arXiv research

Sharp bounds found for energy in projective space mappings.

We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimi…

Extends K-energy to complexified Kähler classes for scalar curvature study.

Optimizes energy of mappings from complex projective spaces.

Proves positive energy conjecture for a specific metric class.

A new parametric method studies Willmore flows and energy quantization.

We show that the non pluripolar product of positive currents is a bimeromorphic invariant. Under some natural assumptions, we show that the (weighted) energy associated to big cohomology classes are also bimeromorphic invariants. We compare the weighted energy functionals of currents with respect to different cohomolog…

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_…

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

Study on biharmonic almost complex structures on compact manifolds.

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 …

Study of Willmore energy on sphere sublevel sets and flow singularities.

Optimizes shapes of curves using Möbius energy gradients.

Geodesics found in a metric space of m-subharmonic functions.

A common problem in a high energy physics experiment is extracting a signal from a much larger background. Posed as a classification task, there is said to be an imbalance in the number of samples belonging to the signal class versus the number of samples from the background class. In this work we provide a brief overv…

New order defined for conformal classes, impacts Bartnik's conjecture.

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 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…

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…

In this paper, we give a result on the properness of the K-energy, which answers a question of Song-Weinkove in any dimensions. Moreover, we extend our previous result on the properness of K-energy to the case of modified K-energy associated to extremal Kahler metrics.

New invariant for 4D hypersurfaces ensures smooth critical points.

The paper is devoted to study the Dirichelet energy of moving frames on 2-dimensional tori immersed in the euclidean $3\leq m$-dimensional space. This functional, called Frame energy, is naturally linked to the Willmore energy of the immersion and on the conformal structure of the abstract underlying surface. As first …

In this paper, we give a criterion for the properness of the K-energy in a general Kahler class of a compact Kahler manifold by using Song-Weinkove's result. As applications, we give some Kahler classes on $\mathbb{C}\mathbb{P}^2\#3\overline {\mathbb{C}\mathbb{P}^2}$ and $\mathbb{C}\mathbb{P}^2\#8\overline {\mathbb{C}\…

EAGC boosts GCD by regulating gradient entanglement, improving known and novel category separability.

Study the stability of membranes using Helfrich energy and second variation formula.

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…

Study of critical points for 4D conformally invariant curvature energies.

We study the notion of $p$-quasihomotopy in Newtonian classes of mappings and link it to questions concerning lifts of Newtonian maps, under the assumption that the target space is nonpositively curved. Using this connection we prove that every $p$-quasihomotopy class of Newtonian maps contains a minimizer of the $p$-e…

We establish an energy quantization result for sequences of Willmore surfaces when the underlying sequence of Riemann surfaces is degenerating in the moduli space. we notably exhibit a new residue which quantifies the potential loss of energy in collar regions. Thanks to these residues, we also prove compactness of Wil…

The higher-power derivative terms involved in both Faddeev and Skyrme energy functionals correspond to $σ_2$-energy, introduced by Eells and Sampson. The paper provides a detailed study of the first and second variation formulae associated to this energy. Some classes of (stable) critical maps are outlined.

We show that on Kahler manifolds M with c_1(M)=0 the Calabi flow converges to a constant scalar curvature metric if the initial Calabi energy is sufficiently small. We prove a similar result on manifolds with c_1(M)<0 if the Kahler class is close to the canonical class.

New energy model avoids self-intersections in curve optimization.

Study minimizes Willmore energy with constraints on surface properties.

Uniform volume estimate for Kähler metrics in big cohomology classes.

New findings on Mabuchi energy and stability of manifolds.

Let $E_f$ be the energy of some knot $τ$ for any $f$ from certain class of functions. The problem is to find knots with extremal values of energy. We discuss the notion of the locally perturbed knot. The knot circle minimizes some energies $E_f$ and maximizes some others. So, is there any energy such that the circle ne…

Study p-Willmore disks with boundary energies, finding equilibrium configurations.

We extend short-time existence and stability of the Dirichlet energy flow as proven in a previous paper by the authors to a broader class of energy functionals. Furthermore, we derive some monotonely decreasing quantities for the Dirichlet energy flow and investigate an equation of soliton type. In particular, we show …

We consider a class of time dependent finite energy multi-soliton solutions of the U(N) integrable chiral model in $(2+1)$ dimensions. The corresponding extended solutions of the associated linear problem have a pole with arbitrary multiplicity in the complex plane of the spectral parameter. Restrictions of these exten…

Extends finite entropy measures in Kähler geometry.

In this paper, we prove that the Kahler Ricci flow converges to a Kahler Einstein metric when E_1 energy is small. We also prove that E_1 is bounded from below if and only if the K energy is bounded from below in the canonical class.

Study null energy condition impacts on special hypersurfaces in static spacetimes.

Optimal Euclidean structure minimizes energy in weighted toroidal graphs.

Inspired by the work of Leung-Wan, we study the mean curvature flow in hyperkähler manifolds starting from hyper-Lagrangian submanifolds, a class of middle dimensional submanifolds, which contains the class of complex Lagrangian submanifolds. For each hyper-Lagrangian submanifold, we define a new energy concept called …

Given a compact polarized Kähler manifold $X\hookrightarrow\mathbb{CP}^N$, the space of Bergman metrics on $X$, parameterized by $\mathrm{SL}(N+1,\mathbb{C})$, corresponds to a dense set in the space of Kähler potentials in the Kähler class as $N\to\infty$. Critical points of the $k$th K-energy functional, which is def…

We show that K-energy minimizing movements agree with smooth solutions to Calabi flow as long as the latter exist. As corollaries we conclude that in a general Kahler class long time solutions of Calabi flow minimize both K-energy and Calabi energy. Lastly, by applying convergence results from the theory of minimizing …

Paper proves various types of varieties minimize a specific energy.

According to Pixton, there are Morse-Smale diffeomorphisms of the 3-sphere which have no energy function, that is a Lyapunov function whose critical points are all periodic points of the diffeomorphism. We introduce the concept of quasi-energy function for a Morse-Smale diffeomorphism as a Lyapunov function with the le…