class of partial differential equations that share properties similar to Hamiltonian (ordinary) differential equations. Two rigidity results are discussed: a non-squeezing theorem and a version of the cuplength result for quadratic Hamiltonians on cotangent bundles. The proof of the latter uses a generalization of Floer curves, for which the necessary Fredholm and compactness results will be proven.

math.SG↗

arXiv2024

An approach to Hamiltonian Floer theory for maps from surfaces

In $n$-dimensional classical field theory one studies maps from $n$-dimensional manifolds in such a way that classical mechanics is recovered for $n=1$. In previous papers we have shown that the standard polysymplectic framework in which field theory is described, is not suitable for variational techniques. In this paper, we introduce for $n=2$ a Lagrange-Hamilton formalism that allows us to define a generalization of Hamiltonian Floer theory. As an application, we prove a cuplength estimate for our Hamiltonian equations that yields a lower bound on the number of solutions to Laplace equations with nonlinearity. We also discuss the relation with holomorphic Floer theory.

math.SG↗

arXiv2023

From Euclidean field theory to hyperkähler Floer theory via regularized polysymplectic geometry

Hamiltonian Floer theory plays an important role for finding periodic solutions of Hamilton's equation, which can be seen as a generalization of Newton's equation. Generalizing Newton's equation to Laplace's equation with non-linearity, we show, building on the work of Ginzburg and Hein, that this role is taken over by the hyperkähler Floer theory of Hohloch, Noetzel, and Salamon. Apart from establishing $C^0$-bounds in order to be able to deal with noncompact hyperkähler manifolds, the core ingredient is a regularization scheme for the polysymplectic formalism due to Bridges, which allows us to link Euclidean field theory with hyperkähler Floer theory. As a concrete result, we prove a cuplength estimate.

math.SG↗

arXiv2022

Regularized polysymplectic geometry and first steps towards Floer theory for covariant field theories

It is the goal of this paper to present the first steps for defining the analogue of Hamiltonian Floer theory for covariant field theory, treating time and space relativistically. While there already exist a number of competing geometric frameworks for covariant field theory generalizing symplectic geometry, none of them are readily suitable for variational techniques such as Hamiltonian Floer theory, since the corresponding action functionals are too degenerate. Instead, we show how a regularization procedure introduced by Bridges leads to a new geometric framework for which we can show that the finite energy $L^2$-gradient lines of the corresponding action functional, called Floer curves, converge asymptotically to space-time periodic solutions. As a concrete example we prove the existence of Floer curves, and hence also of space-time periodic solutions, for a class of coupled particle-field systems defined in this new framework.

math.SG↗