Estimates the growth of Morse index for free boundary minimal hypersurfaces.
problem Estimating the Morse index of free boundary minimal hypersurfaces.
method Adapting Song's method to the free boundary case.
result The Morse index grows linearly with the sum of Betti numbers.
Proves smoothness of minimal surfaces near polyhedral boundaries.
problem Smoothness of free-boundary minimal surfaces near polyhedral domains.
method Allard-type regularity theorem for minimal surfaces in convex polyhedra.
result Minimal surfaces are C1,α graphical over a free-boundary plane if close to it. New minimal discs and annuli found in ellipsoids.
problem Constructing minimal surfaces in ellipsoids.
method Equivariant variational methods.
result At least three distinct embedded free boundary minimal annuli in ellipsoids.
The paper develops a theory for free boundary minimal surfaces with genus at least one.
problem Finding minimal surfaces with specific genus and boundary conditions.
method Using sweepouts of surfaces of genus g≥1 and m≥1 ideal boundary components, the paper constructs a min-max theory for free boundary minimal surfaces.
result The width for the area functional can be achieved by a bubble tree limit of branched genus g free boundary minimal surfaces with nodes.
The study proves the existence of free boundary minimal disks in convex regions.
problem Proving the existence of free boundary minimal disks in convex regions.
method Based on a multiplicity-one theorem for the free boundary Simon-Smith min-max theory.
result Existence of at least three embedded free boundary minimal disks in strictly convex domains with nonnegative Ricci curvature.
New minimal surfaces found in ball with boundary constraints.
problem Finding minimal surfaces with boundary conditions.
method Equivariant differential geometry approach.
result A family of free boundary minimal surfaces in the unit ball.
In this paper we establish a connection between free boundary minimal surfaces in a ball in R3 and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.
The paper studies free boundary minimal surfaces with many boundaries and their convergence to closed minimal surfaces.
problem Sharp isoperimetric inequalities for Steklov eigenvalues on surfaces with many boundary components.
method Maximization of Steklov eigenvalues and convergence analysis of free boundary minimal surfaces.
result Free boundary minimal surfaces converge to closed minimal surfaces in the boundary sphere as the number of boundary components increases.
Constructs minimal annuli with free boundary in hyperbolic 3-space.
problem Finding minimal surfaces with boundary in hyperbolic geometry.
method Constructs families of non-rotational minimal annuli with shared symmetry.
result Bifurcates from hyperbolic catenoids, forming a countable collection.
Study of free boundary minimal Möbius bands in spherical caps.
problem Characterizing minimal surfaces with free boundary in spherical caps.
method Analyzing spectral properties and geometric constraints.
result Proves that any free boundary minimal Möbius band in spherical caps must be intrinsically rotationally symmetric.
In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few years due to the seminal work of Fraser and Schoen on the extremal Steklov eigenvalue problem. We review several different techniques of const…
The study bounds the topology of free boundary minimal surfaces in 3D manifolds.
problem Understanding the topology of free boundary minimal surfaces in compact 3D manifolds.
method Establishing general bounds on the topology via min-max methods and analyzing varifolds.
result The first Betti number is lower semicontinuous in the limit of min-max sequences.
Study ancient solutions to free boundary mean curvature flow in convex manifolds.
problem Understanding ancient solutions to free boundary mean curvature flow in convex manifolds.
method Establish rigidity results and construct foliations to describe ancient solutions.
result Ancient solutions to free boundary mean curvature flow are rigid and exhaust all possibilities under certain conditions.
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.
The critical catenoid is uniquely determined by certain symmetries of its boundary.
problem Uniqueness of free boundary minimal annuli in a half-ball.
method Symmetry analysis and boundary conditions.
result An embedded free boundary minimal annulus with specific symmetries is congruent to the critical catenoid.
The paper proves infinitely many free boundary minimal hypersurfaces in compact manifolds.
problem Existence of free boundary minimal hypersurfaces in compact Riemannian manifolds.
method Adaptations of A. Song's work and Marques-Neves' resolution to Yau's conjecture, combined with Li-Zhou's regularity theorem.
result Proves the existence of infinitely many almost properly embedded free boundary minimal hypersurfaces in compact manifolds.
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…
In this paper we investigate free boundary minimal surfaces in the unit ball in Euclidean 3-space, and by using holomorphic techniques we prove that intersection curves of free boundary minimal surfaces with the unit sphere are all circles.
Researchers found multiple surfaces with same topological and symmetry properties.
problem Determining unique free boundary minimal surfaces from topology and symmetry.
method Provided pairs of non-isometric surfaces with same genus, boundary components, and symmetry group.
result Infinitely many pairs of surfaces with same topology and symmetry group exist.
Study on unique minimal hypersurfaces in rotational domains.
problem Existence of compact free-boundary minimal hypersurfaces in rotational domains.
method Integral identity for compact free-boundary minimal hypersurfaces, applied to rotational domains.
result Existence of minimal hypersurfaces in rotational domains without topological restrictions.
Embedded surfaces in a ball have any genus and connected boundary.
problem Existence of embedded free boundary minimal surfaces with specific properties.
method Min-max techniques applied to the unit ball in R3. result Existence of embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
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.
Study on free boundary minimal hypersurfaces in Schwarzschild space, proving zero Morse index for certain hypersurfaces.
problem Analyzing free boundary minimal hypersurfaces in the Riemannian Schwarzschild space.
method Variational methods and geometric analysis.
result Zero Morse index for certain free boundary rotationally symmetric totally geodesic hypersurfaces in the Riemannian Schwarzschild space.
Two minimal hypersurfaces in a ball intersect in any half-ball.
problem Intersection properties of minimal hypersurfaces in a ball.
method Analyzing the intersection of two minimal hypersurfaces in a unit Euclidean ball.
result Intersection point in any half-ball, strong Frankel property.
In this note we investigate free boundary minimal surfaces in the Euclidean 3-space, and by using holomorphic techniques developed by Fraser and Schoen we prove that the free boundary minimal annulus is the critical catenoid.
The paper constructs free boundary minimal surfaces in product spaces using eigenvalue methods.
problem Constructing free boundary minimal surfaces in product spaces of balls.
method Extremal eigenvalue approach involving mixed Steklov-Neumann eigenvalues.
result No absolute maximum exists for the problem in product spaces.
No free boundary Möbius bands exist in a 3D ball.
problem Proving the non-existence of Möbius bands with free boundaries in a 3D ball.
method Analytical proof based on geometric properties.
result Proves the non-existence of free boundary Möbius bands in the unit three-ball.
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.
We prove surfaces are unknotted with specific properties.
problem Unknottedness of free boundary minimal surfaces and self-shrinkers.
method Introduced concepts of boundary graph and graph at infinity to prove unknottedness.
result Proved surfaces are unknotted with specific properties.
Paper studies minimal surfaces in curved spaces, proving existence and properties.
problem Existence and properties of minimal surfaces with free boundaries.
method Degree theory, Banach manifold construction, Fredholm map analysis.
result Space of partially free boundary minimal half disks is a Banach manifold.
In the Euclidean unit three-ball, we construct compact, embedded, two-sided free boundary minimal surfaces with connected boundary and prescribed high genus, by a gluing construction tripling the equatorial disc. Aside from the equatorial disc itself, these are the first examples in the three-ball of compact free bound…
Study proves rigidity of minimal hypersurfaces in specific manifolds.
problem Proving rigidity of complete free boundary minimal hypersurfaces.
method Warped θ-bubble method, generalizing capillary surfaces. result No complete two-sided stable free boundary immersions in unit ball of R4. Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.
problem Proving multiplicity one for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
method Developed existence and regularity theory for free boundary hypersurfaces with prescribed mean curvature, including Morse index bounds.
result Proved multiplicity one theorem for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
The paper generalizes free boundary min-max theory to equivariant settings.
problem Existence of minimal hypersurfaces with free boundary in manifolds with boundary.
method Free boundary min-max theory generalized to equivariant settings.
result Existence of nontrivial smooth almost properly embedded G-invariant minimal hypersurfaces with free boundary. Study on minimal surfaces with free boundary in a half-space, improving index estimates.
problem Non-existence of index two embedded minimal surfaces with free boundary in a half-space.
method Improved estimates of Neumann and Dirichlet indices, simplified proof of lower bounds.
result Answered Ambrozio et al.'s question and provided new lower bounds.
Minimal surfaces reflect across spheres, proving annulus uniqueness.
problem Uniqueness of free boundary minimal annuli in balls.
method Reflection principle applied to minimal surfaces meeting spheres at 90 degrees.
result Every embedded free boundary minimal annulus in a ball is the critical catenoid.
Paper proves flatness of anisotropic minimal graphs in half-spaces.
problem Anisotropic minimal graphs with free boundaries in half-spaces.
method Proves flatness using linear growth conditions.
result Anisotropic minimal graphs in half-spaces are flat if they have at most one-sided linear growth.
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 finds the optimal metrics for free boundary minimal surfaces in spherical caps.
problem Optimizing metrics for free boundary minimal surfaces in spherical caps.
method Introducing functionals based on eigenvalues of Steklov-type problems and proving maximizers are induced by immersions.
result Maximizing metrics are induced by free boundary minimal immersions in geodesic balls of a round sphere.
Proves existence of mean curvature flow with surgery for free boundary surfaces.
problem Existence of mean curvature flow with free boundary.
method New approach for flows with surgery in free boundary setting.
result Flow converges to stable minimal surfaces without surgeries for large t.
The paper proves smoothness of almost-minimizers' boundaries near the free boundary.
problem Minimizing degenerate area functionals with weighted boundary conditions.
method Epsilon-regularity theorem applied to almost-minimizers.
result Almost-minimizers' boundaries are C1,γ0-smooth, orthogonal to the boundary Ω. The study finds at least 2 free-boundary minimal disks in convex 3-balls.
problem Finding minimal disks in convex 3-balls.
method Combining mean curvature flow, min-max theory, and degree theory.
result Existence of at least 2 free-boundary minimal disks in convex 3-balls for generic metrics.
Constructs minimal surfaces in a 3-ball using PDE gluing.
problem Finding minimal surfaces in a 3-ball with boundary constraints.
method PDE gluing construction of discrete free boundary minimal annuli.
result Discrete family of non-rotational free boundary minimal annuli in a unit 3-ball.
In this paper we prove that a flat free-boundary minimal n-disk, n≥3, in the unit Euclidean ball Bn+1 is the unique compact free boundary minimal hypersurface in the unit Euclidean ball which the squared norm of the second fundamental form is less than either 4n2 or 4∣x∣2(n−2)2. Mor…
Study finds new minimal surfaces in Schwarzschild space.
problem Existence of non-totally geodesic minimal surfaces in Schwarzschild space.
method Family of properly embedded free boundary minimal hypersurfaces of revolution.
result Existence of new minimal surfaces with circular boundaries in Schwarzschild space.
We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-mi…
Paper proves surfaces with specific symmetries have the first Steklov eigenvalue.
problem Proving surfaces with certain symmetries have the first Steklov eigenvalue.
method Analyzing surfaces with reflection planes and genus zero.
result Surfaces with n distinct reflection planes have the first Steklov eigenvalue. Study investigates minimal surfaces in spherical caps, extending previous findings.
problem Characterizing minimal surfaces with free boundaries and capillary conditions in spherical caps.
method Extending previous half-space intersection properties to warped products and capillary minimal surfaces in high codimension.
result Established a dual operation relating free boundary and capillary minimal surfaces.