Lu's conjecture proven for minimal surfaces in codimension two.
problem Proving Lu's conjecture for minimal surfaces in codimension two.
method Analyzing eigenvalues of Lu's fundamental matrix and squared norm of the second fundamental form.
result Lu's second-gap conjecture holds for minimal surfaces in codimension two.
New minimal surfaces found with spherical curvature lines.
problem Finding minimal surfaces with specific curvature lines.
method Constructing surfaces parametrized by rhombic lattices.
result Found new examples of minimal annuli with free boundaries.
Paper finds invariant solutions for Plateau problem in hyperbolic space.
problem Finding minimal surfaces with specific symmetries in hyperbolic space.
method Proved existence of foliations by invariant minimal surfaces, used to solve the Plateau problem.
result Existence of invariant minimal surfaces solving the asymptotic Plateau problem.
Flat minimal tori counterexamples refute Lu's second-gap conjecture.
problem Lu's second-gap conjecture about minimal surfaces in higher codimensions.
method Constructing closed embedded counterexamples for minimal surfaces.
result Constant values of S+λ2 realized by flat minimal tori are dense in (2,3), refuting the conjecture. Study anisotropic obstacle problem for minimal surfaces using Cahn-Hoffman transform.
problem Anisotropic obstacle problem for minimal surfaces.
method Cahn-Hoffman transform to convert to isotropic problem with generalized Robin boundary condition.
result Optimal regularity of the solution and C1,1 regularity of the free boundary. Compact theorem for minimal surfaces with lower injectivity radius.
problem Proving compactness of minimal surfaces with lower injectivity radius.
method Variant of Choi--Schoen compactness theorem, focusing on injectivity radius.
result Proved compactness theorem for minimal surfaces.
Closed Riemannian 4 or 5-manifolds contain branched immersed closed minimal surfaces.
problem Existence of classical minimal surfaces in 4 and 5-manifolds
method Harmonic replacement method
result Proves the existence of branched immersed closed minimal surfaces in 4 and 5-manifolds
Characterizes kernel of linearization for minimal surfaces problem
problem Characterizing kernel of linearization for minimal surfaces problem
method Show kernel consists of potential fields and TT fields
result In whole-space Euclidean decomposition, kernel consists of potential fields and TT fields
Unique minimal translation surfaces found in Heisenberg group.
problem Classifying minimal translation surfaces in Heisenberg group.
method Complete classification through generating curves analysis.
result Uniqueness of minimal translation surfaces up to isometries.
Minimal surfaces in hyperbolic space have a renormalized area criterion.
problem Minimal surfaces in hyperbolic space
method Renormalized area criterion
result Y must be a totally geodesic disk
Derives Lagrangian for minimal surfaces, proving tangential variations vanish.
problem Variational calculus for minimal surfaces.
method Lagrangian formulation, pullback covariant derivative, geometric argument.
result Tangential variations vanish for minimal surfaces.
The paper proves the existence of certain minimal surfaces in specific manifolds.
problem Existence of minimal surfaces with specific properties in Riemannian manifolds.
method Proof of existence using Morse index, bumpy metrics, and cyclic coverings.
result Connected, immersed Morse index one, closed minimal hypersurfaces with unbounded volumes.
The study proves properties of capillary graphs in half-spaces.
problem Characterizing capillary minimal graphs in half-spaces.
method Analyzing tangent cones and regular set properties.
result Capillary minimal graphs in low dimensions or specific cone conditions are linear.
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.
Unified rigidity theorem for Plateau surfaces in Bn.
problem Rigidity of free-boundary minimal surfaces in Bn. method Analyzing conformal free-boundary minimal immersions of Plateau model cones.
result Every conformal free-boundary minimal immersion of the flat T-cone into Bn is congruent to the flat T-cone. Study on minimal surfaces and their Gauss maps intersecting a specific hypersurface.
problem Understanding intersections of complete minimal surfaces and a Fermat hypersurface.
method Established modified defect relations for the Gauss map of a complete minimal surface.
result Finite total curvature of a complete minimal surface if it intersects a specific hypersurface.
Paper proves Fujimoto's conjecture for even m ≥ 4.
problem Proving Fujimoto's conjecture for even dimensions.
method Using a special planar network in the theory of positive matrices.
result Best possible number for even m ≥ 4 is proven.
Machine learning predicts minimal surfaces for knots, supporting a conjecture.
problem Predicting minimal surfaces for knots in hyperbolic space.
method Physics-Informed Neural Networks (PINNs) to solve minimal surface equation.
result Computational minimal surfaces align with Fine's Conjecture.
Study of degenerating maps to Riemannian manifolds, proving asymptotic limits and existence of minimal cylinders.
problem Blow-up analysis and behavior of maps from degenerating surfaces to compact manifolds.
method Blow-up analysis, generalized energy identities, neck asymptotic limits, evolution system for minimal cylinders.
result Asymptotic limits of necks are geodesics or geodesic-like curves, confirming conjectures.
Constructs continuous families of minimal surfaces and holomorphic immersions.
problem Creating continuous families of minimal surfaces and holomorphic immersions.
method Continuous family of complex structures and conformal minimal immersions.
result Continuous families of proper Jb-conformal minimal immersions and holomorphic null immersions. 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.
Optimal Liouville theorem for minimal disks in any codimension.
problem Characterizing harmonic functions on minimal disks in high-dimensional spaces.
method Analyzing harmonic functions and using Liouville's theorem.
result Optimal Liouville theorem for minimal disks in any codimension.
Analyzes a critical spherical catenoid in hyperbolic space, proving its index and nullity.
problem Proves the strong form of Medvedev's Morse index conjecture for a specific minimal surface.
method Analytic local resolution via reductions and Sturm shooting-count arguments.
result Analytic local resolution of the strong Medvedev conjecture for the critical spherical catenoid.
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.
Improved flatness in annuli using PDE methods.
problem Flatness improvement in annuli.
method PDE-based approach adapted to exterior domains.
result Alternative proof of minimal surface end-structure and asymptotics.
Distance between evolving hypersurfaces is a PDE solution.
problem Tracking the distance between evolving hypersurfaces.
method Elliptic and parabolic PDEs, mean curvature flow.
result Local Harnack inequalities for the distance between evolving hypersurfaces.
Minimal surfaces found in 4D space.
problem Minimal surfaces in 4D space.
method Reduced biharmonic equation to ODEs, excluded non-minimal solutions.
result Biharmonic rotational surfaces in 4D are minimal.
Study on minimal surfaces with closed curvature lines in 3D space.
problem Investigating complete non-orientable minimal surfaces with specific curvature properties.
method Analyzing complete non-orientable minimal surfaces of finite total curvature in R3 with ends foliated by closed lines of curvature. result There are no such surfaces with one end, proving a rigid situation.
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.
Study counts minimal Lagrangians in hyperbolic surfaces with precise growth rate.
problem Counting minimal Lagrangians in hyperbolic surfaces.
method Uses Mirzakhani functions to show growth rate and proves rigidity of area spectrum.
result Number of minimal Lagrangians grows as A6(k−1). The paper proves conditions for minimal surfaces to be holomorphic and stable.
problem Conditions for stable minimal surfaces to be holomorphic.
method Developed a method of constructing variations to prove the equivalence.
result Holomorphicity and stability conditions for minimal surfaces.
Study harmonic surfaces in 3D space, proving superposition principle.
problem Understanding harmonic surfaces in R3. method Using harmonic Enneper immersions and superposition principle.
result Minimal and maximal surfaces can be decomposed into harmonic components.
New theorem for nonlocal minimal surfaces in any dimension.
problem Classical half-space theorems for minimal surfaces.
method Established a new half-space theorem.
result Result holds in any dimension.
Develops explicit formulas for minimal immersions in 5D space.
problem Minimal surface models in 5D space.
method Holomorphic null curves in C^5.
result Concrete formulas for conformal minimal immersions in R^5.
Constructs graphs with singularities in a special space.
problem Creating graphs with specific singularities in a unique space.
method Using Weierstrass representation for minimal surfaces.
result Constructs entire singly periodic graphs with isolated cone-like singularities.
New minimal surfaces in spheres with complex topologies from capillarity.
problem Constructing minimal surfaces in spheres with rich topologies.
method General construction of embedded minimal and constant mean curvature surfaces in Sn using capillary hypersurfaces. result Non-trivial sphere bundles over various base spaces, including Stiefel manifolds and complex quadrics.
Minimal surfaces with complex branching structures constructed using various methods.
problem Construct minimal surfaces with specific branching sets.
method Three constructions using perturbation, barrier methods, and bifurcation theory.
result Minimal surfaces with stratified branching sets constructed as graphs of two-valued functions.
New minimal surfaces found in spheres and hyperbolic spaces.
problem Constructing minimal submanifolds in even-dimensional spheres and hyperbolic spaces.
method Using complex-valued harmonic morphisms.
result Explicit examples of minimal submanifolds in S4 and H4. Paper proves Simon's third gap conjecture for minimal surfaces in spheres.
problem Investigating the third gap problem in Simon's conjecture for minimal surfaces in unit spheres.
method Developed refined third-order Simons-type integral identities and established new lower bounds for curvature terms.
result Obtained positive gap results for the squared norm of the second fundamental form throughout the interval \(\left[\frac{5}{3},\frac{9}{5}
ight]\).
Proves conditions for minimal surfaces in complex hyperbolic space.
problem Conditions for finite energy equivariant minimal surfaces in complex hyperbolic space.
method Analyzes peripheral holonomy and uses Higgs bundles.
result Explicit parametrization and construction of minimal surfaces.
Study properties of solutions with singularities in the negative cone.
problem Properties of solutions with singularities in the negative cone.
method Proved PDE for trace and normal derivatives, showed hypersurface is minimal for k=2.
result Hypersurface is minimal for k=2 and satisfies certain PDE.
Minimal surfaces in spheres constructed from symmetry reductions of ODEs.
problem Construct minimal surfaces in spheres with symmetry.
method Doubling links of free-boundary minimal cones in R^(p+q+3) with bi-orthogonal symmetry.
result Existence of minimal embeddings of S^p × S^q × S^1 in S^(p+q+2).
Generalizes Bryant's correspondence to gauge-theoretical settings.
problem Establishing a correspondence between geometric objects.
method Introduces gauge-theoretical structures and differential systems.
result Generalizes classical and hyperbolic Bryant correspondences.
The study constructs new minimal surfaces with more ramified values than previously known.
problem Understanding minimal surfaces with finite total curvature and specific ramification properties.
method Systematic construction of meromorphic functions on punctured spheres.
result New minimal surfaces with νg=2.5 and Dg=1 on the four-punctured sphere. Two classification theorems for Willmore surfaces in S² × S².
problem Classifying Willmore surfaces in S² × S².
method Analytical proofs for minimal and product type surfaces.
result Classification of Willmore surfaces in S² × S².
Study proves no minimal surfaces can be contained in certain half-spaces or cones.
problem Prohibiting minimal surfaces from certain geometric configurations.
method Analyzes weighted minimal surfaces in R3 with height-dependent weights. result No proper surfaces can be contained in specific half-spaces or cones.
Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.
problem Proves existence and uniqueness of minimal capillary cones with bi-orthogonal symmetry.
method Solves a nonlinear free boundary equation parametrized by the contact angle and uses monotonicity properties.
result Demonstrates that minimizing capillary hypersurfaces can have singularities in codimension 7.
Paper classifies minimal graph transformations into new families of surfaces.
problem Classifying minimal graph transformations into new families of surfaces.
method Formulated and solved a coupled system of partial differential equations, reduced to solving an ordinary differential equation.
result Established rigorous equivalence to a modified problem for a harmonic function, yielding new families of minimal surfaces.
Upper bounds for second Robin eigenvalue on Riemannian surfaces.
problem Bounding the second Robin eigenvalue of Schrödinger operators on Riemannian surfaces.
method Geometric upper bound via Hersch balancing argument on capped surfaces.
result Sharp geometric restrictions for minimal surfaces in negatively curved manifolds.
Lu conjecture proven for minimal 2-spheres and surfaces under certain conditions.
problem Discreteness of constant scalar curvatures of compact minimal submanifolds in unit spheres.
method Refined Simons' first gap theorem and Yau's theorems for high-codimensional submanifolds.
result Lu's conjecture for minimal 2-spheres and surfaces proved under inequality conditions.
Study on high-codimensional minimal surfaces in hyperbolic space.
problem Understanding high-codimensional minimal surfaces in hyperbolic space.
method Investigating asymptotic behavior and boundary regularity of area-minimizing currents.
result Established boundary regularity results for high-codimensional minimal surfaces near their asymptotic boundaries.
The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.
problem Counting minimal surfaces in 3-manifolds with positive Ricci curvature.
method An enumerative min-max theorem linking surface counts to topological properties.
result Every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.
Study on spheres with minimal equators.
problem Classifying metrics on spheres with minimal equators.
method Survey and discussion of related problems.
result Classification of metrics on spheres with minimal equators.
Unified treatment of spacelike and timelike minimal surfaces via Liouville equation.
problem Investigating minimal surfaces in Lorentz-Minkowski space.
method Complex and paracomplex analysis, Möbius-type transformations, pseudo-isometries.
result Unified approach to both spacelike and timelike minimal surfaces.
Study finds surfaces in spherical caps that maximize modified energy.
problem Geometry of surfaces with free boundaries and capillary conditions.
method Monotonicity formulae and energy maximization analysis.
result Capillary minimal surfaces maximize a modified energy in their conformal orbit.
Paper constructs and proves existence of chiral triply-periodic minimal surfaces.
problem Existence of chiral triply-periodic minimal surfaces of genus 4.
method Construction based on dual qtz--qzd nets, existence proof using Weierstrass method on branched torus.
result Family of chiral triply-periodic minimal surfaces of genus 4 constructed and proved to exist.
Study finds conditions for minimal surfaces in noncompact spaces.
problem Existence of minimal surfaces in noncompact spaces with nonnegative scalar curvature.
method Conditions on a 2-sphere and its boundary.
result Existence of a closed minimal surface homologous to the boundary.
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.
Minimal surface doublings have specific index and nullity values.
problem Analyzing the properties of minimal surface doublings.
method Analytical proof of index and nullity values.
result Minimal surface doublings have index 2γ+5=2m+7 and nullity 6. Constructs minimal surfaces over Pitot quadrilaterals using harmonic diffeomorphisms.
problem Construct minimal surfaces over Pitot quadrilaterals.
method Develops a fully explicit framework using harmonic diffeomorphisms and Weierstrass data.
result Constructs a unique minimal surface \(Σ^\diamond\) that maximizes Gaussian curvature.