We consider -dimensional integer rectifiable currents which are almost area minimizing and show that their tangent cones are everywhere unique. Our argument unifies a few uniqueness theorems of the same flavor, which are all obtained by a suitable modification of White's original theorem for area minimizing currents…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,291 papers · 148 categories
Trend · papers per month
48 results for “almost area-minimizing currents”
We analyze the asymptotic behavior of a -dimensional integral current which is almost minimizing in a suitable sense at a singular point. Our analysis is the second half of an argument which shows the discreteness of the singular set for the following three classes of -dimensional currents: area minimizing in Rie…
We construct Lipschitz -valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of -dimensiona…
We construct a branched center manifold in a neighborhood of a singular point of a -dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of -dimensional curren…
(Log-)epiperimetric inequality and regularity over smooth cones for almost Area-Minimizing currentsmath.AP
New epiperimetric inequality for cones with improved regularity results.
problem Regularity of almost area-minimizing currents at singular points.
method Flowing in radial direction for cones with isolated singularities.
result New ε-regularity result for almost area-minimizing currents.
Unique tangent cones found for boundary points of 2D almost-minimizing currents.
problem Characterizing boundary points of two-dimensional almost-minimizing currents.
method Combining epiperimetric inequality and almost-monotonicity formula.
result Tangent cones at singular boundary points are unique.
This is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents. Here we perform the second main step in the analysis of the singularities, namely the construction of a center manifold, i.e. an approximate average of the sheets of an almost flat area minimizing cu…
Study on singularities in area-minimizing currents, proving unique tangent cones and rectifiability.
problem Understanding singularities in area-minimizing currents.
method Fine excess decay theorems and almost monotonicity of a frequency function.
result Unique tangent cones and countably -rectifiable singular set.
Paper proves existence of area-minimizing submanifolds on almost any manifold with fractal singular sets.
problem Existence of area-minimizing submanifolds with fractal singular sets.
method Constructing and proving existence on almost any smooth manifold.
result Existence of area-minimizing submanifolds with fractal singular sets on almost any smooth manifold.
Proves unique continuation for area minimizing currents.
problem Ensuring area minimizing currents match minimal surfaces.
method Analyzes infinite order contact between currents and minimal surfaces.
result Currents and minimal surfaces coincide in a neighborhood.
Allard-type regularity theory for area minimizing currents at boundaries with arbitrary multiplicitymath.AP
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.
New examples show flat singular sets can be arbitrarily complex.
problem Understanding the structure of singular sets in almost-minimizing currents.
method Construction of specific examples of area almost-minimizing currents.
result Flat singular sets can contain any closed empty interior subset of a plane.
Constructs area-minimizing submanifolds with fractal singularities.
problem Area-minimizing submanifolds with fractal singular sets.
method Integral currents, mod v currents, stable stationary varifolds.
result Sharp dimensionwise solution to Almgren's conjecture.
Proves interior singular set dimension for area-minimizing currents in smooth submanifolds.
problem Interior singular set dimension of area-minimizing currents.
method Analyzes area-minimizing currents within a -submanifold.
result Interior singular set dimension cannot exceed .
This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren's partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis deducing the regularity of area minimizing currents from that of Dir-minimizing multip…
Partial boundary regularity for area-minimizing currents at tangential boundary points.
problem Boundary regularity of co-dimension one area-minimizing currents.
method Proof closely follows Hardt and Simon's boundary regularity result.
result Partial regularity of tangent cones, uniqueness of tangent cone.
Study on flat singularities of area-minimizing currents in codimension one.
problem Understanding flat singularities of area-minimizing currents in codimension one.
method Analyzing the structure of two-dimensional mod(q) area-minimizing currents near flat singularities.
result Currents are -perturbations of radially homogeneous special multiple-valued functions.
Homologically area-minimizing surfaces mod have at worst codimension 2 singular sets asymptoticallymath.DG
New results show area-minimizing surfaces have fewer singularities than expected.
problem Understanding the singularities of area-minimizing surfaces in homology classes.
method Sharp regularity theorem for area-minimizing currents in finite coefficient homology.
result For large , area-minimizing mod currents are integral currents with a singular set of codimension at least 2.
Study proves uniqueness of tangent cones for area-minimizing currents in higher codimensions.
problem Understanding the fine structure of singular points in area-minimizing currents.
method Analysis of tangent cones and application of previous work.
result Uniqueness of tangent cones at -a.e. points in the support of area-minimizing currents.
Rectifies flat singular points for area-minimizing currents.
problem Understanding singularities of area-minimizing currents.
method Analyzes countably -rectifiable singular points with flat tangent cones.
result The set of singular density- points is countably -rectifiable and has finite upper Minkowski content.
Upper bound on singular set dimension for area-minimizing currents.
problem Bounding the dimension of singular points in area-minimizing currents.
method Using upper Minkowski dimension and properties of blow-up scales.
result Upper Minkowski bound of for the interior singular set.
Study area minimizing currents in conformal cones, solving Dirichlet problems.
problem Area minimizing currents in conformal cones.
method Minimizing problem of area functionals, Dirichlet problem of minimal surface equations.
result Existence and uniqueness of area minimizing currents in a wide class of conformal manifolds.
This a survey on a series of recent papers in collaboration with Emanuele Spadaro on the regularity of area-minimizing currents in codimension higher than .
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.
The study bounds Hausdorff measure of flat singular points in area-minimizing currents.
problem Bounding Hausdorff measure of flat singular points in area-minimizing currents.
method Proving locally finite -dimensional Hausdorff measure and Minkowski content bounds.
result The set of flat singular points has locally finite -dimensional Hausdorff measure.
Study on flat singular points of area-minimizing currents, defining a singularity degree.
problem Understanding the structure of singular points in area-minimizing integral currents.
method Analysis of vanishing sequences of scales around a singular point, defining a singularity degree.
result The singularity degree is independent of the chosen vanishing sequence and has interesting properties.
New metric creates a surface with infinite topology.
problem Constructing a surface with infinite topology.
method Constructing a Riemannian metric and a curve, then finding an area-minimizing surface.
result The area-minimizing surface has infinite topology.
In a series of papers, including the present one, we give a new, shorter proof of Almgren's partial regularity theorem for area minimizing currents in a Riemannian manifold, with a slight improvement on the regularity assumption for the latter. This note establishes a new a priori estimate on the excess measure of an a…
Study area-minimizing currents with specific boundary properties.
problem Understanding the structure of area-minimizing currents with tangentially immersed boundaries.
method Analyzes -dimensional area-minimizing currents with boundary constraints.
result Near the boundary of the current, the structure is either uncontrolled or a union of controlled hypersurfaces.
Rectifies flat singular points of area-minimizing currents with singularity degree > 1.
problem Rectifying flat singular points of area-minimizing currents with singularity degree > 1.
method Subdividing singular points based on singularity degree and proving rectifiability of points with singularity degree > 1.
result The set of points with singularity degree > 1 is (m-2)-rectifiable.
Solves a problem posed by Brezis and Mironescu about least mass of area-minimizing currents.
problem Least mass of area-minimizing currents with a given boundary.
method Demonstrates the value of the least mass and compares it to the infimum of areas of smoothly immersed submanifolds.
result The least mass of area-minimizing currents equals the infimum of areas of smoothly immersed submanifolds with the same boundary.
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 submanifolds with arbitrary boundary multiplicity.
result Tangent cones are unique at density boundary points.
Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.
problem Reduces the dimension of the singular set of area-minimizing hypersurfaces.
method Perturbs a smooth hypersurface to minimize the Minkowski dimension of the singular set.
result The singular set of the perturbed minimizing current has Minkowski dimension less than n-9.
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.
We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…
We introduce and study co-dimension one area-minimizing locally rectifiable currents with tangentially immersed boundary: is locally a finite sum of orientable co-dimension two submanifolds which only intersect tangentially with equal orientation. We show that any such is supported in a s…
Study area minimizing currents in Riemannian manifolds, proving unique structure and decay.
problem Area minimizing currents in Riemannian manifolds with moduli.
method Structural results and uniqueness theorem, inspired by Simon's techniques.
result Uniqueness and decay towards tangent cones for area minimizing currents.
Study on high-codimensional minimal surfaces in hyperbolic space.
problem Understanding high-codimensional minimal surfaces in hyperbolic space.
method Investigating asymptotic behavior and boundary regularity of area-minimizing currents.
result Established boundary regularity results for high-codimensional minimal surfaces near their asymptotic boundaries.
Generic Metrics on Preclude Linearly Stable Singular Tangent Cones of Area-Minimizing Boundariesmath.DG
The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.
problem Preventing stable tangent cones for area-minimizing boundaries under specific metrics.
method Developed a perturbation theorem and used spectral theory and compactness arguments.
result A residual set of metrics on precludes linearly stable tangent cones for area-minimizing boundaries.
Unique solutions found for Plateau problems in smooth and continuous calibrations.
problem Finding unique solutions to the Plateau problem for specific types of currents.
method Boundary regularity theory for area-minimizing currents and unique continuation argument.
result Every compactly supported smoothly or continuously calibrated integral current is the unique solution to the Plateau problem for its boundary data.
This lecture notes are an expanded version of the course given at the ERC-School on Geometric Measure Theory and Real Analysis, held in Pisa, September 30th - October 30th 2013. The lectures aim to explain the main steps of a new proof of the partial regularity of area minimizing integer rectifiable currents in higher …
Analyzes singularities of area minimizing hypersurfaces modulo p, completing the structure analysis.
problem Analyzing singularities of area minimizing hypersurfaces modulo p.
method Combining epiperimetric inequalities, analysis of homogeneous minimizers, and blow-up procedures.
result Completes the structure analysis of area minimizing hypersurfaces modulo p for all cases.
The paper proves generic transversality and regularity for minimal submanifolds and area-minimizing currents.
problem Transversality and regularity of minimal submanifolds and area-minimizing currents.
method Proves transversality and regularity for generic metrics using Baire category and minimization properties.
result Generic metrics ensure transversality and regularity for minimal submanifolds and area-minimizing currents.
It is well known that in compact local Lipschitz neighborhood retracts in Euclidean space flat convergence for integer rectifiable currents amounts just to weak convergence. In the present paper we extend this result to integral currents in complete metric spaces admitting a local cone type inequality. These include in…
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 singular boundary points of is -rectifiable.
The paper proves existence and partial regularity for Legendrian area-minimizing currents.
problem Existence and partial regularity of Legendrian area-minimizing currents.
method Local minimization and application to the Legendrian Plateau problem.
result Existence and partial regularity of solutions to the Legendrian Plateau problem.
The paper proves prevalent existence and partially determines moduli space of area-minimizing surfaces with fractal singular sets.
problem Existence and moduli space of area-minimizing surfaces with fractal singular sets.
method Proof of prevalent existence, determination of moduli space, refinement of strata.
result Sharp results on moduli space and refinement of strata, showing fractal singularities do not completely dissolve under generic perturbations.
In analogy with Almgren's Theorem for area minimizing currents of general dimension and codimension, we prove that an -dimensional semicalibrated current in a -dimensional manifold, semicalibrated by a -form, has singular set of Hausdorff dimension at most .