Study on ball widths and minimal submanifolds in space forms.
problem Understanding widths of balls and minimal submanifolds.
method Analyzing the area of equatorial balls and related bounds for minimal submanifolds.
result Lower bounds for the area of free boundary minimal submanifolds.
No stable minimal submanifolds in certain conformal domains.
problem Stability of minimal submanifolds in conformal domains.
method Analyzing sectional curvatures and boundary convexity.
result No compact stable free boundary minimal submanifolds exist.
Free boundary minimal submanifolds with boundaries on concentric spheres
problem Finding minimal submanifolds with boundaries on concentric spheres in Euclidean space
method Using a Steklov problem with an indefinite weight
result Exact Morse index of an m-dimensional flat annulus in an n-dimensional spherical shell Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.
problem Characterize free boundary minimal submanifolds in geodesic balls of hyperbolic and spherical spaces.
method Define and analyze functionals related to critical metrics and spectral indices.
result Critical metrics of defined functionals arise from free boundary minimal immersions in geodesic balls of hyperbolic and spherical spaces.
New functionals defined for free boundary minimal submanifolds in higher dimensions.
problem Characterizing metrics for free boundary minimal submanifolds in geodesic balls.
method Introducing and studying new functionals Θr,i and Ωr,i for higher-dimensional free boundary minimal submanifolds. result Critical metrics for these new functionals are the metrics induced by free boundary minimal immersions.
Paper proves new isoperimetric inequality for minimal submanifolds with free boundary.
problem Proving a relative isoperimetric inequality for minimal submanifolds with free boundary.
method Generalized restricted normal cones, ABP method, Brendle's approach.
result Optimal relative isoperimetric inequality for minimal submanifolds with free boundary on convex surfaces.
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.
We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.
In this paper, we compute the Morse index for a free boundary minimal submanifold from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. As an application, we show that the Morse index…
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…
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…
We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary minimal submanifolds in the ball we show that the boundary volume is reduced up…
Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the energy index and the area index of a free-boundary minimal surface of a Riemannian manifold with boundary, and show that the area index is controlled from above by the area and the topology of the surface. Combining these results with work of…
The purpose of the present paper is to show that the components of the unit normal of any minimal surface with free boundary in the unit ball, are eigenfunctions associated with the eigenvalue −2, for some (new) natural eigenvalue problem for the Jacobi operator; this fact has analytic (spectral) consequences for fre…
We perform a replacement procedure in order to produce a free boundary minimal surface whose area achieves the min-max value over all disk sweepouts of a manifold whose boundary lie in a submanifold. Our result is based on a proof of the convexity of the energy for free boundary harmonic maps and a generalization of Co…
Survey of rigidity and gap phenomena in sphere-ball submanifolds.
problem Rigidity and gap phenomena in submanifolds of sphere and ball.
method Comparison of techniques, pinching and gap theorems, Morse index and topology.
result Free boundary condition in ball forces stronger rigidity than in sphere.
Study on exact Lagrangian submanifolds in unit ball with Legendrian boundary.
problem Exact Lagrangian submanifolds with Legendrian boundary in unit ball.
method Uses Liouville form and boundary unique continuation for differential forms.
result Equatorial n-disk rigidity for compact exact Lagrangian self-similar submanifolds. New methods compare Steklov eigenspaces of free boundary minimal surfaces in balls.
problem Comparing Steklov eigenspaces of free boundary minimal surfaces.
method Developed new methods to compare span of coordinate functions with Steklov eigenspace.
result Proved congruence of free boundary minimal annuli in 3D unit ball.
We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a three dimensional constant curvature submanifold and is totally umbilic. These res…
Let N be a complete, homogeneously regular Riemannian manifold of dimension greater than 2 and let M be a compact submanifold of N. Let Σ be a compact orientable surface with boundary. We show that for any continuous f:(Σ,∂Σ)→(N,M) for which the induced homomorphism on certain fundamental gro…
The paper proves cohomology vanishing for a specific type of minimal submanifolds in a weighted Euclidean ball.
problem Proving cohomology vanishing for free boundary f-minimal submanifolds in Gaussian-weighted Euclidean balls. method The proof uses a weighted Hardy inequality, cancellation in the weighted Weitzenböck curvature operator, and a boundary reduction.
result The space of tangential f-harmonic p-forms vanishes, leading to Hp(M;R)=0. In this paper, we prove that there exists a universal constant C, depending only on positive integers n≥3 and p≤n−1, such that if Mn is a compact free boundary submanifold of dimension n immersed in the Euclidean unit ball Bn+k whose size of the traceless second fundamental form is less…
Let L be a special Lagrangian submanifold of a compact, Calabi-Yau manifold M with boundary lying on the symplectic, codimension 2 submanifold W. It is shown how deformations of L which keep the boundary of L confined to W can be described by an elliptic boundary value problem, and two results about minimal…
Given a domain Ω of Rm+1 and a k-dimensional non-degenerate minimal submanifold K of $\pa Ω$ with 1≤k≤m−1, we prove the existence of a family of embedded constant mean curvature hypersurfaces which as their mean curvature tends to infinity concentrate along K and intersecting ∂Ω …
Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min-max theory which we develop in this article for the free boundary variational problem. In particular, we develop a modified Birkhoff curve shortening process to ac…
Flow maps into minimal surfaces with free boundary.
problem Mapping surfaces into minimal submanifolds with free boundary.
method Combining Plateau-flow and Teichmüller harmonic flow to achieve half-harmonic maps.
result Flow produces a branched minimal immersion as time tends to infinity.
Minimal surfaces' boundary points are always smooth.
problem Boundary regularity of minimal surfaces.
method Proving all boundary points are regular submanifolds.
result Boundary points of minimal surfaces are regular.
The paper proves a Wulff inequality for minimal submanifolds with boundary in Euclidean space.
problem Proving a Wulff inequality for minimal submanifolds with boundary.
method Associating a nonnegative anisotropic weight to the boundary of minimal submanifolds and proving the inequality.
result The Wulff inequality constant is independent of the weights and depends only on m and n. Study submanifolds in hyperbolic space, focusing on their boundary and Laplace operator.
problem Understanding the geometry and regularity of submanifolds in hyperbolic space.
method Analyzing asymptotic geometry and regularity properties near the ideal boundary, computing essential spectra.
result Computed essential spectra of the Laplace operator on certain submanifolds.
The study defines a boundary problem for Lagrangian submanifolds and proves a theorem.
problem Boundary problem for Lagrangian submanifolds.
method Introduced a boundary problem analogous to free boundary hypersurfaces and capillary hypersurfaces.
result Proved a Lagrangian version of Nitsche (or Hopf) type theorem.
Let N be a smooth (n+l)-dimensional Riemannian manifold. We show that if V is an area-stationary union of three or more C1,μ n-dimensional submanifolds-with-boundary Mk⊂N with a common boundary Γ, then Γ is smooth and each Mk is smooth up to Γ (real-analytic in the case N is real-anal…
Study on deforming calibrated submanifolds with boundary constraints.
problem Deforming calibrated submanifolds with boundary constraints in Riemannian manifolds.
method Extends McLean's deformation theory for closed compact submanifolds to include boundaries.
result Results extend McLean's theory to include boundaries, allowing for more flexible submanifold deformations.
Study shows boundary curves of free boundary minimal surfaces are circles.
problem Characterizing boundary curves of free boundary minimal surfaces.
method Holomorphic techniques applied to the unit ball in Euclidean 3-space.
result Intersection curves of free boundary minimal surfaces with the unit sphere are all circles.
Study proves the critical catenoid is the free boundary minimal annulus.
problem Characterizing free boundary minimal surfaces.
method Holomorphic techniques from Fraser and Schoen.
result Proves the critical catenoid is the free boundary minimal annulus.
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. Inspired by a Blaschke's work about analytic convex surfaces, we study {\em shadow boundaries} of Riemannian submanifolds M, which are defined by a parallel vector field along M. Since a shadow boundary is just a closed subset of M, first, we will give a condition that guarantee its smoothness. It depends on the …
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.
The paper proves Lp-Sobolev inequalities for minimal submanifolds.
problem Proving Lp-Sobolev inequalities for minimal submanifolds. method Optimal mass transport theory on Euclidean submanifolds.
result Asymptotically sharp and codimension-free Sobolev constant for p≥2. 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.
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.