Examples of area-minimizing graphs with low regularity in a specific group.
problem Finding area-minimizing graphs with low regularity in a sub-Finsler Heisenberg group.
method Providing examples of entire area-minimizing horizontal graphs with prescribed singular sets.
result Examples of area-minimizing graphs that are locally Lipschitz but not necessarily smoother.
Study area-minimizing subgraphs in integer lattices.
problem Finding the most efficient subgraphs in integer lattices.
method Formulated functions of bounded variations, classified subgraphs in 2D, proved properties in higher dimensions.
result Classified area-minimizing subgraphs in 2D integer lattice up to isomorphisms.
Every graph can be represented as a singular set of a special surface.
problem Representing any finite graph as the singular set of a compact 3D surface.
method Constructing a calibrated 3-dimensional homologically area minimizing surface with a special Lagrangian form.
result The singular set of the surface is precisely the given graph.
We give new examples of entire area-minimizing t-graphs in the subriemannian Heisenberg group H^1. Most of the examples are locally lipschitz in Euclidean sense. Some regular examples have prescribed singular set consisting of either a horizontal line or a finite number of horizontal halflines extending from a given po…
In this paper we study the area minimizing problem in some kinds of conformal cones. This concept is a generalization of the cones in Eulcidean spaces and the cylinders in product manifolds. We define a non-closed-minimal (NCM) condition for bounded domains. Under this assumption and other necessary conditions we estab…
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 C1,α-perturbations of radially homogeneous special multiple-valued functions. 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 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…
We prove that a strictly stable minimal Ch2 intrinsic graph G is locally area-minimizing, i.e. given any Ch1 graph S with the same boundary, Area(G)<Area(S) unless G=S. As a consequence we show the existence and the uniqueness of C∞ minimal graphs with prescribed small boundary datum…
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…
New forms calibrate minimal graphs in arbitrary dimensions.
problem Calibrating minimal graphs in arbitrary codimension.
method Constructing closed forms from minimal graphs and estimating their comass.
result Conditions ensuring minimal graphs are calibrated and area-minimizing.
Hasse principle applied to area-minimizing submanifolds across different homology types.
problem Understanding the behavior of area-minimizing submanifolds in various homology contexts.
method Extending the Hasse principle from number theory to geometric variational problems.
result Recovering information about area-minimizing submanifolds in integral homology from those in real and mod n homology. Study shows area-minimizing submanifolds are mostly rough, not smooth.
problem Understanding the smoothness of area-minimizing submanifolds.
method Proved non-smoothness by contradiction and established Hausdorff dimension bounds.
result Area-minimizing submanifolds are not generically smooth, resolving a conjecture.
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.
The study finds conditions for area-minimizing cones over submanifolds.
problem Conditions for area-minimizing cones over submanifolds.
method General configuration results for area-minimizing cones.
result Cone over the minimal product of submanifolds and spheres are area-minimizing.
New limits of minimal surface systems have surprising large interior parts.
problem Minimal surface system limits with large interior vertical and non-minimal portions.
method Construction of limits with smallest possible dimension and codimension.
result Limits of minimal surface systems can have surprising large interior parts.
Extends isoparametric foliations and area-minimizing cones in product manifolds.
problem Generalizing isoparametric foliations and area-minimizing cones in SnimesSn. method Analyzes isoparametric foliations and area-minimizing cones, extending known results.
result Extends known area-minimizing cones to codimension-two cases, yielding infinitely many families of area-minimizing subcones.
This paper proves all Pfaffian varieties are area-minimizing except hypersurfaces.
problem Proving area-minimizing property of Pfaffian varieties.
method Analyzing families of minimal real matrix varieties and proving area-minimizing property.
result All Pfaffian varieties are area-minimizing except hypersurfaces.
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.
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.
The paper proves the existence of area-minimizing hypersurfaces in AF manifolds of higher dimensions.
problem Existence of area-minimizing hypersurfaces in AF manifolds with arbitrary dimension and ends.
method Positive mass theorem for AF manifolds with arbitrary ends and global behavior for hypersurfaces in AF manifolds of dimension ≤ 8.
result Existence and behavior of area-minimizing hypersurfaces in AF manifolds of higher dimensions.
Study on existence and structure of P-area surfaces in Heisenberg group.
problem Existence and structure of P-area minimizing surfaces in the Heisenberg group.
method Characterization of existence and structure using an underlying vector field N, proving existence even without satisfying boundary conditions, and applying Barrier condition.
result Existence of P-area minimizing surfaces under certain conditions, providing new understanding of the Heisenberg group.
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.
We extend the results of Hardt and Simon on area-minimizing cones to prove that isolated singularities of stationary one-sided area-minimizing hypersurfaces can be locally perturbed away on the side that they are minimizing.
It has been 40 years since Lawson and Osserman introduced the three minimal cones associated with Dirichlet problems in their 1977 Acta paper [LO77]. The first cone was shown area-minimizing by Harvey and Lawson in the celebrated paper [HL82]. In this paper, we confirm that the other two are also area-minimizing. In fa…
Proves minimality of tensor varieties, generalizing previous results.
problem Finding minimality conditions for tensor varieties.
method Using Lawlor's curvature criterion and deriving Lawlor's ODE.
result Proves minimality of a class of tensor varieties except for one case.
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 C2,α-submanifold. result Interior singular set dimension cannot exceed m−2. We show that area minimizing polyhedral surfaces are saddle.
Solves area-minimizing surface problem for finite curves in H^2xR.
problem Asymptotic Plateau problem for area-minimizing surfaces.
method Complete solution for finite curves in $\BHH$.
result Fairly complete solution for finite curves in $\BHH$.
This paper proves area-minimizing cones over products of Grassmannian manifolds.
problem Proving area-minimizing cones over products of Grassmannian manifolds.
method Using Hermitian orthogonal projectors and carefully computing the Jacobian.
result Cone over minimal products of Grassmannian manifolds are area-minimizing.
Singularities of area minimizing hypersurfaces can be smoothed in dimensions 9 and 10.
problem Singularities of area minimizing hypersurfaces.
method Perturbation of singularities.
result Singularities can be perturbed away in dimensions 9 and 10.
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.
Classifies area-minimizing surfaces in R^4 as algebraic.
problem Classifying entire area-minimizing surfaces in R^4.
method Using quadratic area growth and holomorphic polynomials to cut out surfaces.
result Entire 2-dimensional area-minimizing or stable surfaces in R^4 are algebraic.
We show that every area-minimizing hypercone and every oriented Lawlor cone in [Law91] can be realized as a tangent cone at a point of some homologically area-minimizing singular compact submanifold. In particular this generalizes the result of N. Smale [Sma99].
Entire area-minimizing surfaces of density 2 are planar or quadratic
problem Classifying entire area-minimizing surfaces
method Using density and algebraic properties
result All such surfaces are planar or algebraic
We show the area-minimality property of all homogeneous area-minimizing hypercones in Euclidean spaces (classified by Lawlor) following Lawson's original idea in his 72' Trans. A.M.S. paper "The equivariant Plateau problem and interior regularity". Moreover, each of them enjoys (coflat) calibrations singular only at th…
Rectifies flat singular points for area-minimizing currents.
problem Understanding singularities of area-minimizing currents.
method Analyzes countably (m−2)-rectifiable singular points with flat tangent cones. result The set of singular density-Q points is countably (m−2)-rectifiable and has finite upper Minkowski content. Study area-minimizing hypersurfaces in singular manifolds with nonnegative scalar curvature.
problem Characterize singularities of area-minimizing hypersurfaces in singular ambient manifolds.
method Analyze tangent cones with nonnegative scalar curvature and prove codimension bounds.
result Singular set has codimension at least 3, with an example showing sharpness.
Study shows how certain metrics can be split into warped products.
problem Understanding conditions under which metrics can be split as warped products.
method Investigating warped area-minimizing hypersurfaces and spectral Ricci/scalar curvature bounds.
result Metrics can be locally split as warped products under specific curvature conditions.
Study area-minimizing hypersurfaces in manifolds with controlled curvature.
problem Characterize area-minimizing hypersurfaces in manifolds with Ricci curvature bounds.
method Apply Cheeger-Colding theory and blow-up techniques to analyze hypersurfaces.
result Proves continuity of volume functions and existence of area-minimizing limits.
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…
We show that for a generic nullhomotopic simple closed curve C in the boundary of a compact, orientable, mean convex 3-manifold M with trivial second homology, there is a unique area minimizing disk D embedded in M where the boundary of D is C. We also show that the same is true for absolutely area minimizing surfaces.
We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, w…
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 v, area-minimizing mod v currents are integral currents with a singular set of codimension at least 2. We prove prove a bridge principle at infinity for area-minimizing surfaces in the hyperbolic space H3, and we use it to prove that any open, connected, orientable surface can be properly embedded in H3 as an area-minimizing surface. Moreover, the embedding can be constructed in such a way that t…
In this paper, by constructing area-nonincreasing retractions, we prove area-minimizing properties of some cones over minimal embeddings of R-spaces.
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 Hm−2-a.e. points in the support of area-minimizing currents. We prove the existence of global minimizers of Allen-Cahn equation in dimensions 8 and above. More precisely, given any strictly area-minimizing Lawson's cones, there are global minimizers whose nodal sets are asymptotic to the cones. As a consequence of Jerison-Monneau's program we establish the existence of many co…