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.
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.
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. 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.
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.
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.
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. Paper classifies critical points in half-space with new distance function.
problem Classifying critical points in half-space with capillary CMC hypersurfaces.
method New shifted distance function for capillary problem in half-space.
result Proves Alexandrov-type theorem for singular capillary CMC hypersurfaces.
Study of flow in hyperbolic space with capillary boundary.
problem Optimizing hypersurfaces with capillary boundary in hyperbolic space.
method Mean curvature flow with capillary boundary, preserving volume and energy.
result Flow converges to a truncated umbilical hypersurface.
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.
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 proves rigidity of critical points in hydrophobic capillary systems.
problem Rigidity of critical points in hydrophobic capillary systems.
method Proves rigidity among sets of finite perimeter in the half space, extending to full hydrophobic regime.
result Rigidity of critical points proven in hydrophobic capillary systems.
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.
In this paper, we study a mean curvature type flow with capillary boundary in the unit ball. Our flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases an energy functional E. We show that it has the longtime existence and subconverges to spherical caps. As an applic…
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 paper classifies capillary graphs on manifolds with Ricci lower bounds.
problem Understanding capillary graphs on manifolds with Ricci lower bounds.
method Gradient estimate for positive CMC graphs on manifolds with Ricci lower bounds.
result Classification of capillary graphs over specific domains.
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. Stable capillary surfaces in weighted balls are disks.
problem Finding the shape of isoperimetric regions in weighted balls.
method Stability analysis and Hsiang symmetrization.
result Interior boundaries of isoperimetric regions in weighted balls are disks.
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.
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.
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.
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). 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.
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.
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.
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.
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.
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.
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.
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). Proves quantitative Alexandrov theorem for capillary surfaces.
problem Proving a quantitative version of the Alexandrov theorem for capillary hypersurfaces.
method Quantitative analysis of Montiel-Ros-type argument.
result Generalizes Julin-Niinikoski's result to capillary case.
Study of capillary Gauss curvature flow shrinking to a point.
problem Finite time shrinkage of capillary hypersurfaces.
method Introduced a Gauss curvature type flow for capillary hypersurfaces.
result Normalized flow converges to a soliton.
Study convex capillary hypersurfaces with prescribed curvature in a spherical cap.
problem Prescribed curvature problem for convex capillary hypersurfaces.
method Reformulated as Hessian quotient equation with Robin boundary condition.
result Existence of strictly convex capillary hypersurface with prescribed curvature.
New flow for capillary surfaces converges to spherical caps.
problem Optimizing capillary surfaces in space forms.
method Constrained mean curvature flow.
result Flow converges to spherical caps globally.
Stability of capillary hypersurfaces with higher order mean curvature.
problem Stability of capillary hypersurfaces with constant higher order mean curvature.
method Generalization of classical stability theory for capillary hypersurfaces.
result Results on stability for capillary hypersurfaces with higher order mean curvature.
The paper solves a specific Minkowski problem for capillary hypersurfaces.
problem Finding capillary convex bodies with prescribed dual curvature measures.
method Reduction to a Monge-Ampère type equation with Robin boundary condition.
result Existence and uniqueness of a smooth solution for θ∈(0,2π). Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.
problem Extending capillary convex body results to anisotropic setting.
method Developed theory for anisotropic capillary convex bodies in half-space and established Alexandrov-Fenchel inequality for mixed volumes.
result Established a general Alexandrov-Fenchel inequality for mixed volumes of anisotropic capillary convex bodies, weakening and extending previous results.
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.
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.
Paper solves inequalities for capillary hypersurfaces in half-spaces.
problem Finding inequalities for convex capillary hypersurfaces in half-spaces.
method Introduced quermassintegrals and constructed a new locally constrained curvature flow to prove convergence to spherical caps.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
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 proves inequality for capillary hypersurfaces in a wedge.
problem Proving a best version of Heintze-Karcher inequality for capillary hypersurfaces.
method Utilized Heintze-Karcher method and modified parallel hypersurfaces.
result Classified capillary constant mean curvature hypersurfaces hitting the edge in a wedge.
Paper proves rigidity results for anisotropic capillary hypersurfaces.
problem Rigidity of anisotropic capillary hypersurfaces.
method New Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces.
result Uniqueness of solution to anisotropic Orlicz-Christoffel-Minkowski problem.
The paper examines stable capillary hypersurfaces in hyperbolic space.
problem Stability of capillary hypersurfaces with free boundary on a horosphere.
method Analysis of umbilical and totally geodesic hypersurfaces using stability criteria.
result Umbilical and totally geodesic hypersurfaces are the only stable capillary hypersurfaces with boundary on a horosphere.