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.
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. 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. 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.
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). 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.
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.
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.
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 on convex capillary hypersurfaces with Lp curvature in half-space.
problem Prescribed Lp curvature for convex capillary hypersurfaces.
method Reduction to Hessian quotient equation with Robin boundary condition.
result Existence and uniqueness of smooth admissible solutions.
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.
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.
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.
Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.
problem Understanding the evolution of hypersurfaces with capillary boundaries.
method Locally constrained mean curvature flow for star-shaped hypersurfaces in the half-space.
result Established new Alexandrov-Fenchel inequalities for convex hypersurfaces with capillary boundary.
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.
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.
Study inverse curvature flows for capillary hypersurfaces in a unit ball.
problem Understanding the behavior of capillary hypersurfaces under inverse curvature flows.
method Investigate inverse curvature flows for strictly convex, capillary hypersurfaces in the unit Euclidean ball.
result Establish existence and convergence results for inverse curvature flows.
Proves uniqueness of capillary disks in 3D domains.
problem Proving uniqueness of capillary disks in three-dimensional domains modeled by elliptic PDEs.
method Using elliptic PDEs and properties of surfaces in 3D domains, generalizing Nitsche's and Hopf's theorems.
result Generalizes Nitsche's result for capillary constant mean curvature disks in the Euclidean ball.
Study of manifolds with specific curvature properties using capillary surfaces.
problem Obtaining geometric properties of manifolds with nonnegative scalar curvature and strictly mean convex boundary.
method Use of stable capillary surfaces and Urysohn width to study geometric properties.
result Obtained an obstruction to filling 2-manifolds by 3-manifolds.
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.
Study shows how curved surfaces evolve smoothly to spherical shapes.
problem Evolution of curved surfaces with capillary boundaries.
method Volume-preserving curvature flow with power mean curvature speed.
result Convex initial hypersurfaces evolve to spherical caps over time.
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.
The paper proves the existence of capillary geodesics on Riemannian 2-disks.
problem Existence of capillary geodesics on Riemannian 2-disks with specific conditions.
method Analytical proof and examples.
result Existence of capillary geodesics with contact angle θ ∈ (0, π/2).
Paper proves short-term existence of fractional mean curvature flow.
problem Existence of solutions for fractional mean curvature flow with capillary boundary conditions.
method Fixed point argument.
result Short time existence of solutions for fractional mean curvature flow.
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.
The article proves inequalities for capillary hypersurfaces in hyperbolic space.
problem Proving inequalities for capillary hypersurfaces in hyperbolic space.
method Constructing a new locally constrained inverse curvature flow.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space.
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.
Study a flow in a ball that preserves volume and converges to spherical caps.
problem Preserving volume in a flow with a capillary boundary.
method Mean curvature flow with capillary boundary.
result The flow has longtime existence and converges to spherical caps.
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.
New Minkowski inequality for capillary surfaces in half-space.
problem Establishing a new Minkowski inequality for capillary surfaces.
method Flow of inverse mean curvature type for capillary hypersurfaces in a half-space.
result Derive a new Minkowski-type inequality for star-shaped and mean convex capillary hypersurfaces.
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.
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.
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.
The paper studies a flow for convex capillary hypersurfaces in a ball, proving smooth convergence to a spherical cap.
problem Analyzing the behavior of convex capillary hypersurfaces under mean curvature flow.
method Introduced mean curvature flow for hypersurfaces in the unit Euclidean ball with capillary boundary. Proved smooth convergence to a spherical cap for strictly convex initial hypersurfaces.
result The flow preserves strict convexity and converges smoothly to a spherical cap for all positive time.
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.
Paper proves a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.
problem Proving a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.
method Using a locally constrained nonlinear curvature flow to preserve the n-th quermassintegral and decrease the k-th quermassintegral. result Obtained the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in Bn+1. 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…
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.
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 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). 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.
Paper extends Wente's result to anisotropic capillary surfaces in half-spaces.
problem Extending Wente's result to anisotropic capillary surfaces.
method New Heintze-Karcher inequality and Minkowski formula.
result Anisotropic capillary hypersurfaces in half-spaces are Wulff shapes.
Paper proves inequality for capillary hypersurfaces with new proof.
problem Proving a Heintze-Karcher type inequality for hypersurfaces with capillary boundary.
method Using a mixed boundary value problem in Reilly type formula to establish the inequality.
result New proof of Alexandrov type theorem for capillary hypersurfaces.
We generalize a theorem by J. Choe on capillary surfaces for arbitrary 3-dimensional spaces of constant curvature. The main tools in this paper are an extension of a theorem of H. Hopf due to S.-S. Chern and two index lemmas by J. Choe.
The liquid shape between two vertical parallel plates in a gravity field due to capillary forces is studied. When the physical system achieves its mechanical equilibrium, the capillary surface has mean curvature proportional to its height above a horizontal reference plane and it meets the vertical walls in a prescribe…
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. 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 inequalities for convex capillary hypersurfaces in a half-space.
problem Proving inequalities for convex capillary hypersurfaces in a half-space.
method Locally constrained inverse curvature flow with spherical cap convergence.
result Proves a complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.