The study proves that certain stable minimal hypersurfaces must be cylindrical.
problem Characterizing stable minimal hypersurfaces in Euclidean space.
method Analyzing the density at infinity and using stable area minimizing hypercone properties.
result Stable minimal hypersurfaces with specific conditions are cylindrical.
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…
Characterizes harmonic morphisms preserving minimal submanifolds and finds novel area-minimising hypercones.
problem Understanding harmonic morphisms and their relationship to minimal submanifolds.
method Characterization of harmonic morphisms as weakly horizontally conformal maps preserving minimal submanifold equations, derivation of reduction properties for other co-dimensions, application to find novel area-minimising hypercones.
result Novel family of degree 4 area-minimising hypercones in R^m, m≥32.
Paper constructs flows converging to cones and foliations.
problem Understanding mean curvature flow convergence to cones and foliations.
method Constructs a family of mean curvature flows converging to cones and foliations under specific conditions.
result Flow converges to area minimizing, strictly stable hypercone and Hardt-Simon foliation of the cone.
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.
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].
The paper studies stability and minimizing properties of higher codimensional surfaces in Euclidean space.
problem Stability and minimizing properties of higher codimensional surfaces in Euclidean space.
method Analyzes surfaces associated with the weighted area-functional and proves stability and minimization properties under specific conditions.
result Minimal cones with globally flat normal bundles are f-stable, and highly singular determinantal varieties and Pfaffian varieties are f-minimizing. We prove that the density of a topologically nontrivial, area-minimizing hypercone with an isolated singularity must be greater than the square root of 2. The Simons' cones show that this is the best possible constant. If one of the components of the complement of the cone has nontrivial kth homotopy group, we prove a …
In this paper we considerably extend the class of known α-minimizing hypercones using sub-calibration methods. Indeed, the improvement of previous results follows from a careful analysis of special cubic and quartic polynomials.
Construct locally minimizing (1,2)-clusters with prescribed asymptotic geometry.
problem Minimizing clusters with prescribed asymptotic geometry.
method Develop a refined construction using the Hardt-Simon foliation.
result Produce a countably infinite family of distinct locally minimizing clusters asymptotic to a singular area-minimizing hypercone.
Extends a Liouville theorem for stable minimal hypersurfaces.
problem Stable minimal hypersurfaces in cylindrical cones.
method Analyzes the density and foliation of minimal hypersurfaces.
result Shows that stable minimal hypersurfaces are cylindrical.
Hardt-Simon proved that every area-minimizing hypercone C having only an isolated singularity fits into a foliation of Rn+1 by smooth, area-minimizing hypersurfaces asymptotic to C. In this paper we prove that if a stationary n-varifold M in the unit ball $B_1 \subset \mathbb{R}^…
Study compares nodal sets of solutions to the Allen-Cahn equation.
problem Comparing nodal sets of solutions to the Allen-Cahn equation with conical asymptotics.
method Maximum principle for linearized operator on unbounded domains.
result Positive phase uniquely determines the solution and enforces global ordering.
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. 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.
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.
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.
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…
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.
We prove that a strictly stable constant-mean-curvature hypersurface in a smooth manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L^1 neighborhood.
Paper proves stable minimal surfaces in 3D are flat.
problem Understanding stable minimal surfaces in 3D.
method Analyzes quadratic area growth and stability conditions.
result Stable minimal Plateau surfaces in 3D are flat.
Smooths out complex shapes into simpler forms.
problem Transforming complex shapes into simpler, smooth forms.
method Perturbing minimizing hypercones and viscosity mean convex cones into smooth, properly embedded hypersurfaces.
result Properly embedded smooth minimizing hypersurfaces and self-expanders are achieved.
The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.
problem Preventing stable tangent cones for area-minimizing boundaries under specific metrics.
method Developed a perturbation theorem and used spectral theory and compactness arguments.
result A residual set of metrics on Sn+1 precludes linearly stable tangent cones for area-minimizing boundaries. Study estimates hypersurface areas in curved spaces, with applications to spectrum bounds.
problem Estimating areas of stable hypersurfaces in curved spaces.
method Analyzes stable CMC hypersurfaces in Riemannian manifolds with specific curvature conditions.
result Derives upper bounds for the bottom spectrum of hypersurfaces.
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.
Study proves area estimates for stable capillary hypersurfaces with nonpositive Yamabe invariant.
problem Estimating areas of stable capillary hypersurfaces with nonpositive Yamabe invariant.
method Proves area estimates using stable capillary hypersurfaces in Riemannian manifolds.
result Local rigidity result for embedded, J-energy-minimizing hypersurfaces. Minimal surfaces in 4D space are stable if their Gauss map spherical area is less than 2π.
problem Stability of minimal surfaces in 4D space.
method Geometric criteria based on the Gauss map of minimal surfaces in terms of the spherical area.
result Minimal surfaces in 4D space are stable if their Gauss map spherical area is less than 2π.
The study shows properties of stable anisotropic minimal hypersurfaces in 4D space.
problem Characterizing stable anisotropic minimal hypersurfaces in R4. method Analyzing the intrinsic cubic volume growth and interior volume upper bounds for stable anisotropic minimal hypersurfaces.
result Explicit estimates of constants for stable anisotropic minimal hypersurfaces in R4. In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces which satisfy a uniform area bound. Our result is a natural generalization of the celebrated Schoen-Simon-Yau interior curvature estimates up to the free boundary. A direct corollary of our curvature estimates is…
Generic scarring occurs along stable minimal hypersurfaces in 3-7 dimensional manifolds.
problem Understanding scarring behavior of minimal hypersurfaces along stable ones.
method Analyzing a generic metric on a manifold to show scarring of minimal hypersurfaces.
result Closed, embedded minimal hypersurfaces scarring along stable ones, with diverging area and Morse index.
We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain natural geometric asymptotic constraints, in terms of certain associated varifold…
Study improves curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.
problem Curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.
method Iteration argument based on uniform area bound.
result Improved curvature estimate for stable marginally outer trapped hypersurfaces.
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.
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.
Let M be a 3-manifold (possibly with boundary). We show that, for any positive integer g, there exists an open nonempty set of metrics on M for each of which there are stable compact embedded minimal surfaces of genus g with arbitrarily large area. This extends the result of Colding and Minicozzi for g=1.
Develops theory for stable capillary minimal hypersurfaces in half-space.
problem Regularity and compactness of stable capillary minimal hypersurfaces.
method Integral curvature estimate and tilt excess function.
result Generalized Bernstein theorem for stable capillary minimal hypersurfaces.
Curvature estimate for stable free boundary minimal hypersurfaces in wedge-shaped manifolds.
problem Estimating curvature of stable free boundary minimal hypersurfaces in wedge-shaped manifolds.
method Compactness theorem and Schoen-Simon-Yau estimates.
result Curvature estimate for free boundary minimal hypersurfaces in wedge-shaped manifolds.
For a family of spherical minimal catenoids C_a in the hyperbolic 3-space, there exist two constants 0<a_c<a_l such that the following are true: (1) C_a is an unstable minimal surface with index one if a<a_c, (2) C_a is a stable minimal surface if a>=a_c, and (3) C_a is a least area minimal surface in the sense of Meek…
The author proves that there is an open non empty set of metrics on any 3-manifold such that there exists a family of stably embedded minimal 2-spheres whose area is unbounded. This generalizes the work of T. Colding and W. Minicozzi who have shown an analogous result for the torus and B. Dean who showed the positive g…
Paper proves properties of minimal hypersurfaces in specific solitons.
problem Characterizing minimal hypersurfaces in shrinking gradient Ricci solitons.
method Analyzes stable minimal hypersurfaces with specific curvature conditions.
result Minimal hypersurfaces in these solitons have zero second fundamental form and normal Ricci curvature.
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.
The study proves local rigidity of minimal 2-spheres in electrovacuum spacetimes.
problem Proving local rigidity of minimal 2-spheres in electrovacuum spacetimes under certain conditions.
method Analyzing electrovacuum spacetimes and using constraints on charged Hawking mass and area minimization.
result Local rigidity of minimal 2-spheres in electrovacuum spacetimes, with isometric neighborhoods to specific spacetimes.
Compact hypersurfaces minimize area in convex cones with free boundary.
problem Finding compact hypersurfaces minimizing area in convex cones with free boundary.
method Minimizing an anisotropic area functional under a volume constraint.
result Compact hypersurfaces are contained in a Wulff-shape.
The paper studies g-stability of surfaces with boundary and derives area estimates.
problem Investigating g-stability of surfaces with boundary. method Analyzing geometric properties and deriving area estimates.
result Derives area estimates and determines the topology of the surface.
We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals--Cahn--Hilliard energy functionals with perturbation parameter tending to 0 is supported by an embedded smooth stable minimal hypersurface in low dimensions and …
It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. We also prove minim…
We consider surfaces of class C1 in the 3-dimensional sub-Riemannian Heisenberg group H1. Assuming the surface is area-stationary, i.e., a critical point of the sub-Riemannian perimeter under compactly supported variations, we show that its regular part is foliated by horizontal straight lines. In cas…
As discussed in the paper, in a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inquality. Namely, its area must be bounded above by 4π/c, where c>0 is a lower bound on a natural energy momentum term. In this note we cons…