arXiv research

We prove that capillary surfaces converge to a specific energy density as the angle approaches zero.

The study shows how energy density of harmonic maps dominates in $n$-Fuchsian fibers, leading to unique minimal surfaces.

An anisotropic surface energy is the integral of an energy density that depends on the normal at each point over the considered surface, and it is a generalization of surface area. The minimizer of such an energy among all closed surfaces enclosing the same volume is unique and it is (up to rescaling) so-called the Wul…

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

Study on harmonic maps from surfaces with energy bounds and neck domains.

The study provides optimal estimates for surfaces close to constant mean curvature.

Develops spherical density-equalizing maps for closed surfaces.

Developed an ellipsoidal density-equalizing map for genus-0 closed surfaces.

Study magnetic Laplacians on hyperbolic surfaces, revealing three regimes of eigenfunction behavior.

Develops a PDE approach to constructing nontrivial anisotropic surfaces.

Method flattens complex surfaces with consistent density and shape.

The calculation of minimum energy paths for transitions such as atomic and/or spin re-arrangements is an important task in many contexts and can often be used to determine the mechanism and rate of transitions. An important challenge is to reduce the computational effort in such calculations, especially when ab initio …

For a fixed smooth map $u_0$ between two Riemann surfaces $Σ$ and $S$ with non-zero degree, we consider the energy function on Teichmüller space $\mc{T}$ of $Σ$ that assigns to a complex structure $t\in \mc{T}$ on $Σ$ the energy of the harmonic map $u_t:Σ_t:=(Σ,t) \to S$ homotopic to $u_0$. We prove that the energy fun…

New findings confirm parallels to De Giorgi's conjecture for phase transitions in higher dimensions.

We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with larg…

The eigenvalue problem for the Sen--Witten operator on closed spacelike hypersurfaces is investigated. The (square of its) eigenvalues are shown to be given exactly by the 3-surface integral appearing in the expression of the total energy-momentum of the matter+gravity systems in Witten's energy positivity proof. A sha…

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…

Neural density estimators are flexible families of parametric models which have seen widespread use in unsupervised machine learning in recent years. Maximum-likelihood training typically dictates that these models be constrained to specify an explicit density. However, this limitation can be overcome by instead using …

Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local spectral densities, energy densities, global eigenvalue densities, and tot…

Gaussian process regression loses locality in high dimensions, affecting molecular energy surface fitting.

Minimum energy paths for transitions such as atomic and/or spin rearrangements in thermalized systems are the transition paths of largest statistical weight. Such paths are frequently calculated using the nudged elastic band method, where an initial path is iteratively shifted to the nearest minimum energy path. The co…

Efficiently samples and learns densities with symmetries using equivariant methods.

The paper classifies biharmonic quadratic maps between spheres, proving their energy density properties.

Kernel ridge regression is used to approximate the kinetic energy of non-interacting fermions in a one-dimensional box as a functional of their density. The properties of different kernels and methods of cross-validation are explored, and highly accurate energies are achieved. Accurate {\em constrained optimal densitie…

We show that the energy density of critical points of a class of conformally invariant variational problems with small energy on the unit 2-disk B_1 lies in the local Hardy space h^1(B_1). As a corollary we obtain a new proof of the energy convexity and uniqueness result for weakly harmonic maps with small energy on B_…

In this paper we contribute a novel algorithm family, which generalizes many unsupervised techniques including unnormalized and energy models, and allows us to infer different statistical modalities (e.g. data likelihood and ratio between densities) from data samples. The proposed unsupervised technique, named Probabil…

Study on JNR monopoles, focusing on their spectral curves and energy density.

Stable solutions to a specific equation are one-dimensional.

A new training method for normalizing flows without samples.

New method trains deep neural networks for non-interacting kinetic-energy functionals in DFT.

iEFM trains CNF models from unnormalized densities efficiently.

Study investigates singularity formation in $\alpha$-Yang-Mills-Higgs fields on spheres.

Proves density and mass theorems for specific initial data sets.

The paper proves short-time existence of $α$-Dirac-harmonic map flow and applications.

We give a derivation of the Einstein equation for gravity which employs a definition of the local energy density of the gravitational field as a symmetric second rank tensor whose value for each observer gives the trace of the spatial part of the energy-stress tensor as seen by that observer. We give a physical motivat…

The Clifford torus minimizes Willmore energy closely for small perturbations.

Study on the $L^p$ behavior of helix curves' energy density.

RNE provides a flexible framework for diffusion models, enabling inference-time control and energy-based training.

iDEM generates samples from Boltzmann densities without data.

Adaptive multi-stage density ratio estimation improves learning of latent space EBM.

Sharp criteria for 2-varifolds to be induced by smooth immersions.

For Ginzburg-Landau vortices, energy quantization holds only when density is less than 2.

When working with asymptotically hyperbolic initial data sets for general relativity it is convenient to assume certain simplifying properties. We prove that the subset of initial data sets with such properties is dense in the set of physically reasonable asymptotically hyperbolic initial data sets. More specifically, …

The Möbius energy, defined by O'Hara, is one of the knot energies, and named after the Möbius invariant property which was shown by Freedman-He-Wang. The energy can be decomposed into three parts, each of which is Möbius invariant, proved by Ishizeki-Nagasawa. Several discrete versions of Möbius energy, that is, corres…

The paper establishes curvature inequalities and rigidity results for surfaces in Riemannian and Lorentzian geometry.

In this paper, we introduce a new energy density function $\mathscr Y$ on the projective bundle $\mathbb{P}(T_M)\>M$ for a smooth map $f:(M,h)\>(N,g)$ between Riemannian manifolds $\mathscr Y=g_{ij}f^i_αf^j_β\frac{W^αW^β}{\sum h_{γδ} W^γW^δ}.$ We get new Hessian estimates to this energy density and obtain various new…

Bidirectional bounds stabilize training of energy-based models.

Researchers created an accurate kinetic energy functional for materials modeling.