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.
Study on energy-minimizing structures in complex geometry.
problem Existence and regularity of harmonic almost complex structures.
method Inspired by harmonic map theory, proving results similar to Schoen-Uhlenbeck and Cheeger-Naber.
result Proved existence and regularity similar to harmonic map theory.
Minimal elastic networks minimize energy and length at fixed angles.
problem Finding optimal network configurations under elastic constraints.
method Minimizing a combination of elastic energy and length.
result Existence and regularity of minimizers with prescribed angles.
Solves Plateau problem in metric spaces by minimizing energy and area.
problem Solving the Plateau problem in metric spaces.
method Minimizing energy and area in the context of proper metric spaces.
result Established regularity results for energy minimizers under quadratic isoperimetric inequality.
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.
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.
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 …
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 on minimal energy problems for Riesz kernels on manifolds.
problem Minimal energy problems for strongly singular Riesz kernels on manifolds.
method Formulation of natural regularization using Hadamard's partie finie integral operator and analysis of measures with finite energy in Sobolev space.
result The minimal energy problem admits a unique solution and is related to discrete minimal energy problems.
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.
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.
Improved bounds on singular set dimensions for energy-minimizing maps.
problem Bounding the Hausdorff dimension of singular set for harmonic maps.
method Combining Shi's epsilon-regularity theorem with compactness results for energy-minimizing maps.
result Improved bound on Hausdorff dimension of singular set, assuming bounded energy at all scales.
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…
Study minimizes energy functionals with completely monotone kernels, finding analytic solutions.
problem Optimal portfolio liquidation under transient price impact.
method Characterizes minimizers via Fredholm integral equations of the second type.
result Minimizers are analytic and have power series development in even powers of distance.
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.
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.
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.
The paper studies graphs minimizing Dirichlet energy with analytic boundaries, confirming a conjecture about singularities.
problem Understanding the singularities of area-minimizing currents with real analytic boundaries.
method Analyzing multi-valued graphs with real analytic interfaces that minimize Dirichlet energy.
result Dirichlet energy-minimizers with analytic boundary singularities are discrete in 2 dimensions, confirming a conjecture by B. White.
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 …
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.
Extended Möbius energy formula for generalized O'Hara's energies.
problem Maintaining Möbius invariance in O'Hara's energies.
method Extended cosine formula for generalized O'Hara's energies.
result Condition for right circle minimization under length-constraint.
Study of higher-dimensional Willmore energies via minimal submanifold asymptotics.
problem Understanding conformally invariant generalizations of Willmore energy.
method Derives and studies a new energy functional for submanifolds, connects it to minimal submanifold asymptotics in Poincare-Einstein spaces.
result Explicitly identifies the energy for four-dimensional submanifolds and studies its variational properties.
Lower bound found for energy on specific Lagrangian tori in complex projective space.
problem Finding a lower bound for the energy functional on Lagrangian tori in CP2. method Analyzing the energy functional on a family of Hamiltonian minimal Lagrangian tori.
result Proved that the energy of certain Hamiltonian minimal Lagrangian tori is strictly larger than the Clifford torus.
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 disproves compactness for high-energy Willmore immersions and finds minimal bubbles on Willmore surfaces.
problem Compactness for high-energy Willmore immersions of Willmore energy above 16π. method Explicit construction of minimal bubbles and analysis of limit sequences of Willmore immersions.
result Compactness for immersed Willmore tori of energy below 12π is proven. 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.
Study on curves minimizing bending energy in confined spaces.
problem Finding optimal shapes of curves within bounded domains.
method Existence, regularity, and structural properties of minimizers proven.
result Existence of minimizers, convexity of minimizers in convex domains, and examples of non-convex minimizers.
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.
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.
The Reeb field is an energy minimizer on certain Sasakian 3-manifolds.
problem Finding energy minimizers in Sasakian 3-manifolds.
method Characterization of minimizers using eigenvalues and volume preserving diffeomorphisms.
result The Reeb field is a minimizer on some Sasakian manifolds but unstable on others.
Existence of p-energy minimizers linked to Newtonian maps on nonpositively curved spaces.
problem Existence of p-energy minimizers in homotopy classes of Newtonian maps. method Linking p-quasihomotopy to Newtonian maps and using hyperbolic fundamental group properties. result Every p-quasihomotopy class of Newtonian maps contains a minimizer of the p-energy. 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.
The study finds optimal minimum distances for Green's energy points on compact manifolds.
problem Finding optimal minimum distances for Green's energy points on compact Riemannian manifolds.
method Analyzing point configurations minimizing discrete energy with the Green's function for the Laplacian.
result Every point in a minimizing configuration lies inside a harmonic ball, and the minimum distance has optimal asymptotic order.
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.
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.
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.
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.
The paper proves strict convexity of the Mabuchi functional for geodesics connecting energy minimizers.
problem Proving strict convexity of the Mabuchi functional for geodesics.
method Explicit formula for the complex Hessian of the weighted log-Bergman kernel, and proof by showing geodesics must be non-degenerate and smooth.
result Strict convexity of the Mabuchi functional along geodesics connecting energy minimizers.
Researchers define and prove existence of minimizers for generalized Willmore functionals.
problem Existence of area constrained minimizers for generalized Willmore functionals.
method Compactness result for branched, immersed, stratified surfaces; direct minimization; introduction of haunted surfaces.
result Existence of area constrained minimizers for generalized Willmore functionals.
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.
Complex wrinkling patterns emerge in non-Euclidean elastic sheets due to energy minimization.
problem Understanding hierarchical buckling patterns in non-Euclidean elastic sheets.
method Minimizing elastic energy to explain complex wrinkling patterns.
result Branch-point singularities are key to generating complex wrinkling patterns.
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.
Study on energy of smooth and singular distributions on manifolds.
problem Energy calculation for smooth and singular distributions on manifolds.
method Derive lower bounds and find minimizers for energy functionals.
result Lower bounds and minimizers for energy of distributions found.