arXiv research

Study on finite entropy and energy in Kähler geometry.

Synthetic approach to pluripotential theory measures finite energy.

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.

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

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.

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

Study finite-energy metrics over complex manifold degenerations.

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…

Geodesics in non-Archimedean metrics are continuous.

New energy measure for isolated systems in general relativity.

Extends finite entropy measures in Kähler geometry.

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

The paper classifies energy-minimizing sets in specific domains.

Proves conditions for minimal surfaces in complex hyperbolic space.

The paper studies quaternionic Monge-Ampère equations in weighted energy classes.

Proves convergence of gradient Ricci shrinkers with uniform bounds.

We introduce different Finsler metrics on the space of smooth Kähler potentials that will induce a natural geometry on various finite energy classes $\mathcal E_{\tilde χ}(X,ω)$. Motivated by questions raised by R. Berman, V. Guedj and Y. Rubinstein, we characterize the underlying topology of these spaces in terms of c…

Study calculates the elastic energy of curves on a sphere.

The paper proves a finite diffeomorphism theorem for manifolds with lower Ricci curvature and bounded energy.

Gradient Ricci solitons can be extended to non-gradient Ricci solitons using energy function.

We study biharmonic maps between Riemannian manifolds with finite energy and finite bi-energy. We show that if the domain is complete and the target of non-positive curvature, then such a map is harmonic. We then give applications to isometric immersions and horizontally conformal submersions.

The paper proves the existence of finite-energy solutions to a specific elliptic equation on Riemannian manifolds.

A new parametric method studies Willmore flows and energy quantization.

Study free energy in spherical spin glasses, proving universality dichotomy.

We consdier in dimension four weakly convergent sequences of approximate biharmonic maos into sphere with bi-tension fields bounded in $L^p$ for some $p>1$. We prove an energy identity that accounts for the loss of Hessian energies by the sum of Hessian energies over finitely many nontrivial biharmonic maps on $\mathbb…

Asymptotic behavior of energy of a harmonic map defined on an asymptotically hyperbolic manifold is considered. Using the growth of energy, we show that a harmonic map defined on some asymptotically hyperbolic manifolds has to be constant if the total energy is finite, or if the map approaches a point fast enough, in t…

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…

We consider in dimension four weakly convergent sequences of approximate biharmonic maps to a Riemannian manifold with bi-tension fields bounded in $L^p$ for $p>\frac43$. We prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivial biharmonic ma…

The paper proves existence of solutions to the Allen-Cahn equation on certain Riemannian manifolds.

In this paper we are concerned with the learnability of energies from data obtained by observing time evolutions of their critical points starting at random initial equilibria. As a byproduct of our theoretical framework we introduce the novel concept of mean-field limit of critical point evolutions and of their energy…

On $(X,ω)$ compact Kähler manifold, given a model type envelope $ψ\in PSH(X,ω)$ (i.e. a singularity type) we prove that the Monge-Ampère operator is an homeomorphism between the set of $ψ$-relative finite energy potentials and the set of $ψ$-relative energy measures endowed with their strong topologies given as the coa…

We consider the energy of smooth generalized distributions and also of singular foliations on compact Riemannian manifolds for which the set of their singularities consists of a finite number of isolated points and of pairwise disjoint closed submanifolds. We derive a lower bound for the energy of all $q$-dimensional a…

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

We develop a method for preserving pseudoholomorphic curves in contact 3-manifolds under surgery along transverse links. This makes use of a geometrically natural boundary value problem for holomorphic curves in a 3-manifold with stable Hamiltonian structure, where the boundary conditions are defined by 1-parameter fam…

Microgrids (MGs) are small, local power grids that can operate independently from the larger utility grid. Combined with the Internet of Things (IoT), a smart MG can leverage the sensory data and machine learning techniques for intelligent energy management. This paper focuses on deep reinforcement learning (DRL)-based…

Meta-materials simulation sped up with energy surrogates.

For a bounded smooth domain in the plane and smooth boundary data we consider the minimisation of the Willmore functional for graphs subject to Dirichlet or Navier boundary conditions. For $H^2$-regular graphs we show that bounds for the Willmore energy imply area and diameter bounds. We then consider the $L^1$-lower s…

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…

Compactness theory for biharmonic maps on degenerating Einstein manifolds.

This paper generalizes neural transport learning for free energy estimation in arbitrary state spaces.

Study complex Monge-Ampère equations on compact Kähler manifolds.

New metric spaces for geodesic rays in cohomology classes.

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

Given a closed submanifold, or a compact regular domain, in euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two different regularization techniques (Hadamard's finite part and analytic continuatio…

Study on surfaces minimizing elastic energy with boundary constraints.

Finite-time blow-up in Yang-Mills flow for small energy initial connections.

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp $L^q$ regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of $\mathbb{R}^n$, which imply asymptotic behavior of the solutions at i…