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.
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. 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.
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.
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.
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.
Minimal graphs in Heisenberg group are stable and area-minimizing.
problem Existence and uniqueness of minimal graphs in Heisenberg group.
method Proof of stability and area-minimizing property of minimal graphs.
result Existence and uniqueness of smooth minimal graphs with small boundary.
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.
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.
In this paper, by constructing area-nonincreasing retractions, we prove area-minimizing properties of some cones over minimal embeddings of R-spaces.
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.
The study finds large area minimal surfaces in certain manifolds.
problem Existence of minimal surfaces with large area in manifolds.
method Analyzes bumpy closed Riemannian n-manifolds for n between 3 and 7.
result Proves the existence of a sequence of minimal surfaces with unbounded area.
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.
Study minimal annuli in a slab, estimating their area.
problem Estimating the area of minimal annuli in a slab.
method Organized minimal annuli based on winding number, deduced convexity of length function, compared to catenoid waist area.
result Deduced convexity of length function and estimated area of minimal annuli.
Minimal surfaces in hyperbolic space have a renormalized area criterion.
problem Minimal surfaces in hyperbolic space
method Renormalized area criterion
result Y must be a totally geodesic disk
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.
New minimal surfaces grow area very quickly.
problem Understanding minimal surfaces with rapid area growth.
method Examples of minimal immersions in Euclidean space.
result Proper minimal surfaces with rapid area growth found.
New minimal surfaces can have huge area and index.
problem Existence of minimal hypersurfaces with large area and index.
method Analyzing bumpy closed Riemannian manifolds.
result Sequence of minimal hypersurfaces with arbitrarily large area and index.
Minimal area of spun trefoil knot is found in 4D cubical space.
problem Finding the minimum area for a spun trefoil knot in 4D cubical space.
method Defined minimal area of cubical 2-knots, used isotopy to find minimum area for spun trefoil.
result Spun trefoil knot requires a specific minimal area in 4D cubical space.
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.
Minimal surfaces in hyperbolic space have a sharp area bound.
problem Bounding the renormalized area of minimal surfaces.
method Proving an inequality using conformal length of ideal boundary.
result Sharp isoperimetric property of renormalized area.
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.
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 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 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.
Minimal surfaces' area bounds proven equivalent, extending known results.
problem Equivalence of area bounds for minimal surfaces.
method Combining recent breakthroughs, extending known results.
result Equivalence of intrinsic and extrinsic area density bounds for minimal immersions.
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.
Sharp upper bound for minimal graph area in unit ball established.
problem Determining the exact upper limit for the area of minimal graphs intersecting a unit ball.
method Constructing a sequence of minimal graphs via solutions to a Dirichlet problem.
result The areas of constructed minimal graphs tend to the upper bound of 2π. 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.
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.
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 counts minimal surfaces in curved 3D spaces, finding hyperbolic space minimizes area.
problem Counting minimal surfaces in negatively curved 3-manifolds.
method Introduced an asymptotic quantity to count area-minimizing surfaces and showed minimization by hyperbolic metric.
result Hyperbolic metric minimizes the quantity of area-minimizing surfaces in negatively curved 3-manifolds.
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.
Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
problem Calculating the renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
method Derives a Gauss-Bonnet formula for renormalized area in terms of scalar invariants and characterizes minimal hypersurfaces in terms of conformal geometry.
result Derives a formula expressing renormalized area in terms of integrals of scalar Riemannian invariants.
Stationary polyhedral varifolds minimize area in two senses.
problem Minimizing area of polyhedral varifolds.
method Proves minimization of area through specific conditions.
result Stationary polyhedral varifolds minimize area in two senses.
Minimal surfaces in a ball have limited area.
problem Bounding the area of genus zero minimal surfaces in a unit ball.
method Proving an area inequality and showing convergence of saturating sequences.
result The area of each nonflat surface is less than its radial projection, with sharp asymptotic bounds.
New convex programs solve minimal-area problems on Riemann surfaces.
problem Finding the conformal metric of least area with constraints on curve lengths.
method Formulated as a local convex program using calibrations and max flow-min cut theorem.
result Derives new insights and numerical solutions for minimal-area metrics.
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. Sharp area estimates for minimal submanifolds in curved spaces.
problem Estimating the area of minimal submanifolds passing through a specific point.
method Proving sharp area estimates in hyperbolic and spherical spaces.
result Sharp area estimates analogous to Euclidean settings.
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 $\BHH$.
Estimates area and spectrum of stable minimal surfaces in Euclidean and hyperbolic spaces.
problem Estimating the growth of area and spectrum of stable minimal surfaces.
method Elementary argument and stability inequality for Euclidean space; explicit area growth estimate for hyperbolic space; scalar curvature lower bound for spectrum.
result Minimal surfaces in Euclidean space grow like the Euclidean plane, and in hyperbolic space, explicit area growth estimates are derived.
In this paper, we study closed embedded minimal hypersurfaces in a Riemannian (n+1)-manifold (2≤n≤6) that minimize area among such hypersurfaces. We show they exist and arise either by minimization techniques or by min-max methods: they have index at most 1. We apply this to obtain a lower area bound for su…
We show that area minimizing polyhedral surfaces are saddle.
Bounds on the index of free boundary minimal surfaces.
problem Understanding the index of free boundary minimal surfaces.
method Comparison of energy and area indices, combining with previous work.
result Area index is bounded by a linear function of genus and boundary components.