arXiv research

Green's functions and Biot-Savart operators on curved spaces quantify linking numbers.

We study steady-state magnetic fields in the geometric setting of positive curvature on subdomains of the three-dimensional sphere. By generalizing the Biot-Savart law to an integral operator BS acting on all vector fields, we show that electrodynamics in such a setting behaves rather similarly to Euclidean electrodyna…

The paper connects linking numbers to Biot-Savart kernels on compact manifolds.

We introduce here explicit integral formulas for linking, twisting, writhing and helicity on the 3-sphere and in hyperbolic 3-space. These formulas, like their prototypes in Euclidean 3-space, are geometric rather than just topological, in the sense that their integrands are invariant under orientation-preserving isome…

In this first of two papers, we develop a steady-state version of classical electrodynamics on the 3-sphere and in hyperbolic 3-space, including an explicit formula for the vector-valued Green's operator, an explicit formula of Biot-Savart type for the magnetic field, and a corresponding Ampere's Law contained in Maxwe…

Extends Arnold's linking theory to higher dimensions and submanifolds.

We introduce and illustrate a new approach to the unknotting problem via the dynamics of vortex strings in a nonlinear partial differential equation of reaction-diffusion type. To untangle a given knot, a Biot-Savart construction is used to initialize the knot as a vortex string in the FitzHugh-Nagumo equation. Remarka…

New constraints rule out some optimal domains for helicity maximisation.

Neural Networks improve incompressible flow simulations without complex kernels.

The helicity of a vector field is a measure of the average linking of pairs of integral curves of the field. Computed by a six-dimensional integral, it is widely useful in the physics of fluids. For a divergence-free field tangent to the boundary of a domain in 3-space, helicity is known to be invariant under volume-pr…

We consider the $D$-dimensional Euclidean space, $\mathbb{R}^D$, with certain $(D-N)$-dimensional compact, closed and orientable sub-manifolds (which we call \emph{singularity manifolds} and represent by $\widetilde{\mathcal{S}}$) removed from it. We define and investigate the problem of finding a homotopy-like class i…

The paper introduces surfaces with constant solid angle for designing shell structures.

Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three-dimensional (3D) ideal flow (Frisch & Zheligovsky, Commun. Math. Phys., vol. 326, 2014, pp. 499-505, Podvigina et al., J. Comput. Phys., vol. 306, 2016, pp. 320-342). Looking at such invariants with the modern tools …

This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slovák has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for …

The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.

Introduces a new elliptic operator with positive eigenvalue.

Proves Kato inequalities for various conformal operators.

We describe a set of conformally covariant boundary operators associated to the Paneitz operator, in the sense that they give rise to a conformally covariant energy functional for the Paneitz operator on a compact Riemannian manifold with boundary. These operators naturally give rise to a first- and third-order conform…

Study on biharmonic hypersurfaces with specific recurrent operators in Euclidean space.

Local index theorem for chiral geometric operators proved using heat kernel.

Study estimates eigenvalues for concave Hessian operators on convex domains.

The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.

GJMS operators connect geometry, analysis, and physics.

Extends Calabi operator to Riemannian locally symmetric spaces.

New spectral torsion defined for rescaled Dirac operators.

Researchers create new operators from Riemannian invariants.

Proves formal self-adjointness of certain differential operators.

The paper proves new theorems about specific types of operator perturbations.

Study of Dirac-like operators on spin manifolds with large mass parameters.

The study proves inequalities for complex operators on curved spaces.

Study essential spectrum of differential operators on geometrically finite orbifolds.

Study on opers over complex manifolds of dimension one.

Mixtures of neural operators reduce active complexity in operator learning.

Formula for Hadamard coefficients from Green's operators on spacetimes.

Identifies Lorentzian locally symmetric spaces where Calabi operator suffices to determine Killing operator range.

Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.

Method calculates spectra of Rarita-Schwinger operator on symmetric spaces.

Paper generalizes paracomposition and change of variables for paradifferential operators.

The paper revisits and analyzes the tmd-operator in almost Kähler manifolds.

Researchers create a family of conformally covariant operators.

The k-Dirac operator is a differential operator which is natural to geometric structure of a parabolic type. We will give a set of initial conditions for this operator. In the proof of the claim we will need to adapt some parts from the theory of exterior differential systems to the setting of weighted differential ope…

We describe a set of conformally covariant boundary operators associated to the sixth-order GJMS operator on a conformally invariant class of manifolds which includes compactifications of Poincaré--Einstein manifolds. This yields a conformally covariant energy functional for the sixth-order GJMS operator on such manifo…

The paper derives expansions for Green's operators and resolvents using Hadamard methods.

Study natural operators transforming tensor fields, proving all bilinear ones are of order one.

The paper quantizes Kähler manifolds using differential operators.

The paper classifies and proves properties of symmetry breaking operators for specific groups.

New operators generalize Michelsohn's on almost Hermitian manifolds.

Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.