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 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.
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.
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.
Smooth minimizing hypersurfaces in 11D are generic, with singularities in higher dimensions.
problem Finding smooth minimizing hypersurfaces in high dimensions.
method Analyzing the Plateau problem and area minimization in integral homology.
result Smooth minimizing hypersurfaces are generic in 11D, with singularities in higher dimensions.
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.
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.
The study finds generic regularity of minimal hypersurfaces in Riemannian manifolds.
problem Finding regularity of minimal hypersurfaces in Riemannian manifolds.
method Estimate for a one-parameter min-max minimal hypersurface.
result Generic regularity of minimal hypersurfaces in 8-dimensional Riemannian manifolds with positive Ricci curvature.
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.
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.
Study minimal hypersurfaces with bounded area and high Morse index using combinatorial methods.
problem Understanding minimal hypersurfaces with high Morse index and bounded area.
method Combinatorial argument to study Betti numbers and Hausdorff dimension of singular sets.
result Bounds on Betti numbers and Hausdorff measure of singular sets for minimal hypersurfaces.
The paper studies deformations of singular minimal hypersurfaces in dimensions 7 and above.
problem The behavior of singular minimal hypersurfaces in dimensions 7 and above.
method Analyzes the local behavior of minimal hypersurfaces under perturbations and convergence of families of hypersurfaces.
result Existence and smoothness of nearby minimal hypersurfaces under perturbations, uniqueness of homological minimization, and existence of Jacobi fields.
This is the first in a series of papers where we develop new structural elements on singular area minimizing hypersurfaces, the skin structures. They disclose previously unapproachable and largely unexpected geometric and analytic properties of such hypersurfaces.
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.
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 study finds constraints on scalar curvature using maps and potential theory.
problem Largeness constraints in scalar curvature geometry.
method Basic splitting results and potential theory on singular area minimizing hypersurfaces.
result Non-existence of positive scalar curvature metrics on enlargeable manifolds.
Minimal hypersurfaces with cylindrical tangent cones constructed and analyzed.
problem Constructing minimal hypersurfaces with specific geometric properties.
method Constructing minimal hypersurfaces with cylindrical tangent cones and proving unique continuation results.
result Existence and properties of minimal hypersurfaces with cylindrical tangent cones.
Study calculates spectra of minimal hypersurfaces in hyperbolic space.
problem Computing Laplacian spectra of minimal hypersurfaces.
method Analyzes hypersurfaces in hyperbolic space with specific asymptotic data.
result Obtains spectra and extremal properties of the bottom of the spectrum.
We study the intrinsic geometry of area minimizing (and also of almost minimizing) hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. For any such hypersurface we define and construct a so-called S-structure which reveals some unexpected geometric and analytic properties of the …
This is the second in a series of papers where we estab- lish skin structural concepts and results for singular area minimizing hypersurfaces. Here we conformally unfold these spaces to complete Gromov hyperbolic spaces with bounded geometry and we recover their singular set as the Gromov boundary but also as the Marti…
New results on hypersurfaces show no branch points, improving smoothness.
problem Analyzing area minimising hypersurfaces mod p without branch points.
method General analysis of immersed stable minimal hypersurfaces with alternating orientation.
result Area minimising hypersurfaces mod p do not admit immersed branch points.
The study classifies area-maximizing hypersurfaces with singularities and exterior domains.
problem Classifying area-maximizing hypersurfaces with singularities and exterior domains.
method Complete classification for entire area maximizing hypersurfaces with isolated singularities. Construction of an example. Partial result on asymptotic behavior for exterior domains. Solvability of exterior Dirichlet problems.
result Complete classification and partial results on asymptotic behavior for area maximizing hypersurfaces.
This paper solves minimal surface equations near Hardt-Simon foliations.
problem Minimal surfaces near Hardt-Simon foliations.
method Uses gluing methods to construct minimal surfaces.
result Constructs minimal surfaces over Hardt-Simon surfaces and near quadratic cones.
Study proves properties of constant mean curvature hypersurfaces in high-dimensional spaces.
problem Properties of constant mean curvature hypersurfaces in high-dimensional spaces.
method Proves properties of constant mean curvature hypersurfaces using min-max procedure and surgery.
result Every tangent cone at each isolated singularity is area-minimising.
Minimal 7D hypersurfaces degenerate under stability or bounded index constraints.
problem Degeneration of minimal hypersurfaces under stability or bounded index constraints.
method Analysis of sequences of minimal hypersurfaces, parameterization with controlled maps, and topological finiteness results.
result Minimal hypersurfaces can degenerate to singular ones with controlled geometry, topology, and singular set.
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. This is the third in a series of papers on the geometry and analysis of singular area minimizing hypersurfaces. We show how to derive obstruction and structure theories for scalar curvature constraints without imposing dimensional or topological restrictions on the underlying manifold. To this end, we use skin structur…
Low-entropy surfaces can be flowed into spheres and cylinders.
problem Proving mean curvature flow for low-entropy hypersurfaces.
method Low-entropy density drop argument and recent work on hypersurfaces.
result Closed hypersurfaces with entropy ≤ 2 can be flowed into spherical and cylindrical shapes.
Paper proves minimal surfaces near quadratic cones have specific smooth structure.
problem Characterize minimal surfaces near quadratic cones.
method Analyzes n-varifolds in the unit ball close to a minimizing quadratic cone. result Singularities modeled on these cones determine the local structure of nearby minimal surfaces.
Study bounds singular set of minimal hypersurfaces with index control.
problem Estimating singular set size of minimal hypersurfaces.
method Finite index and null singular set conditions on integral varifolds.
result Local measure bounds on singular set and upper Minkowski content.
New proof of Bishop's theorem using soap bubbles with singularities.
problem Proving Bishop's volume comparison theorem for manifolds with Ricci curvature.
method Using isoperimetric hypersurfaces (soap bubbles) with singularities.
result Successfully overcame the challenge of singularities to prove the theorem.
Study quantifies properties of PMC hypersurfaces with area bounds.
problem Understanding the topology and singular set of PMC hypersurfaces.
method Established quantitative topological and singularity properties for PMC hypersurfaces.
result Quantitative bounds on Betti numbers and Minkowski content of singular sets.
Study min-max theory for hypersurfaces with boundary constraints.
problem Finding minimal hypersurfaces with boundary constraints.
method Schoen-Simon-type regularity result for integral varifolds, proving existence of closed hypersurfaces with specific properties.
result Existence of a closed C1,1 hypersurface with codimension ≥7 singular set in the interior. Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.
problem Proves existence and uniqueness of minimal capillary cones with bi-orthogonal symmetry.
method Solves a nonlinear free boundary equation parametrized by the contact angle and uses monotonicity properties.
result Demonstrates that minimizing capillary hypersurfaces can have singularities in codimension 7.
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.
The paper confirms Yau's conjecture for minimal rotational hypersurfaces.
problem Yau's conjecture about the minimal area of certain hypersurfaces.
method Analyzes minimal rotational hypersurfaces to confirm the conjecture.
result The area of compact minimal rotational hypersurfaces is either equal to the unit sphere's area or another specific value.
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. Unique cylindrical tangent cone for Simons' hypersurface found.
problem Uniqueness of cylindrical tangent cones for area-minimizing hypersurfaces.
method Developed a new Lojasiewicz inequality for non-isolated singularities.
result Cylindrical tangent cone for Simons' hypersurface is unique.
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.
In Riemannian manifolds, minimal hypersurfaces with large area exist or have complex structures.
problem Existence of minimal hypersurfaces with arbitrarily large area in closed Riemannian manifolds.
method Almgren-Pitts min-max theory, Marques-Neves ideas, Song's proof of Yau's conjecture, Zhou's resolution of generic multiplicity-one conjecture.
result Existence of minimal hypersurfaces with arbitrarily large area or pathological Cantor set structures in certain manifolds.
Recent work on stable minimal hypersurface singularities.
problem Understanding singularities of stable minimal hypersurfaces.
method Simplifications of technical discussion in previous work.
result Simplified approach to analyzing hypersurface singularities.
We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In particular, we prove that if we are given a sequence of closed minimal hypersurfaces of …
In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a Riemannian manifold (Mn+1,g) of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension 7. We characterize the …
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…
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 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.
The paper proves foliation of area-minimizing hypersurfaces in asymptotically flat manifolds.
problem Proving foliation of area-minimizing hypersurfaces in asymptotically flat manifolds.
method Demonstrates foliation by area-minimizing hypersurfaces, proving the existence of hypersurfaces asymptotic to Cartesian coordinate hyperplanes.
result Verifies a version of the Schoen Conjecture for asymptotically flat manifolds with nonnegative scalar curvature and positive mass.