Study improves boundary smoothness for area-minimizing currents with complex boundaries.
problem Boundary regularity for area-minimizing currents with arbitrary multiplicity.
method Generalization of Allard's boundary regularity theorem.
result Derivation of structural consequences from the generalized theorem.
Bi-Lipschitz proof for 2-varifolds near critical Allard condition.
problem Proving bi-Lipschitz homeomorphism for 2-varifolds near critical Allard condition.
method Analyzing 2-varifolds with critical Allard condition and small mean curvature.
result 2-varifold is bi-Lipschitz homeomorphic to a flat disk.
Paper proves Allard's theorem in Alexandrov spaces.
problem Proving Allard's theorem in non-collapsed Alexandrov spaces.
method Developed an intrinsic proof for Riemannian manifolds, then extended to Alexandrov spaces using approximation theorem.
result Explicit constants for constants in terms of geometric data.
New, shorter proofs for varifolds and flows with improved decay of flatness.
problem Proving regularity theorems for varifolds and flows with bounded first variation and forcing.
method Decay of flatness via weighted monotonicity formulas and viscosity approach.
result Improved proofs with decay of flatness and characterization of blow-ups.
We give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and an…
In this paper, we study the critical case of the Allard regularity theorem. Combining with Reifenberg's topological disk theorem, we get a critical Allard-Reifenberg type regularity theorem. As a main result, we get the topological finiteness for a class of properly immersed surfaces in Rn with finite Willm…
Study on vortex sheet formation in Abelian gauge theories.
problem Understanding vortex sheet formation in Abelian gauge theories.
method Inspired by Allard's regularity theory, constructs approximate solutions and analyzes their perturbations.
result Establishes a geometric framework and regularity theory for the limiting defect set.
Proves smoothness of minimal surfaces near polyhedral boundaries.
problem Smoothness of free-boundary minimal surfaces near polyhedral domains.
method Allard-type regularity theorem for minimal surfaces in convex polyhedra.
result Minimal surfaces are C1,α graphical over a free-boundary plane if close to it. In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are themselves smooth and C2,α-close to the given sub manifold. We show also a version …
Shows smoothness of varifolds with specific boundary angles.
problem Regularity of varifolds with prescribed contact angles.
method Analyzes varifolds with bounded first variation and prescribed contact angles, proving smoothness.
result Support of varifold is a C1,γ hypersurface near the boundary. We adapt the method of Simon [JDG '93] to prove a C1,α-regularity theorem for minimal varifolds which resemble a cone C02 over an equiangular geodesic net. For varifold classes admitting a "no-hole" condition on the singular set, we additionally establish C1,α-regularity near the cone $\bf{C}_0^2 \ti…
Develops new strategy for Hessian estimates in Lagrangian mean curvature equation.
problem Interior Hessian estimates for solutions with prescribed Lipschitz phases.
method Allard-type regularity theorem, geometric measure theory, geometry of Lagrangian graphs, De Giorgi-Nash-Moser iteration.
result Sharp interior Hessian estimates for solutions with critical and supercritical phases.
We analyze a notion of multiple valued sections of a vector bundle over an abstract smooth Riemannian manifold, which was suggested by W. Allard in the unpublished note "Some useful techniques for dealing with multiple valued functions" and generalizes Almgren's Q-valued functions. We study some relevant properties o…
The paper proves regularity for varifolds with bounded anisotropic mean curvature.
problem Regularity of varifolds with bounded anisotropic mean curvature.
method Local anisotropic regularity theorem and touching balls approach.
result Varifolds can be covered by countably many C2-regular submanifolds. The study finds anisotropic minimal surfaces in 3-manifolds with smooth boundaries.
problem Finding smooth anisotropic minimal surfaces in closed 3-manifolds.
method Min-max construction with elliptic integrands, uniform upper bound for density ratios.
result Obtains a smooth anisotropic minimal surface in a closed 3-manifold.
A 3D area-minimizing current in R^5 has a 2-fold essential singularity.
problem Understanding singularities in area-minimizing currents.
method Construction of a specific current with prescribed boundary properties.
result The boundary regularity theory is dimensionally sharp.
Study shows unique tangent cones for area-minimizing currents at boundary points.
problem Uniqueness of tangent cones for area-minimizing currents with arbitrary multiplicity.
method Analysis of area minimizing currents in C2 submanifolds with arbitrary boundary multiplicity. result Tangent cones are unique at density Q/2 boundary points. Paper provides estimates for varifolds with critical mean curvature.
problem Estimating tilt-excess on varifolds with critical mean curvature.
method Generalizing Lipschitz approximation and Sobolev-Poincaré estimates to almost-integral rectifiable varifolds.
result VMO-type estimates for quadratic tilt-excess on varifolds with critical mean curvature.
Let G be a k-step Carnot group. We prove an isoperimetric-type inequality for compact C^2-smooth immersed hypersurfaces with boundary, involving the horizontal mean curvature of the hypersurface. This generalizes an inequality due to Michael and Simon, and Allard, independently. Some applications are discussed.
Sharp generalization of boundary regularity for area minimizing currents with arbitrary multiplicity.
problem Boundary regularity of area minimizing currents with multiplicity.
method Sharp generalization of Allard's boundary regularity theorem to higher multiplicity settings.
result The set of density Q/2 singular boundary points of T is Hm−3-rectifiable. Consider a sequence of minimal varieties M_i in a Riemannian manifold N such that the boundary measures are uniformly bounded on compact sets. Let Z be the set of points at which the areas of the M_i blow up. We prove that Z behaves in some ways like a minimal variety without boundary: in particular, it satisfies the s…
We give two structural conditions on a codimension 1 integral n-varifold with first variation locally summable to an exponent p>n that imply the following: whenever each orientable portion of the C1-embedded part of the varifold (which is non-empty by the Allard regularity theory) is stationarity and the $C^…
The paper proves existence of special surfaces in 3D manifolds with constant curvature.
problem Existence of surfaces with constant anisotropic mean curvature.
method Min-max theory applied to elliptic integrands in 3D Riemannian manifolds.
result Existence of smooth surfaces with at most one singular point.
The study examines stationary integral varifolds near multiplicity 2 planes, proving regularity under specific conditions.
problem Understanding the structure of stationary integral varifolds near multiplicity 2 planes.
method Investigates the structure of varifolds close to planes with multiplicity 2, proving an ε-regularity theorem under certain conditions.
result In B1/2(0), V is represented by the graph of a Lipschitz 2-valued function over P0 with small Lipschitz constant; all tangent cones at singular points are unique and comprised of stationary unions of 4 half-planes. Local minimality proven for stable free-boundary minimal hypersurfaces.
problem Proving local minimality for stable free-boundary minimal hypersurfaces.
method Using relative current setting and strict stability, proving local minimality among relative cycles.
result Local minimality of stable free-boundary minimal hypersurfaces in a small tubular neighborhood.
Corrects errors and completes missing parts in a paper on tangent cones.
problem Incorrect inequalities and missing definitions in a paper on tangent cones.
method Completely rewrote sections 5.1-5.4, providing missing definitions and proofs.
result Corrected inequalities and completed missing parts in the paper.
The paper proves Lp-Sobolev inequalities for minimal submanifolds.
problem Proving Lp-Sobolev inequalities for minimal submanifolds. method Optimal mass transport theory on Euclidean submanifolds.
result Asymptotically sharp and codimension-free Sobolev constant for p≥2. 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.
Study on unique solutions to one-phase free boundary problems.
problem One-phase free boundary problems with singularities.
method Analyzing solutions at singular points and at infinity using one-homogeneous functions.
result Uniqueness of blowups and rigidity results at infinity.
The paper proves unique blow-down for critical points of a Yang-Mills-Higgs functional.
problem Proving uniqueness of blow-down for critical points of a Yang-Mills-Higgs functional.
method Using an Allard-type improvement of flatness to establish co-dimension-two analogue of Savin's theorem.
result Entire critical points have unique blow-down, two-dimensional in ambient dimensions 2-4 or any dimension assuming local minimizer.
The paper establishes structure theory for stable varifolds with applications to area minimising hypersurfaces.
problem Understanding the structure of stable codimension 1 integral varifolds.
method Develops a structure theory for stable codimension 1 stationary integral varifolds with no classical singularities.
result Establishes local structure properties of area minimising currents mod p, including the uniqueness of tangent cones at points with planar tangent cones.
Moser's Bernstein theorem \cite{moser61} says that an entire minimal graph of codimension 1 with bounded slope must be a hyperplane. An analogous result for arbitrary codimension is not true, by an example of Lawson-Osserman. Here, we show that Moser's theorem nevertheless extends to codimension 2, i.e., a minimal p-…
In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q. Among other results, we shall prove an isoperimetric inequality for the case of a C2-smooth compact hypersurface S with - or without - boundary ∂S; S and ∂S are endowed with their homo…
AIR-Net adapts low-rank regularization dynamically for better image completion.
problem Fixed low-rank regularization limits adaptability to different images.
method AIR-Net uses adaptive and implicit regularization parameterized by a dynamic Laplacian matrix.
result AIR-Net enhances implicit regularization and outperforms fixed methods in non-uniform missing data scenarios.
A 6-regular triangulation for hyperbolic plane created.
problem Creating a 6-regular triangulation for hyperbolic plane.
method Constructed a 6-regular geodesic triangulation.
result A 6-regular geodesic triangulation of the hyperbolic plane was successfully created.
Gradient descent implicitly regularizes neural networks by penalizing large loss gradients.
problem How to optimize deep neural networks without explicit regularization.
method Backward error analysis to calculate implicit gradient regularization and demonstrate its effectiveness empirically.
result Implicit gradient regularization biases gradient descent toward flat minima, improving model robustness and test errors.
Choquet regularization improves exploration in RL.
problem Improving exploration in reinforcement learning.
method Introducing Choquet regularizers to measure and manage exploration, reformulating RL problems and deriving explicit solutions.
result Explicit optimal distributions and Choquet regularizers for various exploratory samplers.
The paper explores optimal regularizers for data sources, linking them to star bodies.
problem Understanding optimal regularizers for data sources.
method Investigates optimal regularizers for data distributions using star bodies and dual Brunn-Minkowski theory.
result Identifies optimal regularizers and assesses amenability to convex regularization.
A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In…
In this article, we describe symplectic and complex toric spaces associated to the five regular convex polyhedra. The regular tetrahedron and the cube are rational and simple, the regular octahedron is not simple, the regular dodecahedron is not rational and the regular icosahedron is neither simple nor rational. We re…
The paper studies convergence rates of Tsallis entropic regularization in optimal transport.
problem Optimal transport with regularization.
method Γ-convergence and quantization/shadow arguments.
result Derives convergence rate of Tsallis entropic regularization.
Study uses property elicitation to understand how fairness regularizers affect optimal decisions.
problem Understanding how fairness regularizers change the optimal decision in predictive algorithms.
method Property elicitation to analyze the relationship between loss, regularization, and optimal decision.
result Necessary and sufficient condition for when a property changes with the addition of a regularizer.
Fiedler regularization uses spectral graph theory to improve neural network performance.
problem Improving neural network performance by penalizing weights based on connectivity.
method Uses the Fiedler value of the neural network's graph as a regularization tool, providing theoretical and computational methods.
result Demonstrates Fiedler regularization's effectiveness in improving neural network performance.
Study on convergence rates for optimal transport with regularization.
problem Convergence analysis of divergence-regularized optimal transport.
method Novel methodology using quantization and martingale couplings.
result Sharp rates for various divergences and transport costs.
We establish continuous maximal regularity results for parabolic differential operators acting on sections of tensor bundles on Riemannian manifolds. As an application, we show that solutions to the Yamabe flow instantaneously regularize and become real analytic in space and time. The regularity result is obtained by i…
Improved optimal regularity for harmonic almost complex structures.
problem Establishing optimal regularity for harmonic almost complex structures.
method Quantitative stratification method and rectifiability of singular strata.
result Optimal regularity theory for energy minimizing harmonic almost complex structures.
Dropout is a simple but effective technique for learning in neural networks and other settings. A sound theoretical understanding of dropout is needed to determine when dropout should be applied and how to use it most effectively. In this paper we continue the exploration of dropout as a regularizer pioneered by Wager,…
Regularization plays an important role in generalization of deep neural networks, which are often prone to overfitting with their numerous parameters. L1 and L2 regularizers are common regularization tools in machine learning with their simplicity and effectiveness. However, we observe that imposing strong L1 or L2 reg…