Triangulates surfaces with bounded energy using diffeomorphisms.
problem Triangulating surfaces with bounded Kolasinski--Menger energy.
method Uses bounded distortion diffeomorphisms of subsets of a plane.
result Triangulation with bounded number of triangles.
Sharp bounds found for energy in projective space mappings.
problem Finding bounds for energy in mappings of real projective spaces.
method Sharp lower and upper bounds for energy in homotopy classes of mappings from real projective space to Riemannian manifolds.
result Characterization of maps that achieve the lower bound for energy and determination of the infimum of energy in a homotopy class.
Lower bounds on geodesic lengths for spheres with Willmore energy.
problem Finding shortest closed geodesics on spheres with Willmore energy.
method Proving a lower bound on geodesic lengths for spheres with Willmore energy below 6π.
result The energy threshold of 6π is optimal and the inequality cannot be extended to higher genus surfaces.
Quantizes Willmore energy in Riemannian manifolds with bounded energy and area.
problem Quantization of Willmore energy in bounded energy and area conditions.
method Uniform boundedness of Willmore energy and area, weak convergence of maps, and conformal structures in compact domain.
result Quantization of Willmore energy holds under specified conditions.
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.
Characterizes complex Hessian equations for bounded energy functions.
problem Understanding degenerate complex Hessian equations for bounded energy functions.
method Proving sublevel set estimates and using Sobolev inequalities.
result Characterization of degenerate complex Hessian equations for bounded (p,m)-energy functions. We give a unified statement and proof of a class of wellknown mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energ…
Researchers establish bounds and continuity of decomposed Möbius energies using cosine formula.
problem Estimating the bounds and continuity of decomposed Möbius energies.
method Using the cosine formula to evaluate upper and lower bounds and modulus of continuity of decomposed energies.
result Affirmative answer to the question of estimating decomposed energies using the cosine formula.
An energy estimate is proved for the Bel--Robinson energy along a constant mean curvature foliation in a spatially compact vacuum spacetime, assuming an L∞ bound on the second fundamental form, and a bound on a spacetime version of Bel--Robinson energy.
AdS uniqueness and black hole energy bounds proven.
problem Proving uniqueness of Anti-de Sitter spacetime and energy bounds for AdS black holes.
method Adapted Wang's proof to static asymptotically locally hyperbolic vacuum metrics and higher-genus horizons.
result Negativity of free energy E−TS for AdS black holes with higher-genus horizons. Signals are submanifolds; bounds on energy calculated.
problem Abstract theory of signal propagation.
method Energy inequalities and bounds calculated for specific signal spaces.
result Upper and lower bounds on energy derived for various signal configurations.
Study lower bounds on modified K-energy on Fano manifolds with Kähler-Ricci solitons.
problem Lower boundedness of modified K-energy on Fano manifolds.
method Extend Tosatti's method to study Fano manifolds with Kähler-Ricci solitons.
result Establish lower bounds on modified K-energy for Kähler-Ricci solitons.
Proves energy quantization for surfaces with bounded index.
problem Energy quantization for Willmore surfaces with bounded index.
method Translated the question to the conformal Gauss map's perspective and showed convergence in specific regions.
result Conformal Gauss map converges to a light-like geodesic in De Sitter space in neck or collar regions.
Study on harmonic maps from surfaces with energy bounds and neck domains.
problem Behavior of harmonic maps with bounded energy on complex domains.
method Analysis of a sequence of harmonic maps in generalized neck domains.
result Upper bound of energy density and study of nullity and index limits.
Local examples of singular connections with bounded energy.
problem Constructing singular Hermitian Yang-Mills connections with bounded energy.
method Local construction of singular connections over B1⊂C3. result Number of essential singular points can be arbitrarily large.
Study bounds CMC surface index in 3-manifolds using energy.
problem Bounding the index of CMC surfaces in 3-manifolds.
method Energy comparison to prove linear upper bound.
result Linear upper bound on CMC surface index.
A universal geometric inequality for bodies relating energy, size, angular momentum, and charge is naturally implied by Bekenstein's entropy bounds. We establish versions of this inequality for axisymmetric bodies satisfying appropriate energy conditions, thus lending credence to the most general form of Bekenstein's b…
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 H2-regular graphs we show that bounds for the Willmore energy imply area and diameter bounds. We then consider the L1-lower s…
Bidirectional bounds stabilize training of energy-based models.
problem Training energy-based models is difficult and prone to instability.
method Propose bidirectional bounds linking to gradient penalty and Jacobi-determinant estimator.
result Significant stabilization and high-quality density estimation achieved.
We study various aspects related to boundary regularity of complete properly embedded Willmore surfaces in H3, particularly those related to assumptions on boundedness or smallness of a certain weighted version of the Willmore energy. We prove, in particular, that small energy controls C1 boundary regularity. We examin…
Estimates for Schrödinger operators on manifolds with bounded Ricci curvature.
problem Quantifying unique continuation for Schrödinger operators on manifolds with specific curvature conditions.
method Proving quantitative unique continuation estimates for Schrödinger operators on manifolds with Ricci curvature bounded below.
result Upper bound for energy range and constant in terms of Ricci curvature and parameters of relatively dense set.
In this paper, we first prove a folklore conjecture on a greatest lower bound of the Calabi energy in all Kähler manifold. Similar result in algebriac setting was obtained by S. K. Donaldson. Secondly, we give an upper/lower bound estimate of the K energy in terms of the geodesic distance and the Calabi energy. This is…
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 Rn. It turns out that due to a smoothing effect any seq…
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…
The Willmore energy of a closed surface in R^n is the integral of its squared mean curvature, and is invariant uner Möbius transformations of R^n. We show that any torus in R^3 with energy at most 8π−delta has a representative under the Möbius action, for which the induced metric and a conformal metric of constant (…
The paper proves a Hawking-type singularity theorem using worldvolume quantum strong energy inequalities.
problem Improving classical singularity theorems with weakened energy conditions.
method Integral Ricci curvature bounds based on worldvolume quantum strong energy inequalities.
result Past geodesic incompleteness proven in cosmological scenarios.
Bayesian model for energy-efficient EV navigation.
problem Limited battery capacity in electric vehicles.
method Bayesian modeling and online learning framework with exploration strategies.
result Established rigorous regret bounds for Thompson Sampling.
Using Perelman's results on Kahler Ricci flow, we prove that the K energy is bounded from below if and only if the F functional is bounded from below in the canonical Kahler class.
The fundamental properties of J-holomorphic maps depend on two inequalities: The gradient inequality gives a pointwise bound on the differential of a J-holomorphic map in terms of its energy. The cylinder inequality stipulates and quantifies the exponential decay of energy along cylinders of small total energy. We …
Stable solutions to a specific equation are one-dimensional.
problem Stability and dimensionality of solutions to the Allen-Cahn equation.
method Analysis of stable solutions with bounded energy density.
result Stable solutions to the Allen-Cahn equation are one-dimensional.
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.
Optimizes energy efficiency in wireless sensor networks with limited information.
problem Maximizing energy efficiency in energy harvesting wireless sensor networks with limited channel state information.
method Modeling as a Multi-Armed Bandits problem and developing an Upper Confidence Bound algorithm.
result Significant gains in energy efficiency compared to benchmark schemes.
New ε-harmonic maps of low degree are rigid under certain energy bounds.
problem Understanding the rigidity of ε-harmonic maps of low degree. method Analysis of ε-harmonic maps and their critical points. result Non-trivial ε-harmonic maps of degree zero exist with energy above 8π. Study on bending energy of surfaces with curvature concentration, deriving new lower bounds.
problem Analyzing the Willmore energy of surfaces with curvature concentration.
method Using isoperimetric inequalities and framed loops, derive new lower bounds for the bending energy.
result Optimal blowup rates of the Willmore energy when curvature is concentrated.
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.
The study extends calibrated geometry to smooth maps and finds energy bounds.
problem Finding energy bounds for smooth maps between Riemannian manifolds.
method Generalizing calibrated submanifolds to smooth maps and applying to energy functional.
result Lower bounds to the energy of smooth maps in homotopy classes.
Proves convergence of gradient Ricci shrinkers with uniform bounds.
problem Compactness and energy concentration in gradient Ricci shrinkers.
method Bubble-tree convergence and local energy analysis.
result No energy concentrates in neck regions, leading to a local diffeomorphism finiteness theorem.
Bayesian model for energy consumption helps electric vehicles navigate efficiently.
problem Limited battery capacity in electric vehicles makes energy efficient navigation challenging.
method Developed an online learning framework using Bayesian models and exploration strategies like Thompson Sampling.
result Established rigorous regret bounds for Thompson Sampling in both single-agent and multi-agent settings.
We address the problem of computing approximate marginals in Gaussian probabilistic models by using mean field and fractional Bethe approximations. As an extension of Welling and Teh (2001), we define the Gaussian fractional Bethe free energy in terms of the moment parameters of the approximate marginals and derive an …
Optimizes energy of mappings from complex projective spaces.
problem Finding energy-minimizing mappings between complex projective spaces and Riemannian manifolds.
method Establishes optimal lower bounds for energy functionals and characterizes optimal mappings.
result Optimal lower bounds for energy functionals are characterized for mappings from real and complex projective spaces.
We prove existence and a.e. regularity of an area minimizing soap film with a bound on energy spanning a given Jordan curve in R^3. The energy of a film is defined to be the sum of its surface area and the length of its singular branched set. The class of surfaces over which area is minimized includes images of disks, …
Study quantizes energy distribution in inhomogeneous phase transitions.
problem Quantifying energy distribution in inhomogeneous Allen-Cahn phase transitions.
method Analysis of varifolds and convergence of integer rectifiable varifolds.
result Equidistribution of energy between Dirichlet and Potential energy in phase field limit.
Geodesics found in a metric space of m-subharmonic functions.
problem Metric structure on energy class of m-subharmonic functions.
method Inspired by Kähler geometry, introduced a metric structure and studied metric convergence.
result Geodesics constructed in a subspace of the complete metric space.
We consider a vector bundle E over a compact Riemannian manifold M=Mn,n≥4,and A is a Yang-Mills connection with L2n curvature FA on E.Then we prove a mean value inequality for the density ∣FA∣2n.This inequality give rise to an energy concentrate principle for seque…
Study gradient flow of phase transitions with fixed contact angle.
problem Understanding phase transitions with fixed contact angle.
method Gradient flow of the Allen-Cahn equation with fixed boundary contact angle.
result Established interior and boundary convergence properties for solutions and energy measures.
The paper proves a finite diffeomorphism theorem for manifolds with lower Ricci curvature and bounded energy.
problem Proving a finite number of diffeomorphism types for manifolds with specific curvature and energy bounds.
method Analyzing the space of closed manifolds with lower Ricci curvature, volume, diameter, and energy bounds.
result The space of manifolds has at most a finite number of diffeomorphism types.
We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. …
We address the problem of computing approximate marginals in Gaussian probabilistic models by using mean field and fractional Bethe approximations. We define the Gaussian fractional Bethe free energy in terms of the moment parameters of the approximate marginals, derive a lower and an upper bound on the fractional Beth…