arXiv research

Proves convergence of gradient Ricci shrinkers with uniform bounds.

New energy measure for isolated systems in general relativity.

Study generalizes Möbius energy to non-smooth sets in arbitrary dimensions.

We consider (locally) energy finite coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for the infinitesimal generator. As examples we discuss Euclidean spaces, Riemannian…

This article considers the quasi-local energy in reference to a general static spacetime. We follow the approach developed by the authors in [19, 20, 7, 9] and define the quasi-local energy as a difference of surface Hamiltonians, which are derived from the Einstein-Hilbert action. The new quasi-local energy provides a…

Study $p$-parabolicity on graphs using various energy functionals.

The paper classifies energy-minimizing sets in specific domains.

Study examines Wang-Yau quasi-local energy in strong fields near apparent horizons.

New calibration energy measures deviation from calibrated geometry, enabling mean curvature flow in infinite volumes.

The article explores Helfrich flow with spontaneous curvature, finding singularities and convergence behaviors.

Proposes a new quasi-local mass for timelike 2-surfaces in spacetimes.

Flow preserves isoperimetric ratio for immersed surfaces.

Investigates energy minimizers and critical points of scale-invariant tangent-point energies for knots.

We give a unified statement and proof of a class of wellknown mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energ…

Study non-local isoperimetric energies on spheres using a Riemannian autocorrelation function.

Paper uses DRL for smart MG energy dispatch, improving stability and performance.

We investigate Yang--Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these spaces introduced by Hausel--Hunsicker--Mazzeo. First referring to the codimension 2 …

Study local minimizers of Ginzburg-Landau functionals in high dimensions, showing energy measures converge to rectifiable measures.

Geometric proof shows regularity of anisotropic minimal surfaces in 2D.

We consider finite energy and $L^2$ differential forms associated with strongly local regular Dirichlet forms on compact connected topologically one-dimensional spaces. We introduce notions of local exactness and local harmonicity and prove the Hodge decomposition, which in our context says that the orthogonal compleme…

Study on finite entropy and energy in Kähler geometry.

We study a moduli space of ASD connections over $S^3\times \mathbb{R}$. We consider not only finite energy ASD connections but also infinite energy ones. So the moduli space is infinite dimensional in general. We study the (local) mean dimension of this infinite dimensional moduli space. We show the upper bound on the …

In this paper we study the steepest descent $L^2$-gradient flow of the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, $λ_1$-weighted surface area, and $λ_2$-weighted enclosed volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'.…

Develops tests for conditional symmetry under group actions.

In [13], a new quasi-local energy is introduced for spacetimes with a non-zero cosmological constant. In this article, we study the small sphere limit of this newly defined quasi-local energy for spacetimes with a negative cosmological constant. For such spacetimes, the anti de-Sitter space is used as the reference for…

In this article we introduce and investigate a new two-parameter family of knot energies $TP^{(p,q)}$ that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will firs…

Synthetic approach to pluripotential theory measures finite energy.

In this article, we study the small sphere limit of the Wang-Yau quasi-local energy defined in [18,19]. Given a point $p$ in a spacetime $N$, we consider a canonical family of surfaces approaching $p$ along its future null cone and evaluate the limit of the Wang-Yau quasi-local energy. The evaluation relies on solving …

In this article, we consider the limit of quasi-local conserved quantities [31,9] at the infinity of an asymptotically hyperbolic initial data set in general relativity. These give notions of total energy-momentum, angular momentum, and center of mass. Our assumption on the asymptotics is less stringent than any previo…

In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasi-local mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the …

We investigate probability measures with finite pluricomplex energy. We give criteria insuring that a given measure has finite energy and test these on various examples. We show that this notion is a biholomorphic but not a bimeromorphic invariant.

In this article we study the regularity of stationary points of the knot energies $E^α$ introduced by O'Hara in the range $α\in (2,3)$. In a first step we prove that $E^α$ is $C^1$ on the set of all regular embedded closed curves belonging to $H^{(α+1)/2,2}$ and calculate its derivative. After that we use the structure…

We show that for every quasi-isometric map from a Hadamard manifold of pinched negative curvature to a locally compact, Gromov hyperbolic, ${\rm CAT}(0)$-space there exists an energy minimizing harmonic map at finite distance. This harmonic map is moreover Lipschitz. This generalizes a recent result of Benoist-Hulin.

This study examines local co-movements in energy, agriculture, and metal markets using copulas.

Framework calculates positional influence in causal residual Transformers.

We provide a direct proof for the positivity of Chen-Nester-Tung quasi-local energy with analytic reference in spherical symmetry. A hoop-type theorem for this energy is also established. Finally, the relation between Chen-Nester-Tung and Brown-York quasi-local energies will be discussed.

Researchers find optimal configurations of complex knots and links.

In this paper, we show that the Chen-Nester-Tung (CNT) quasi-local energy is closely related to the Wang-Yau (WY) quasi-local mass. As a particular example, we compute the second variation of the CNT quasi-local energy for axially symmetric Kerr-like spacetimes with axially symmetric embeddings at the obvious critical …

The positive mass theorem is one of the fundamental results in general relativity. It states that, assuming the dominant energy condition, the total mass of an asymptotically flat spacetime is non-negative. The Penrose inequality provides a lower bound on mass by the area of the black hole and is closely related to the…

In this paper, we study the critical case of the Allard regularity theorem. Combining with Reifenberg's topological disk theorem, we get a critical Allard-Reifenberg type regularity theorem. As a main result, we get the topological finiteness for a class of properly immersed surfaces in $\mathbb{R}^n$ with finite Willm…

Finite energy solutions of 4-harmonic and ES-4-harmonic maps are trivial.

Lipschitz mappings found between Riemann surfaces with specific properties.

We use a new combinatorial technique to prove the optimal interior partial regularity result for Lp-vectorfields with integer fluxes minimizing the Lp-energy. More precisely, we prove that the minimal vectorfields are Hölder outside a set which is locally finite inside the domain. The results continue the program start…

We modify previous quasi-local mass definition. The new definition provides expressions of the quasi-local energy, the quasi-local linear momentum and the quasi-local mass. And they are equal to the ADM expressions at spatial infinity. Moreover, the new quasi-local energy has the positivity property.

Minimal energy local systems on curves are compact components of character varieties.

Paper proves stability of positive mass theorem for specific types of manifolds.

Let (M, g) be an (n+1) dimensional space-time, with bounded curvature with respect to a bounded framing. If (M, g) is vacuum or satisfies a mild condition on the stress-energy tensor, then we show that (M, g) locally admits coordinate systems in which the Lorentz metric is well-controlled in the (space-time) Sobolev sp…

Study on finite energy SU(2) monopoles on AC 3-manifolds, proving integrality of charge and curvature decay.