arXiv research

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

Flow preserves isoperimetric ratio for immersed surfaces.

We establish some a priori geometric relations on stable minimal surfaces lying inside three-manifolds with scalar curvature uniformly bounded below. The relations are based on a slight generalization of a formula due to Castillon. We apply it to prove non-local rigidity results in the particular sense that they expres…

Smooth minimizers found for Willmore energy surfaces.

Proves strict inequality for minimizers of Willmore energy under isoperimetric constraints.

The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.

Study of Dirac equation with non-local nonlinearity on spheres.

Researchers find a surface with minimum bending energy for any genus and isoperimetric ratio.

The Clifford torus is unique when its isoperimetric ratio is prescribed.

Delaunay tori minimize Willmore energy under isoperimetric constraints.

Sharp inequalities and symmetries on Riemannian surfaces quantified.

Gradient flow method solves isoperimetric inequality for maps.

The isoperimetric ratio of an embedded surface in $R^3$ is defined as the ratio of the area of the surface to power three to the squared enclosed volume. The aim of the present work is to study the minimization of the Willmore energy under fixed isoperimetric ratio when the underlying abstract surface has fixed genus $…

Paper proves a sharp weighted Isoperimetric inequality for substatic manifolds.

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…

The paper analyzes the emergence of almost-honeycomb structures in low-energy planar clusters.

Study minimizers in large volume isoperimetric problems with a new flatness criterion.

We prove that smooth critical points of the Möbius energy parametrized by arc-length are analytic. Together with the main result in \cite{BRS16} this implies that critical points of the Möbius energy with merely bounded energy are not only $C^\infty$ but also analytic. Our proof is based on Cauchy's method of majorants…

The paper proves inequalities for closed surfaces involving mean curvature.

In this article, by following the method in \cite{PT}, combining Willmore energy with isoperimetric inequalities, we construct two examples of singularities under mean curvature flow in $\mathbb{H}^3$. More precisely, there exists a torus, which must develop a singularity under MCF before the volume it encloses decreas…

Study on bending energy of surfaces with curvature concentration, deriving new lower bounds.

Two non-local asymptotic invariants of magnetic fields for the ideal magnetohydrodynamics are introduced. The velocity of variation of the invariants for a non-ideal magnetohydrodynamics with a small magnetic dissipation is estimated. By means of the invariants the spectra of electromagnetic fields are investigated. A …

Study on minimal disks in metric spaces, focusing on branch set structure.

New proofs of Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.

We show that in the setting of proper metric spaces one obtains a solution of the classical two-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area (in the sense of convex geometry) has been chosen appropriately. We prove the quasi-convexity of this new definition o…

We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-…

Global convergence proved for Gursky-Malchiodi $Q$-curvature flow in dimensions $n \geq 5$.

We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume…

New proofs and inequalities for capillarity problems quantify asymmetries.

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.

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

Non-local GNNs improve performance on disassortative graphs.

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

In this paper, we study the moduli spaces of noncollapsed Ricci flow solutions with bounded energy and scalar curvature. We show a weak compactness theorem for such moduli spaces and apply it to study isoperimetric constant control, Kähler Ricci flow and moduli space of gradient shrinking solitons.

Study of flow in hyperbolic space with capillary boundary.

Study of immersions with Willmore energy leading to spherical and catenoid bubbles.

The abstract explores a new wave equation linking quantum mechanics and complex adaptive systems.

Study on blow-up behavior of sign-changing solutions for Yamabe equation.

Paper introduces new fractional Dirac operator and Q-curvature.

Given an elliptic integrand of class $\mathscr{C}^{3}$, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.

Let $(M^n,g)$ be simply connected, complete, with non-positive sectional curvatures, and $Σ$ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in $M$. Let $S$ be an area minimising integral 3-current (resp. flat chain mod 2) such that $\partial S = Σ$. We use a weak mean curvature flow,…

The Allen-Cahn system on manifolds yields multiple phase distributions.

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

Kuwert and Schätzle showed in 2001 that the Willmore flow converges to a standard round sphere, if the initial energy is small. In this situation, we prove stability estimates for the barycenter and the quadratic moment of the surface. Moreover, in codimension one we obtain stability bounds for the enclosed volume and …

It is argued that the enlargement of the gauge group found in non-commutative gauge theory is more fundamentally thought of as a consequence of the non-locality of the construction and that it was already encountered in an earlier discussion of a non-local gauge theory.

Motivated by the HRRT-formula for holographic entanglement entropy, we consider the following question: what are the position and the surface area of extremal surfaces in a perturbed geometry, given their anchor on the asymptotic boundary? We derive explicit expressions for the change in position and surface area, ther…

In this paper, we study two kind of L^2 norm preserved non-local heat flows on closed manifolds. We first study the global existence, stability and asymptotic behavior to such non-local heat flows. Next we give the gradient estimates of positive solutions to these heat flows.

Score matching efficiency tied to distribution isoperimetric properties.