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 removes singularities from area-minimizing surfaces.
problem Removal of singularities from area-minimizing surfaces.
method Extending results on area-minimizing cones to handle isolated singularities.
result Isolated singularities can be locally perturbed away on the minimizing side.
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.
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.
Proves cones over minimal embeddings of R-spaces are area-minimizing.
problem Finding area-minimizing properties of cones over minimal embeddings.
method Constructing area-nonincreasing retractions.
result Proves cones over minimal embeddings of R-spaces are area-minimizing.
Newly confirmed area-minimizing properties of Lawson-Osserman cones.
problem Verifying the area-minimizing property of Lawson-Osserman cones.
method Analyzing cones of type (n, p, 2) constructed in [XYZ].
result All Lawson-Osserman cones of type (n, p, 2) are area-minimizing.
Survey on recent work on area-minimizing currents.
problem Regularity of area-minimizing currents in higher codimensions.
method Collaborative research with Emanuele Spadaro.
result Recent findings on the regularity of area-minimizing currents.
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.
Bound on singular points for area-minimizing surfaces.
problem Understanding singular points on area-minimizing surfaces.
method Provided a bound on the measure of singular points in terms of the boundary geometry.
result An a priori bound on the (n-7)-dimensional measure of the singular set.
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.
Study shows area-minimizing submanifolds are mostly smooth except for specific types.
problem Understanding when area-minimizing submanifolds are smooth in mod 2 homology.
method Proved area-minimizing submanifolds are not generically smooth except for geodesics, minimal surfaces, and minimal hypersurfaces.
result Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces.
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 shows area-minimizing submanifolds are mostly smooth except for geodesics, minimal surfaces, and hypersurfaces.
problem Understanding when area-minimizing submanifolds are smooth in mod 2 homology.
method Proving the mod 2 area-minimizing submanifolds are smooth in specific cases and establishing lower bounds on singular sets.
result Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces.
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.
Flat 3D space if has a big area-minimizing surface.
problem Understanding flat 3D space from certain curved surfaces.
method Analyzing asymptotically flat Riemannian manifolds with non-negative scalar curvature.
result An asymptotically flat Riemannian three-manifold with non-negative scalar curvature is isometric to flat \(\mathbb{R}^3\) if it has an unbounded area-minimizing surface.
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.
Solves area minimizing surface problem in H2xR.
problem Finding area minimizing surfaces in H2xR.
method Identifies Jordan curves bounding these surfaces.
result Identifies specific Jordan curves for area minimization.
Study proves conditions for area-minimizing surfaces in a specific space.
problem Existence and non-existence of area-minimizing surfaces in E(−1,τ). method Analyzes sufficient conditions for curves to be the asymptotic boundary of area-minimizing surfaces.
result Presented sufficient conditions for a curve to admit a solution to the asymptotic Plateau problem.
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. 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.
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.
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 $\BHH$.
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.
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.
We show that area minimizing polyhedral surfaces are saddle.
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.
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
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. 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…
New structures reveal properties of area-minimizing hypersurfaces.
problem Understanding geometric and analytic properties of area-minimizing hypersurfaces.
method Development of skin structures on singular area-minimizing hypersurfaces.
result Disclosed previously unapproachable geometric and analytic properties.
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 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.
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.
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…
The paper solves area minimizing problems in special geometric cones.
problem Area minimizing problems in conformal cones.
method Defining NCM condition, proving existence of minimal graphs, solving in specific cones.
result Existence of minimal graphs in mean convex conformal cones.
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.
Static potentials on flat manifolds imply zero boundary values or non-compact area minimizers.
problem Characterizing static potentials on asymptotically flat manifolds.
method Analyzing the properties of static potentials and area minimizing hypersurfaces.
result Asymptotically flat manifolds with static potentials either have zero boundary values or contain non-compact area minimizers.
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. Smooth cones can be approximated by smooth hypersurfaces.
problem Approximating cones with smooth surfaces.
method Hyperbolic unfoldings and Jacobi field operator potential theory.
result Every area minimizing cone can be approximated by smooth hypersurfaces.
The study proves stability inequalities for specific area-minimizing Lawson cones.
problem Stability of area-minimizing Lawson cones.
method Proof of stability inequalities for Lawson cones with specific parameters.
result Extends results to all area-minimizing Lawson cones.
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.
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…