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.
The paper extends stable minimal hypersurface results to δ-stable hypersurfaces in R^(n+1).
problem Extending stable minimal hypersurface results to δ-stable hypersurfaces.
method Regularity and compactness theorems for immersed δ-stable minimal hypersurfaces in R^(n+1).
result Optimal range of δ for δ-stable hypersurfaces.
Flat stable minimal hypersurfaces in 5D are always flat.
problem Characterizing stable minimal hypersurfaces in higher dimensions.
method Analyzing properties of stable minimal hypersurfaces in \(\mathbf{R}^5\).
result Complete, two-sided stable minimal hypersurfaces in \(\mathbf{R}^5\) are flat.
Sharp upper bound found for stable minimal surfaces.
problem Bounding the diameter of stable minimal surfaces.
method Analyzing three-dimensional Riemannian manifolds with specific curvature conditions.
result Sharp upper bound for the diameter of stable minimal surfaces.
Flat stable minimal hypersurfaces found in 6D space.
problem Existence of stable minimal hypersurfaces in R6. method Adapted Chodosh-Li-Minter-Stryker strategy with volume estimates.
result Complete, two-sided stable minimal hypersurfaces in R6 are flat. The study restricts stable minimal immersions in product spaces to specific configurations.
problem Prohibiting stable minimal immersions in certain product spaces.
method Analyzing stable minimal immersions in products of complex, quaternionic, and octonionic projective spaces.
result The only stable compact minimal immersions in the product of a quaternionic projective space with any other Riemannian manifold are the products of quaternionic projective subspaces with compact stable minimal immersions of the second manifold.
Flat minimal hypersurfaces in 4D space are always flat.
problem Understanding stable minimal hypersurfaces in 4D space.
method Proving stability and completeness lead to flatness.
result Complete, stable minimal hypersurfaces in 4D are flat.
New findings on stable minimal hypersurfaces in curved 4-manifolds.
problem Nonexistence of complete stable minimal hypersurfaces in positively curved 4-manifolds.
method Combination of non-negative sectional curvature and strict positivity of scalar curvature.
result Rigidity of complete stable minimal hypersurfaces in 4-manifolds with positive curvature.
Proof that stable minimal surfaces in 3D are flat.
problem Classification of stable minimal surfaces in R3. method Index theory for Dirac operators on twisted spinor bundles.
result Every complete two-sided stable minimal surface in R3 is flat. Estimates for stable minimal hypersurfaces in Euclidean space.
problem Deriving estimates for stable minimal hypersurfaces.
method Derivation of estimates related to Bernstein theorems.
result Indicates limitations of existing methods for n=6. 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. Characterizes stable minimal capillary surfaces with specific angles.
problem Understanding stable minimal capillary surfaces with near 0 or π angles. method Curvature estimates for sequences of weakly stable minimal capillary surfaces.
result Characterization of tangential limits of stable minimal capillary surfaces.
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.
Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the product of two spheres is obtained. Also, it is proved that the only stable comp…
The study examines stable minimal hypersurfaces in higher dimensions.
problem Characterizing stable minimal hypersurfaces in Rn+1. method Analyzing volume growth and stability conditions.
result Conditions for complete two-sided δ-stable minimal hypersurfaces to be the hyperplane. 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.
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.
Simple proof that stable minimal hypersurfaces in R^4 are hyperplanes.
problem Proving stable minimal hypersurfaces in R^4 are hyperplanes.
method Using spectral Ricci curvature bounds and Green kernel estimates.
result Complete, two-sided stable minimal hypersurfaces in R^4 are hyperplanes.
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.
Proves rigidity of stable minimal hypersurfaces in low dimensions.
problem Rigidity of stable minimal hypersurfaces in low dimensions.
method Conformal method inspired by Fischer-Colbrie.
result No stable minimal hypersurfaces in positively curved closed Riemannian manifolds when dimension is 5 or less.
The study of stable and index compact minimal submanifolds in Berger spheres.
problem Stability and index of compact minimal submanifolds in Berger spheres.
method Analyzing stability and index properties of compact minimal submanifolds in Berger spheres.
result Stable compact minimal submanifolds exist in Berger spheres for specific values of τ, and their classification is provided.
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. 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.
The paper proves vanishing theorems for harmonic forms and spinors on stable minimal hypersurfaces.
problem Vanishing theorems for harmonic forms and spinors on stable minimal hypersurfaces.
method Positive curvature assumptions on the ambient manifold.
result Vanishing of L2-harmonic forms and spinors on stable minimal hypersurfaces. New stable minimal hypersurfaces found in 4-manifolds, proving topology results.
problem Finding stable minimal hypersurfaces with specific topologies in 4-manifolds.
method Geometric measure theory and 4-manifold topology techniques.
result Existence of stable minimal hypersurfaces diffeomorphic to S3 or S2imesS1. The paper explores stable surfaces in Einstein-Maxwell theory, proving mass bounds and nonexistence results.
problem Exploring stable surfaces in static Einstein-Maxwell space-time.
method Using mean-stable surfaces theory to prove properties of lapse functions and mass bounds.
result Proves ADM mass is bounded by Hawking quasi-local mass.
Paper studies curvature of stable surfaces meeting at a common boundary.
problem Stable multiple junction surfaces and their curvature estimates.
method Derived Lp estimate of curvature for stable multiple junction surfaces. result Bernstein Theorem holds for stable multiple junction surfaces in certain cases.
Strictly stable Allen-Cahn hypersurfaces have multiplicity one.
problem Understanding the multiplicity of stable hypersurfaces in Allen-Cahn equations.
method Analyzing strictly stable components without variational assumptions.
result Strictly stable components occur with multiplicity one.
In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface M in the hyperbolic space which has finite L2-norm of the second fundamental form on M. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stabl…
Local minimality proven for stable free-boundary minimal hypersurfaces.
problem Proving local minimality for stable free-boundary minimal hypersurfaces.
method Using relative current setting and strict stability, proving local minimality among relative cycles.
result Local minimality of stable free-boundary minimal hypersurfaces in a small tubular neighborhood.
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…
We provide a new topological obstruction for complete stable minimal hypersurfaces in R^n. For n≥4, we prove that any complete orientable stable hypersurfaces in R^n has only one end. This follows from a more general analytic theorem we prove in the paper.
We construct examples of spherical space forms (S3/Γ,g) with positive scalar curvature and containing no stable embedded minimal surfaces, such that the following happens along the Ricci flow starting at (S3/Γ,g): a stable embedded minimal two-sphere appears and a non-trivial singularity occurs. We also give in d…
The paper proves criticality criteria and spectral splitting theorems for manifolds with Ricci bounds.
problem Understanding criticality and splitting theorems for manifolds with spectral Ricci bounds.
method Proving criticality criteria and spectral splitting theorems for manifolds with more than one end and spectral Ricci bounds.
result New insights into Li-Wang's theory and applications to stable and δ-stable minimal hypersurfaces.
Minimal submanifolds are stable in certain conformal spheres.
problem Stability of minimal submanifolds in conformal spheres.
method Analyzing n-dimensional Riemannian spheres with specific curvature conditions. result Closed stable minimal submanifolds are not found in δ-pinched conformal spheres. 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 note we show that Hamiltonian stable minimal Lagrangian submanifolds of projective space need not have parallel second fundamental form.
Optimal regularity theory for stable minimal hypersurfaces with small singular set.
problem Optimal regularity of stable minimal hypersurfaces with small singular set.
method Analysis of stable minimal hypersurfaces in a specific domain with small singular set.
result Optimal size assumption on the non-immersed singular set guarantees optimal regularity.
Flat minimal hypersurfaces found in wedge-shaped domains.
problem Finding minimal surfaces in wedge-shaped domains.
method Proving stability and flatness of C1,1-to-edge minimal hypersurfaces. result Stable minimal hypersurfaces are flat in wedge-shaped domains.
The study proves stable minimal immersions in positively curved manifolds are totally geodesic.
problem Proving stable minimal immersions in positively curved manifolds are totally geodesic.
method Formulating stable Bernstein type theorems in certain positively curved ambient manifolds.
result Proves stable minimal immersions in positively curved manifolds are totally geodesic.
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 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.
We prove that a stable minimal hypersurface of an open ball having a singular set of locally finite codimension 2 Hausdorff measure which is weakly close to a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior. Applications including a compactness theorem for a class of immersed stable minimal h…
We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently smal…
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…
The paper constructs stable minimal hypersurfaces with specific singularities.
problem Creating minimal hypersurfaces with controlled singularities.
method Constructing hypersurfaces with a given singular set in a modified Euclidean space.
result Embedded minimal hypersurfaces with stable properties and specified singularities.
Anisotropic min-max theory constructs stable minimal surfaces in 3-manifolds.
problem Constructing stable anisotropic minimal surfaces in 3-manifolds.
method Anisotropic min-max theory, removable singularity theorems.
result Constructs stable anisotropic minimal surfaces in 3-manifolds without singularities.
Minimal Lagrangians in certain curved spaces are stable under specific flows.
problem Stability of minimal Lagrangians in Kähler-Einstein manifolds of non-positive curvature.
method Proved stability under Lagrangian mean curvature flow.
result Equivalence between linear and dynamical stability for C1-close Lagrangians.