arXiv research

We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Moebius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a ``nice'' way, and th…

The study extends calibrated geometry to smooth maps and finds energy bounds.

New invariant for 4D hypersurfaces ensures smooth critical points.

Smooth minimizers found for Willmore energy surfaces.

Smooth isotopy on cube saves energy with extra dimensions.

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

We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman's $ν$-entropy. We then investigate some properties the energy shares with these constants, in particular its relationship with the $κ$-noncollapsing property. Finally, we us…

Study on smoothness of 4D Willmore-type hypersurfaces.

Proves rigidity for specific initial data sets under the dominant energy condition.

Investigates a new four-dimensional energy related to Willmore energy.

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…

We study a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional Euclidean space with an anisotropic energy. An anisotropic energy is the integral of an energy density that depends on the normal at each point over the considered hypersurface. The minimizer of such an energy among all closed hy…

The paper classifies energy-minimizing sets in specific domains.

We investigate a discrete version of the Möbius energy, that is of geometric interest in its own right and is defined on equilateral polygons with $n$ segments. We show that the $Γ$-limit regarding $L^{q}$ or $W^{1,q}$ convergence, $q\in [1,\infty]$ of these energies as $n\to\infty$ is the smooth Möbius energy. This re…

Study on energy of maps from K3 surface to flat orbifold.

Smooth dec initial data sets may not extend to smooth spacetimes.

We prove that the critical points of various energies such as the area, the Willmore energy, the frame energy for tori...etc among possibly branched immersions constrained to evolve within a smooth sub-manifold of the Teichmüller space satisfy the corresponding constrained Euler Lagrange equation. We deduce that critic…

Smooth flows for physical systems with smooth energies and forces.

Let X be a smooth, linearly normal n dimensional complex projective variety. Assume that the projective dual of X has codimension one with defining polynomial D(X). In this paper the log of the norm of D(X) is expressed as the restriction to the Bergman metrics of an energy functional on X. We show how, for smooth plan…

Proves compact Cauchy horizons have constant surface gravity under null energy condition.

We investigate the functional determinant of the laplacian on piece-wise flat two-dimensional surfaces, with conical singularities in the interior and/or corners on the boundary. Our results extend earlier investigations of the determinants on smooth surfaces with smooth boundaries. The differences to the smooth case a…

In this short note, we show a uniqueness result of the energy solutions for the Cauchy problem of Schrodinger flow in the whole space $R^n$ provided there is a smooth solution in the energy class.

We prove the energy identity and the no neck property for a sequence of smooth extrinsic polyharmonic maps with bounded total energy.

Study curves evolving by gradient flow of elastic energy, proving existence, smoothing, and convergence.

We investigate the relationship between a discrete version of thickness and its smooth counterpart. These discrete energies are defined on equilateral polygons with $n$ vertices. It will turn out that the smooth ropelength, which is the scale invariant quotient of length divided by thickness, is the $Γ$-limit of the di…

We prove the analyticity of smooth critical points for O'Hara's knot energies $\mathcal{E}^{α,p}$, with $p=1$ and $2<α< 3$, subject to a fixed length constraint. This implies, together with the main result in \cite{BR13}, that bounded energy critical points of $\mathcal{E}^{α,1}$ subject to a fixed length constraint ar…

In this paper we study the energy function associated to fourth order equations of critical growth on smooth compact conformally flat manifolds of dimension greater or equal than 5.

Graph neural networks over-smooth when layers increase, reducing discriminative power.

Triangulates surfaces with bounded energy using diffeomorphisms.

The Moebius energy of a knot is an energy functional for smooth curves based on an idea of self-repelling. If a knot has a thick tubular neighborhood, we would intuitively expect the energy to be low. In this paper, we give explicit bounds for energy in terms of the ropelength of the knot, i.e. the ratio of the length …

New elastic energy for irregular curves defined through polygonal approximations.

A new framework SIMBA improves graph classification performance on size-imbalanced datasets.

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…

New insights into simple kernel smoothing reveal surprising asymptotics.

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

We study the stability of closed, not necessarily smooth, equilibrium surfaces of an anisotropic surface energy for which the Wulff shape is not necessarily smooth. We show that if the Cahn Hoffman field can be extended continuously to the whole surface and if the surface is stable, then the surface is, up to rescaling…

For every $g\in\mathbb{N}_0$ and $ε>0$, we construct a smooth genus $g$ surface embedded into the unit ball with area $8π$ and Willmore energy smaller than $8π+ ε$. From this we deduce that a minimising sequence for Willmore's energy in the class of genus $g$ surfaces embedded in the unit ball with area $8π$ converges …

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

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

New model predicts energy prices volatility by smoothing time variation and persistence.

Graph convolutions can enhance high frequencies, leading to over-sharpening.

EnCF improves data assimilation for implicit, non-smooth observations.

Proves weight polytope matches with energy vectors in toric varieties.

Study smooth linear statistics on random covers of hyperbolic surfaces, showing central limit and variance results.

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…

A uniqueness result in the inverse problem for an inhomogeneous hyperbolic system on a real vector bundle over a smooth compact manifold, based on energy measurements for improperly known sources, is established.

For a smooth curve $γ$, we define its elastic energy as $E(γ)= \frac 12 \int_γ k^2 (s) ds$ where $k(s)$ is the curvature. The main purpose of the paper is to prove that among all smooth, simply connected, bounded open sets of prescribed area in $\mathbb{R}^2$, the disc has the boundary with the least elastic energy. In…

We consider branes $N$ in a Schwarzschild-$\text{AdS}_{(n+2)}$ bulk, where the stress energy tensor is dominated by the energy density of a scalar fields map $\f:N\ra \mc S$ with potential $V$, where $\mc S$ is a semi-Riemannian moduli space. By transforming the field equation appropriately, we get an equivalent field …