arXiv research

Paper proves stability of multi-dimensional rarefaction waves in gas dynamics.

New inequality shows energy growth and decay in geometric problems.

The paper establishes scattering theory for wave equations on Schwarzschild spacetime.

Stable solutions to a specific equation are one-dimensional.

Derives energy-momentum tensor from Standard Model, examines energy conditions.

Uniform small energy regularity for fractional geometric problems proved.

New neural method calculates EMD for particle physics data.

Study shows TAP free energy minimization provides better posterior inference in high-dimensional linear models.

We consider the wave equation on a closed Riemannian manifold. We observe the restriction of the solutions to a measurable subset $ω$ along a time interval $[0, T]$ with $T>0$. It is well known that, if $ω$ is open and if the pair $(ω,T)$ satisfies the Geometric Control Condition then an observability inequality is sat…

We prove a Gamma-convergence result for a family of bending energies defined on smooth surfaces in $\mathbb{R}^3$ equipped with a director field. The energies strongly penalize the deviation of the director from the surface unit normal and control the derivatives of the director. Such type of energies for example arise…

We study a class of fourth-order geometric problems modelling Willmore surfaces, conformally constrained Willmore surfaces, isoperimetrically constrained Willmore surfaces, bi-harmonic surfaces in the sense of Chen, among others. We prove several local energy estimates and derive a global gap lemma.

Geometric Occam's Razor shapes deep learning solutions.

Unified geometric description of Kepler flow across all energies.

We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in $\mathbb{R}^2$ introduced by Rivière. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmoni…

L-GATr transforms high-energy physics data using geometric algebra and Lorentz symmetry.

In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded $m$-dimensional Lipschitz submanifolds in ${\mathbb{R}}^n$. It turns out that due to a smoothing effect any seq…

The Einstein equations in wave map gauge are a geometric second order system for a Lorentzian metric. To study existence of solutions of this hyperbolic quasi diagonal system with initial data on a characteristic cone which are not zero in a neighbourhood of the vertex one can appeal to theorems due to Cagnac and Dossa…

In the present work, torsion energy is defined. Its law of conservation is given. It is shown that this type of energy gives rise to a repulsive force which can be used to interpret supernovae type Ia observations, and consequently the accelerating expansion of the Universe. This interpretation is a pure geometric one …

The Palais-Smale condition is proven for various knot energies.

New geometric interpretation of discrete Willmore energy using rolling spheres connection.

In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis, the blow-up analysis usually strongly utilizes conformal invariance, which yields …

Geometric inequality linking Dirichlet and bienergy for maps between Riemannian manifolds.

Equations of motion of low-energy string effective actions can be conveniently described in terms of generalized geometry and Levi-Civita connections on Courant algebroids. This approach is used to propose and prove a suitable version of the Kaluza-Klein-like reduction. Necessary geometrical tools are recalled.

Proves a higher-order positive energy theorem for stationary solutions in fourth-order gravity.

Study on helix curves and their Möbius energy asymptotics.

Researchers solve a complex equation to embed graphs with negative curvature.

The present chapter gives an overview on results for discrete knot energies. These discrete energies are designed to make swift numerical computations and thus open the field to computational methods. Additionally, they provide an independent, geometrically pleasing and consistent discrete model that behaves similarly …

Study of Dirac fields on Kerr spacetimes using peeling method.

In this paper, we analyzed the physical meaning of scalar curvatures for a generalized Riemannian space. It is developed the Madsen's formulae for pressures and energy-densities with respect to the corresponding energy-momentum tensors. After that, the energy-momentum tensors, pressures, energy-densities and state-para…

In this paper we consider a geometric variant of Hofer's symplectic energy, which was first considered by Eliashberg and Hofer in connection with their study of the extent to which the interior of a region in a symplectic manifold determines its boundary. We prove, by a simple geometric argument, that both versions of …

This paper details a series of experiments in searching for minimal energy configurations for knots and links using the computer program KnotPlot. The most interesting phenomena found in these experiments is the dependence of the trajectories of energy descent upon the initial geometric conditions of the knotted embedd…

The higher-power derivative terms involved in both Faddeev and Skyrme energy functionals correspond to $σ_2$-energy, introduced by Eells and Sampson. The paper provides a detailed study of the first and second variation formulae associated to this energy. Some classes of (stable) critical maps are outlined.

Our aim in this paper is to investigate some geometrical properties of Berger Spheres i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields. We determine all vector fields which are critical points for the energy functional restricted to vector fields. We also see that do not exist any v…

We use a min-max procedure on the Allen-Cahn energy functional to construct geodesics on closed, 2-dimensional Riemannian manifolds, as motivated by the work of Guaraco. Borrowing classical blowup and curvature estimates from geometric analysis, as well as novel Allen-Cahn curvature estimates due to Wang-Wei, we manage…

FEAT estimates free energy using adaptive transports.

This is the second in a series of three papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sob…

Unified approach to various energy conditions in spacetime geometry.

A geometric construction for obtaining a prolongation of a connection to a connection of a bundle of connections is presented. This determines a natural extension of the notion of canonical energy-tensor which suits gauge and gravitational fields, and shares the main properties of the energy-tensor of a matter field in…

Study finds bound on energy of minimal spheres on complex manifolds.

The biharmonic flow and Willmore flow are studied in higher dimensions using geometric evolution equations.

The paper proves inequalities for closed surfaces involving mean curvature.

We consider the oscillator group equipped with a bi-invariant Lorentzian metric, and then some geometrical properties of this group i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields are obtained. We also determine all vector fields which are critical points for the energy functional …

Extends denoising and score estimation to energy models via Tweedie's formula.

Uniform bounds prove connection between Kähler metrics and RCD spaces.

We prove that a random group of the graph model associated with a sequence of expanders has fixed-point property for a certain class of CAT(0) spaces. We use Gromov's criterion for fixed-point property in terms of the growth of n-step energy of equivariant maps from a finitely generated group into a CAT(0) space, to wh…

An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis is on several new stability conditions, such as K-stability, Donaldson's infinite…

Study contact instantons and Legendrian links, proving energy inequalities.

Modified Gibbs-Helmholtz equation geometric models for thermodynamics.