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.
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 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.
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.
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. 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.
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.
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. 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.
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. 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.
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.
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. 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.
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.
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.
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. 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.
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.
The paper classifies stable free boundary minimal hypersurfaces outside a ball.
problem Classifying stable free boundary minimal hypersurfaces outside a ball.
method Proved a Bôcher type result for positive Jacobi functions and used a symmetrization procedure.
result Stable free boundary minimal hypersurfaces outside a ball are catenoidal.
New method proves rigidity of minimal hypersurfaces in curved 4-manifolds.
problem Proving rigidity of minimal hypersurfaces in curved 4-manifolds.
method Combining nonnegative 2-intermediate Ricci curvature and strict positivity of scalar curvature, extending Chodosh-Li-Stryker method.
result Rigidity of two-sided free boundary stable minimal hypersurfaces in 4-manifolds with bounded geometry and weakly convex boundary.
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…
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.
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 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.
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…
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.
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.
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.
Uniqueness proven for stable hypersurface tangent cones.
problem Stability and uniqueness of tangent cones for stable hypersurfaces.
method Analysis of isolated singularities and tangent cones of stable minimal hypersurfaces.
result Uniqueness of tangent cones with integer multiplicities.
Paper defines minimal hypersurfaces in Euclidean and Riemannian spaces.
problem Characterizing minimal hypersurfaces in different spaces.
method Analyzes conditions for hypersurfaces to be minimal or stable.
result Minimal and stable hypersurfaces are hyperplanes in Euclidean spaces and totally geodesic submanifolds in Riemannian manifolds.
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.
Study proves rigidity of certain hypersurfaces in 5- and 6-manifolds.
problem Proving rigidity of stable minimal hypersurfaces in 5- and 6-manifolds.
method Nonnegative 3-intermediate Ricci curvature combined with uniformly positive k-triRic curvature.
result No complete noncompact stable minimal hypersurface in a closed 5-dimensional manifold with positive sectional curvature.
Paper proves no specific CMC hypersurfaces in hyperbolic space.
problem Existence of complete noncompact CMC hypersurfaces in hyperbolic space.
method Uses μ-bubbles developed by Gromov and Chodosh-Li-Stryker.
result No complete noncompact CMC hypersurfaces with mean curvature > 1 in hyperbolic space.
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.
Proves intersection properties of minimal hypersurfaces in various spaces.
problem Intersection properties of minimal hypersurfaces in different geometric settings.
method Two approaches: classifications of stable minimal hypersurfaces and conformal change with comparison geometry.
result Intersection properties for minimal hypersurfaces in specific geometric settings, including free boundary minimal hypersurfaces.
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.
New definition of stable (r+1)-th capillary hypersurfaces proposed.
problem Stability of capillary hypersurfaces in different geometries.
method Defining stable (r+1)-th capillary hypersurfaces as smooth local minimizers of a new energy functional under volume-preserving and contact angle-preserving variations. result Generalization of stability results to (r+1)-th capillary hypersurfaces. 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.
Study on stable minimal hypersurfaces under Ricci curvature constraints.
problem Stability of weighted minimal hypersurfaces under Ricci curvature bounds.
method Derive geometric consequences and prove a Schoen-Yau type criterion.
result Structure theorem for three-dimensional weighted manifolds of non-negative Ricci curvature.
Let (Y,g) be a compact Riemannian manifold of positive scalar curvature (psc). It is well-known, due to Schoen-Yau, that any closed stable minimal hypersurface of Y also admits a psc-metric. We establish an analogous result for stable minimal hypersurfaces with free boundary. Furthermore, we combine this result wit…
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. We study the global behavior of (weakly) stable constant mean curvature hypersurfaces in general Riemannian manifolds. By using harmonic function theory, we prove some one-end theorems which are new even for constant mean curvature hypersurfaces in space forms. In particular, a complete oriented weakly stable minimal h…
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 …
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…