Paper proves unique energy-minimizing curves in constrained spaces.
problem Uniqueness of energy-minimizing curves in constrained spaces.
method Investigated energy-minimizing curves with fixed endpoints in a constrained space.
result Proved that the set of points for which the energy-minimizing curve is not unique has no interior points.
Paper proves various types of varieties minimize a specific energy.
problem Minimizing a specific energy in various types of varieties.
method Analyzes various polarized varieties over Q. result Various types of varieties minimize the Arakelov K-energy.
The paper studies harmonic graphs in the Heisenberg group and their properties.
problem No analogous theorem exists for H-minimal surfaces in the Heisenberg group. method Introduced intrinsic Dirichlet energy and studied its critical points (contact harmonic graphs).
result Calibration condition and construction of energy-minimizing graphs with various singularities.
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 …
Study energy-minimizing maps in projective spaces, proving sharp bounds.
problem Finding optimal mappings in projective spaces.
method Proving lower bounds and characterizing energy-minimizing maps.
result Sharp lower bounds and characterization of energy-minimizing maps.
The paper classifies energy-minimizing sets in specific domains.
problem Classifying volume-constraint local energy-minimizing sets.
method Proved a Poincaré-type inequality for stable sets.
result Relative boundary of energy-minimizing sets is smooth.
Study finds surfaces in spherical caps that maximize modified energy.
problem Geometry of surfaces with free boundaries and capillary conditions.
method Monotonicity formulae and energy maximization analysis.
result Capillary minimal surfaces maximize a modified energy in their conformal orbit.
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…
Smooth minimizers found for Willmore energy surfaces.
problem Finding minimizers for Willmore energy surfaces.
method Existence and smoothness established through axially symmetric surfaces with prescribed isoperimetric ratio.
result Existence and smoothness of minimizers proven.
Optimal Euclidean structure minimizes energy in weighted toroidal graphs.
problem Finding the optimal Euclidean structure for weighted toroidal graphs.
method Minimizing Dirichlet energy over all possible Euclidean structures and realizations within a fixed homotopy class.
result The optimal Euclidean structure induces a weighted Delaunay decomposition.
Researchers find optimal configurations of complex knots and links.
problem Finding the most efficient configurations of complex knots and links.
method Minimizing Möbius and Minimum Distance energies by describing them with a small number of free parameters.
result Optimal geometries for Hopf links, Borromean rings, and chain links are found.
The paper proves a Fenchel theorem for Gauss maps and shows circles and disks minimize certain energies.
problem Finding minimizers of nonlocal curvature energies.
method Combining Fenchel-type theorems with geometric analysis techniques.
result Circles and disks minimize specific energy functionals.
In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasi-local mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the …
Study bounds the index of minimal submanifolds using energy measures and Yang-Mills-Higgs equations.
problem Bounding the index of codimension 2 minimal submanifolds.
method Second inner variation of energy, convergence of energy measures, and stress-energy tensors.
result Bound the Morse index of the submanifold by the index of critical points.
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 Lq or W1,q convergence, q∈[1,∞] of these energies as n→∞ is the smooth Möbius energy. This re…
The paper finds curves minimizing elastic energy pinned at endpoints.
problem Finding curves that minimize elastic energy with fixed endpoints.
method Applying the shooting method to identify and classify critical points.
result Critical points consist of wavelike elasticae, and minimizers have no loops or interior inflection points.
The paper proves Γ-convergence of discrete tangent-point energies to continuous energies and ropelength, with applications to biarc curves.
problem Proving convergence of discrete tangent-point energies to continuous energies and ropelength.
method Using biarc curves and interpolation, the paper proves Γ-convergence of discretized tangent-point energies to the continuous tangent-point energies and ropelength functional. result Discrete almost minimizing biarc curves converge to ropelength minimizers and minimizers of continuous tangent-point energies.
Minimal submanifolds are found as energy concentration sets in variational problems.
problem Understanding the structure of minimal submanifolds in codimension two.
method Purely variational approach, extending previous work on geodesics.
result Non-degenerate minimal submanifolds can be derived from critical maps of the Ginzburg-Landau functional.
We study the converse to the statement that instantons are minimizers of the Yang--Mills energy in four dimensions. We show that given an energy minimizing connection, A, the curvature of A takes values in a subbundle of the adjoint bundle which decomposes as a sum of instantons.
Investigates energy minimizers and critical points of scale-invariant tangent-point energies for knots.
problem Finding and characterizing minimizers and critical points of scale-invariant tangent-point energies for closed curves.
method Develops convergence and regularity theories based on fractional Sobolev spaces and new energy functionals.
result Minimizing sequences converge to locally critical embeddings in all but finitely many points, and locally critical embeddings are regular.
Study on surfaces minimizing elastic energy with boundary constraints.
problem Finding stable configurations of surfaces with elastic boundaries and surface energy.
method Investigation of critical surfaces with mean curvature and spontaneous curvature, coupled to boundary elastic energy.
result Characterization and minimization of surface energy for specific topological shapes.
EMIX minimizes surprise in multi-agent reinforcement learning.
problem Surprise and approximation bias in multi-agent reinforcement learning.
method Energy-based MIXer (EMIX) for minimizing surprise across multiple agents.
result EMIX demonstrates consistent stable performance in challenging StarCraft II scenarios.
Introduces Causal Energy Minimization to understand Transformer layers.
problem Empirical parameterization of Transformer blocks remains largely unexplored.
method Causal Energy Minimization framework that recasts Transformer layers as optimization steps on conditional energy functions.
result Identifies design space for Transformer layers including weight sharing and energy-based interpretations.
Study minimizes Willmore energy with constraints on surface properties.
problem Minimizing Willmore energy under specific surface properties.
method Adapting Keller-Mondino-Rivière, Bauer-Kuwert, and Ndiaye-Schätzle methods.
result Existence of smooth minimizers for a broad range of constraints.
The paper characterizes gaps in minimal foliations on tori using energy criteria.
problem Characterizing gaps in minimal foliations on tori.
method Introduced an energy to study min-max theory and applied it to Almgren-Pitts min-max theory.
result For a generic metric, if a lamination contains a gap, there exists a non-area-minimizing minimal hypersurface inside the gap.
Minimal energy local systems on curves are compact components of character varieties.
problem Characterizing local systems on surfaces with minimal energy.
method Study of minimal energy local systems on surfaces of genus g with d punctures.
result Minimal energy local systems form compact components of real relative character varieties.
This work develops machine learning for micromagnetic energy minimization.
problem Minimizing Gibbs free energy in full 3D micromagnetic simulations.
method Advanced machine learning techniques, including Physics-Informed Neural Networks (PINNs) and Extreme Learning Machines (ELMs), with reformulated bounds and optimization schemes.
result Competitive performance of machine learning methods compared to traditional numerical approaches.
We study minimal energy problems for strongly singular Riesz kernels on a manifold. Based on the spatial energy of harmonic double layer potentials, we are motivated to formulate the natural regularization of such problems by switching to Hadamard's partie finie integral operator which defines a strongly elliptic pseud…
Quantitative stability for nearly minimizing Yamabe metrics.
problem Understanding the stability of nearly minimizing metrics in Riemannian geometry.
method Proving quantitative closeness of nearly minimizing metrics to minimizing metrics in a specific sense.
result The distance between nearly minimizing metrics and minimizing metrics is controlled quadratically by the Yamabe energy deficit.
New theorem on 3-manifolds with curvature and convex boundary.
problem Understanding 3-manifolds with specific curvature and boundary properties.
method Analyzes properties of Riemannian 3-manifolds with nonnegative scalar curvature and mean-convex boundary.
result Shows flatness of certain 3-manifolds containing specific geometric objects.
We study the energy functional on the set of Lagrangian tori in CP2 . We prove that the value of the energy functional on a certain family of Hamiltonian minimal Lagrangian tori in CP2 is strictly larger than energy of the Clifford torus.
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 prove that capillary surfaces converge to a specific energy density as the angle approaches zero.
problem Understanding the limiting behavior of capillary surfaces as the angle approaches zero.
method Rigorous limiting analysis and application to curvature estimates.
result Capillary area-density converges to the Weiss energy density as the angle tends to zero.
Motivated by the problem of optimal portfolio liquidation under transient price impact, we study the minimization of energy functionals with completely monotone displacement kernel under an integral constraint. The corresponding minimizers can be characterized by Fredholm integral equations of the second type with cons…
Study on sphere-valued maps, proving energy convergence and current limits.
problem Understanding the behavior of sphere-valued Sobolev maps as their energy grows.
method Proving Gamma-convergence of p-energies to the mass of an integral current. result Jacobian convergence to an area-minimizing current in a cobordism class.
In 1996, Shi generalized the epsilon-regularity theorem of Schoen and Uhlenbeck to energy-minimizing harmonic maps from a domain equipped with a bounded measurable Riemannian metric. In the present work we prove a compactness result for such energy-minimizing maps. As an application, we combine our result with Shi's th…
Investigates a new four-dimensional energy related to Willmore energy.
problem Exploring a new conformally invariant energy in four dimensions.
method Computed first variation, applied Noether's theorem, investigated other generalizations.
result Critical points of the new energy are smooth and do not include minimal hypersurfaces.
Optimal flat ribbons can be created from nonplanar curves with minimal energy.
problem Finding the optimal shape of flat ribbons from nonplanar curves.
method Direct method of the calculus of variations.
result Optimal flat ribbons can be created with minimal bending energy, but they may have isolated planar points.
New proof of harmonic map uniqueness with analytic targets.
problem Uniqueness of energy-minimizing harmonic maps with analytic targets.
method Symmetric (log)-epiperimetric inequality for harmonic maps with analytic targets.
result Tangents at infinity of energy-minimizing harmonic maps are unique.
Study shows TAP free energy minimization provides better posterior inference in high-dimensional linear models.
problem Deviation from true posterior mean and underestimation of posterior uncertainty in variational inference.
method Minimization of TAP free energy in a high-dimensional asymptotic framework, showing geometric and statistical properties.
result Local minimizer of TAP free energy provides consistent estimate of posterior marginals and correctly calibrated posterior inference.
In this paper, we consider multi-valued graphs with a prescribed real analytic interface that minimize the Dirichlet energy. Such objects arise as a linearized model of area minimizing currents with real analytic boundaries and our main result is that their singular set is discrete in 2 dimensions. This confirms (and p…
A conformally invariant generalization of the Willmore energy for compact immersed submanifolds of even dimension in a Riemannian manifold is derived and studied. The energy arises as the coefficient of the log term in the renormalized area expansion of a minimal submanifold in a Poincare-Einstein space with prescribed…
We introduce a notion of generalized Willmore functionals motivated by the Hawking energy of General Relativity and bending energies of membranes. An example of a bending energy is discussed in detail. Using results of Y. Chen and J. Li, we present a compactness result for branched, immersed, haunted, stratified surfac…
Let Ef 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 Ef and maximizes some others. So, is there any energy such that the circle ne…
Improves MRI-based brain surface reconstruction with minimal deformation energy loss.
problem Ensuring optimal deformation energy and consistency in learning-based cortical surface reconstruction.
method Design and implementation of a Minimal Energy Deformation (MED) loss in the V2C-Flow model.
result Significant improvements in training consistency and reproducibility without sacrificing reconstruction accuracy and topological correctness.
The paper analyzes discrete approximations to minimize curve length in Euclidean space.
problem Minimizing the length of curves between two sets in Euclidean space.
method Finite differences and numerical integration for discrete approximations.
result The squared length of the reconstructed curve converges to the squared minimal length with rate O(N−1/2). We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface M. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Mañé critical value of the universal abelian cov…
Characterizes neutral deformation modes of minimal surfaces.
problem Understanding the energy content of deformation modes of minimal surfaces.
method Analyzes the energy content of stretching, drilling, and bending modes of minimal surfaces.
result All isometries of a minimal surface are globally neutral and give rise to soft elasticity.