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.
Study on minimizing singular capillary cones with stability and instability results.
problem Minimizing singular capillary cones with free boundary.
method Stability criterion à la Jerison-Savin, Simons-type inequality for convex, homogeneous, symmetric functions of principal curvatures, boundary condition specific to capillary setting.
result Minimizing cones with non-sign-changing mean curvature are flat in dimensions up to 4, and non-trivial axially symmetric cones are unstable in dimensions up to 6.
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).
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.
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 show that a capillary surface in a solid cone, that is, a surface that has constant mean curvature and the boundary of surface meets the boundary of the cone with a constant angle, is radially graphical if the mean curvature is non-positive with respect to the Gauss map pointing toward the domain bounded by the surf…
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.
We prove that capillary surfaces converge to a specific energy density as the angle approaches zero.
problem Understanding the limiting behavior of capillary surfaces as the angle approaches zero.
method Rigorous limiting analysis and application to curvature estimates.
result Capillary area-density converges to the Weiss energy density as the angle tends to zero.
Improved estimates for singularities in capillary surfaces.
problem Understanding the singularities of minimizing capillary hypersurfaces.
method Improved estimates based on connections to the one-phase Bernoulli problem.
result The singular set is of codimension at least 4, improving for specific angles.
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.
Winterbottom shape minimizes capillary functional under volume constraint.
problem Finding the shape that minimizes capillary functional under volume constraint.
method Proved minimality of Winterbottom shape using anisotropic capillary functional.
result Winterbottom shape is a volume-constraint minimizer of the anisotropic capillary functional.
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.
Paper proves Liouville-type theorems for minimal graphs with capillary boundary.
problem Proves conditions for minimal graphs to be flat over half-spaces with capillary boundaries.
method Uses gradient estimates for mean curvature equation over R+n with capillary boundary condition, adapting maximum principle. result Minimal graphs are flat under specific conditions on growth or boundedness.
Study proves rigidity of capillary surfaces in curved 3D spaces.
problem Proving rigidity of capillary surfaces in curved 3D spaces.
method Local rigidity result for infinitesimally rigid capillary surfaces in Riemannian 3-manifolds with mean convex boundary.
result Bounds on genus, boundary components, and area of compact capillary minimal surfaces with low index.
New varifolds with capillary boundary properties studied.
problem Understanding varifolds with specific boundary conditions.
method Introducing a Radon measure on a Grassmannian bundle as a capillary boundary.
result Structural properties, monotonicity inequality, and integral compactness proved.
Study confirms conjecture: minimal Lagrangian surfaces with Legendrian boundary are rigid.
problem Characterize minimal Lagrangian surfaces with Legendrian capillary boundary.
method Analyzes surfaces in B4 with Legendrian boundary on S3. result Equatorial plane disks and catenoids are the only minimal Lagrangian surfaces with Legendrian capillary boundary.
Proves ε-regularity for capillary surfaces in Riemannian manifolds.
problem Regularity of minimal surfaces with capillary boundary conditions.
method Uniform first variation control and ε-regularity theorems for varifolds.
result Capillary varifolds with bounded mean curvature and close to a capillary half-plane coincide with a C1,α properly embedded hypersurface. Develops slicing method to prove rigidity of scalar curvature on manifolds with boundary.
problem Understanding positive scalar curvature metrics on manifolds with boundary.
method Minimal slicing via capillary hypersurfaces to prove rigidity statements.
result Proves rigidity statement in dimension 4 for specific geometric conditions.
In this paper we prove that a capillary minimal surface outside the unit ball in R3 with one embedded end and finite total curvature must be either part of the plane or part of the catenoid. We also prove that a capillary minimal surface outside the unit ball with one end asymptotic to the end of the Ennep…
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. Paper studies stability of curved surfaces in a half-space.
problem Stability of anisotropic capillary hypersurfaces in a half-space.
method Analyzes weak stability and proves Bernstein-type theorems.
result Compact hypersurfaces are stable if and only if they are a truncated Wulff shape.
The paper studies capillary surfaces, proving stability and curvature estimates.
problem Stability and curvature estimates for capillary surfaces.
method Analyzes the index and geometry/topology of capillary surfaces in 3-manifolds.
result Stable capillary surfaces in a half-space are half-planes.
Study of spacelike discs in Minkowski cones, proving self-similar expansion.
problem Mean curvature flow of spacelike discs in Minkowski cones.
method Analysis of parabolic boundary value problem for self-similar solutions.
result Existence of solutions rescaling to self-similarly expanding solutions.
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. The paper proves formulas for capillary surfaces and applies them to inequalities and area estimates.
problem Understanding capillary surfaces and their properties.
method Established monotonicity formulas for capillary surfaces in half-space and unit ball.
result Extended Li-Yau-type inequalities and optimal area estimates for capillary surfaces.
Study proves a new formula for capillary hypersurfaces and shows a flow converging to a special shape.
problem Understanding the behavior of capillary hypersurfaces in hyperbolic space.
method Developed a volume-preserving flow starting from a star-shaped initial hypersurface and proved its long-time existence and convergence.
result The flow converges to a θ-totally umbilical cap, which is an energy minimizer for a given enclosed volume. 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. Theory for capillary surfaces in 3-manifolds with smooth boundary.
problem Existence and multiplicity of capillary surfaces with given mean curvature and contact angle.
method Min-max theory applied to capillary surfaces in 3-manifolds.
result Existence of nontrivial, smooth, almost properly embedded surfaces with constant mean curvature and contact angle.
The paper bounds the index of CMC surfaces with capillary boundary.
problem Bounding the index of CMC surfaces with capillary boundary.
method Comparison of second variations of area and energy, derived second variation formulae.
result The index is bounded linearly by genus, boundary components, and contact angle.
Paper solves anisotropic capillary Minkowski problem for p ≥ 1.
problem Anisotropic capillary convex bodies and their properties.
method Introduced anisotropic capillary p-sum and computed variations of quermassintegrals. result Solved the anisotropic capillary Lp-Minkowski problem for p≥1. Study anisotropic flow for capillary hypersurfaces, proving new inequalities.
problem Anisotropic capillary hypersurfaces and their properties.
method Anisotropic volume-preserving mean curvature flow, new approach for strictly convex initial hypersurfaces.
result Established new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces.
The paper applies a capillary John ellipsoid theorem to solve capillary curvature problems.
problem Solving capillary curvature problems in Euclidean half-spaces.
method Applying a capillary John ellipsoid theorem to derive non-collapsing estimates and gradient estimates.
result Established existence of solutions to capillary curvature problems in certain ranges of p and q. 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.
New minimal annuli found in unit ball, solving old problems.
problem Constructing free boundary minimal annuli in unit ball.
method Symmetric and foliated by spherical curvature lines.
result First non-embedded free boundary minimal annuli in unit ball.
Paper solves capillary Orlicz-Minkowski problem with new inequalities.
problem Finding capillary convex bodies with prescribed Orlicz surface area measures.
method Continuity method and inequalities to solve the capillary even Orlicz-Minkowski problem.
result Volume-normalized smooth solutions and inequalities established.
Proves conjectured capillary Blaschke-Santaló inequality for certain convex hypersurfaces.
problem Proving a conjectured inequality for specific convex hypersurfaces.
method Analyzing unconditional, strictly convex capillary hypersurfaces.
result The inequality holds for θ∈(0,2π) and has no finite upper bound for θ∈(2π,π). The paper proves rigidity results for capillary hypersurfaces in hyperbolic space.
problem Understanding the rigidity of capillary hypersurfaces in hyperbolic space.
method Proving a Heintze-Karcher type inequality and applying it to Alexandrov type theorems.
result Rigidity results for capillary hypersurfaces, including totally umbilical and totally geodesic cases.
Solves capillary Lp-Christoffel-Minkowski problem in half-space.
problem Capillary surfaces in half-space geometry.
method Non-collapsing estimate for height and capillary support function.
result Extends Christoffel-Minkowski existence result.
Paper introduces capillary Schwarz symmetrization in half-space.
problem Capillary problems in half-space.
method Introduces a special anisotropic gauge to transform capillary symmetrization to convex symmetrization.
result Capillary Schwarz symmetrization in half-space established.
Study flows for capillary Minkowski problems in half-spaces.
problem Existence and behavior of capillary Minkowski problems.
method Anisotropic capillary Gauss curvature flows.
result Long-time existence and asymptotic behavior of flows.
Solves Christoffel-Minkowski problem for capillary convex bodies in Euclidean half-space.
problem Finding capillary convex bodies with prescribed k-th capillary area measure. method Solving a Hessian-type equation with Robin boundary condition.
result Existence and uniqueness of a smooth solution under natural conditions.
Solves capillary curvature problems for specific p values.
problem Capillary curvature problems for −n<p<1 and θ∈(0,2π). method Iterative scheme based on capillary Minkowski problem and capillary curvature image operators.
result Fixed points of capillary curvature image operators correspond to solutions of capillary Lp-Minkowski problem. Study an anisotropic capillary flow to solve capillary Orlicz-Minkowski problem.
problem Solve capillary Orlicz-Minkowski problem without evenness assumption.
method Analyze an anisotropic capillary Gauss curvature flow to prove convergence and establish existence.
result Establish existence result for capillary Orlicz-Minkowski problem without evenness assumption.
Compact capillary hypersurfaces with specific curvatures are proven.
problem Proving compactness of capillary hypersurfaces with prescribed mean curvature.
method Extending curvature varifolds to oriented integral varifolds and applying curvature varifolds with capillary boundary.
result Compactness result for capillary hypersurfaces with mean curvature prescribed by ambient functions.
Paper solves capillary Lp-Minkowski problem for p>1.
problem Finding capillary convex bodies with prescribed Lp-surface area measures.
method Reduction to Monge-Ampère equation with Robin boundary condition.
result Solved capillary Lp-Minkowski problem in smooth category for p>1.
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.
New capillary Christoffel-Minkowski problem solved for half-space.
problem Finding strictly convex capillary hypersurfaces from capillary functions.
method Established analogous result in capillary setting for half-space.
result Capillary functions lead to strictly convex capillary hypersurfaces.
Proves existence of smooth convex solutions to capillary curvature equations.
problem Proving existence of smooth convex solutions to capillary curvature equations.
method Gradient estimate for capillary curvature equations in half-space.
result Existence of even, smooth, strictly convex solutions for all 1<p<k+1 and θ∈(0,π/2).