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 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.
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
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.
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.
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.
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.
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.
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.
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.
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.
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…
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.
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.
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.
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.
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.
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 $\BHH$.
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…
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.
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.
We study an area minimization problem for spacelike zero mean curvature surfaces in four dimensional Lorentz-Minkowski space. The areas of these surfaces are compared of with the areas of certain marginally trapped surfaces having the same boundary values.
We show that in the setting of proper metric spaces one obtains a solution of the classical two-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area (in the sense of convex geometry) has been chosen appropriately. We prove the quasi-convexity of this new definition o…
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.
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 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
Study uses renormalized area to determine metric expansion from minimal surfaces.
problem Recovering the expansion of asymptotically hyperbolic metrics from minimal surfaces.
method Uses renormalized area functional on minimal submanifolds to recover metric expansion.
result Proves rigidity for log-analytic metrics and determines obstruction tensor.
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…
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. The paper shows how heat flow approximates area functional on specific geometric spaces.
problem Approximating the area functional on $\RCD(K,\infty)$ spaces.
method Using heat flow and properties of $\RCD(K,\infty)$ spaces.
result The area functional coincides with its relaxation in $\RCD(K,\infty)$ spaces.
Flat stable minimal hypersurfaces in 5 or 6D are always flat.
problem Characterizing stable minimal hypersurfaces in high-dimensional spaces.
method Proving stability of anisotropic minimal hypersurfaces in R5 and R6 under certain smoothness conditions. result Complete, stable anisotropic minimal hypersurfaces in R5 or R6 are flat if the anisotropic area functional is C4-close to the area functional.