The paper introduces combinatorial Calabi flows to find hyperbolic metrics on surfaces with boundary.
problem Finding hyperbolic metrics on surfaces with totally geodesic boundaries of given lengths.
method Introducing combinatorial Calabi flows and proving their long time existence and global convergence.
result Proves the long time existence and global convergence of combinatorial Calabi flow on surfaces with boundary.
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.
We prove surfaces are unknotted with specific properties.
problem Unknottedness of free boundary minimal surfaces and self-shrinkers.
method Introduced concepts of boundary graph and graph at infinity to prove unknottedness.
result Proved surfaces are unknotted with specific properties.
The paper studies g-stability of surfaces with boundary and derives area estimates.
problem Investigating g-stability of surfaces with boundary. method Analyzing geometric properties and deriving area estimates.
result Derives area estimates and determines the topology of the surface.
Researchers found multiple surfaces with same topological and symmetry properties.
problem Determining unique free boundary minimal surfaces from topology and symmetry.
method Provided pairs of non-isometric surfaces with same genus, boundary components, and symmetry group.
result Infinitely many pairs of surfaces with same topology and symmetry group exist.
Classifies hyperbolic groups with surface-like boundaries.
problem Classifying hyperbolic groups with specific surface-like boundaries.
method Analyzing quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.
result Identifies hyperbolic groups with surface-like boundaries.
Uniformizes surfaces with boundaries, focusing on triple junctions.
problem Uniformization of surfaces with boundaries, especially triple junctions.
method Extends conformal structure results to triple junction surfaces.
result Weak uniformization results for triple junction surfaces.
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. 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.
This paper classifies discrete conformal structures on surfaces with boundary.
problem Classifying discrete conformal structures on surfaces with boundary.
method Axiomatic approach ensuring good geometric structure, classification based on triangulation and axioms.
result Unified and generalized existing discrete conformal structures on surfaces with boundary.
Compact Special Weingarten surfaces with planar convex boundaries are disks.
problem Characterizing Special Weingarten surfaces with specific boundary conditions.
method Proved a Ros-Rosenberg theorem in the context of Special Weingarten surfaces.
result Compact Special Weingarten surfaces with planar convex boundaries are topological disks.
Compactness proven for CMC surfaces with bounded topology and boundary length.
problem Proving compactness of CMC surfaces with specific constraints.
method Graphical Ck compactness proof for surfaces with bounded topology, area, and boundary length. result Space of free boundary CMC surfaces is compact in the Ck graphical sense away from a finite set of points. Simply connected surfaces with large constant mean curvature and free boundaries concentrate at critical points of the boundary's mean curvature.
problem Surfaces with large constant mean curvature and free boundaries.
method Proving concentration at critical points of the boundary's mean curvature.
result Simply connected H-surfaces concentrate at critical points of the boundary's mean curvature.
The paper constructs free boundary minimal surfaces in product spaces using eigenvalue methods.
problem Constructing free boundary minimal surfaces in product spaces of balls.
method Extremal eigenvalue approach involving mixed Steklov-Neumann eigenvalues.
result No absolute maximum exists for the problem in product spaces.
Bi-geodesic mappings preserve distances on hyperbolic surfaces with boundaries.
problem Preserving distances on hyperbolic surfaces with boundaries.
method Proving bijections between geodesics are isometries.
result A bijection between geodesics is an isometry.
Symplectic structure found on projective structures on surfaces with boundary.
problem Deformation space of projective structures on surfaces with boundary.
method Natural symplectic structure on the space, integrating the Adler-Gelfand-Dikii-space of the boundary.
result Space is a Hamiltonian space for the symplectic groupoid.
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.
Study on shortest arcs on hyperbolic surfaces with boundary.
problem Characterize and maximize the length of shortest essential arcs on hyperbolic surfaces with geodesic boundaries.
method Analyze hyperbolic surfaces with multiple boundary components, construct surfaces with large orthosystole, and compare growth rates.
result Orthosystole grows at the same rate as Bavard's upper bound as the genus increases.
Optimal $C^{1, rac{1}{2}}$-regularity for H-surfaces with free boundary.
problem Proving optimal regularity for H-surfaces with free boundary. method Analyzing H-surfaces with free boundary on a C2-manifold. result Optimal $C^{1, rac{1}{2}}$ regularity up to the boundary.
New inequality links surface orthospectrum to boundary length.
problem Establishing a relationship between orthospectrum and boundary length for compact Riemannian surfaces.
method Analyzing compact Riemannian surfaces with a single closed geodesic, establishing a uniform lower bound on boundary length in terms of orthospectrum.
result A uniform lower bound on boundary length in terms of orthospectrum, akin to Basmajian's identity.
Open problems on surfaces with boundary and constant mean curvature.
problem Open problems on compact constant mean curvature surfaces with boundary.
method Survey of current status of open problems.
result Collection of open problems in the theory of surfaces.
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.
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 provides estimates for Steklov eigenvalues of surfaces with boundary.
problem Estimating Steklov eigenvalues of surfaces with boundary components.
method Computable lower bounds for the first non-zero Steklov eigenvalue using geometric quantities specific to manifolds with boundary.
result The geometry of the manifold away from the boundary affects the Steklov eigenvalue.
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.
Constructs minimal surfaces near the boundary of a ball.
problem Creating minimal surfaces close to the boundary of a ball.
method PDE gluing methods to construct FBMS of genus zero.
result Desingularizations of catenoidal annuli and flat discs near the boundary.
In this paper we investigate free boundary minimal surfaces in the unit ball in Euclidean 3-space, and by using holomorphic techniques we prove that intersection curves of free boundary minimal surfaces with the unit sphere are all circles.
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.
Extends rigidity and existence results for discrete conformal structures on surfaces with boundary.
problem Rigidity and existence of discrete conformal structures on surfaces with boundary.
method Axiomatic framework and classification of discrete conformal structures.
result Extends results by Guo-Luo and Guo to a general context.
The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.
problem Finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.
method Introducing combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings.
result Proves longtime existence and global convergence of combinatorial curvature flows.
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.
The study derives formulas for functionals on surface with boundary under harmonic Ricci flow.
problem Understanding functionals along harmonic Ricci flow on surfaces with boundaries.
method Derivation of formulas for functionals under harmonic Ricci flow.
result Established formulas for functionals on surface with boundary.
We solve the problem on flat extensions of a generic surface with boundary in Euclidean 3-space, relating it to the singularity theory of the envelope generated by the boundary. We give related results on Legendre surfaces with boundaries via projective duality and observe the duality on boundary singularities. Moreove…
Study of Ricci flow convergence on surfaces with boundary.
problem Convergence of singular solutions to Ricci flow on compact surfaces with boundary.
method Subsequential convergence analysis of Ricci flow with prescribed geodesic curvature.
result Convergence does not depend on the sign of geodesic curvature of the boundary in the case of rotational symmetry.
In the Euclidean unit three-ball, we construct compact, embedded, two-sided free boundary minimal surfaces with connected boundary and prescribed high genus, by a gluing construction tripling the equatorial disc. Aside from the equatorial disc itself, these are the first examples in the three-ball of compact free bound…
We discuss an alternative approach to the uniformisation problem on surfaces with boundary by representing conformal structures on surfaces M of general type by hyperbolic metrics with boundary curves of constant positive geodesic curvature. In contrast to existing approaches to this problem, the boundary curves of o…
Paper proves surfaces with specific symmetries have the first Steklov eigenvalue.
problem Proving surfaces with certain symmetries have the first Steklov eigenvalue.
method Analyzing surfaces with reflection planes and genus zero.
result Surfaces with n distinct reflection planes have the first Steklov eigenvalue. We study of the shape of a compact singular minimal surface in terms of the geometry of its boundary, asking what type of {\it a priori} information can be obtained on the surface from the knowledge of its boundary. We derive estimates of the area and the height in terms of the boundary. In case that the boundary is a …
Describes curves on surfaces with punctures and boundaries.
problem Representing multiple curves on surfaces with punctures and boundaries.
method Using geometric intersection numbers with embedded curves.
result Each multiple curve can be uniquely described.
The Poincaré series for surfaces with boundary extends to the complex plane.
problem Counting geodesics on surfaces with boundaries.
method Analytic continuation of Poincaré series.
result Poincaré series extend meromorphically to the whole complex plane.
The article proves a new entropy formula for surfaces with boundaries.
problem Entropy formula for surfaces with boundaries.
method Established a monotonicity formula of Hamilton type entropy.
result Entropy functional and W-functional relation studied. Study constant mean curvature surfaces with integrable boundary conditions.
problem Understanding surfaces with constant mean curvature under specific boundary conditions.
method Used generalized Weierstrass representation to determine potentials.
result Determined potentials for surfaces satisfying integrable boundary conditions.
Numerical methods solve Steklov eigenvalue problems to generate free boundary minimal surfaces.
problem Generating free boundary minimal surfaces using Steklov eigenvalue problems.
method Maximizing Steklov eigenvalues over a class of metrics, using conformal uniformization and gradient-based optimization.
result Numerical solutions for free boundary minimal surfaces with various boundary components.
We study immersed surfaces in R3 which are critical points of the Willmore functional under boundary constraints. The two cases considered are when the surface meets a plane orthogonally along the boundary, and when the boundary is contained in a line. In both cases we derive weak forms of the resulting fre…
In homogenous space Sol we study compact surfaces with constant mean curvature and with non-empty boundary. We ask how the geometry of the boundary curve imposes restrictions over all possible configurations that the surface can adopt. We obtain a flux formula and we establish results that assert that, under some restr…
For surfaces without boundary, nonlocal notions of directional and mean curvatures have been recently given. Here, we develop alternative notions, special cases of which apply to surfaces with boundary. Our main tool is a new fractional or nonlocal area functional for compact surfaces.
Study on surfaces with conical singularities and geodesic boundaries, deriving existence results.
problem Existence of conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundaries.
method Variational argument to derive existence results for surfaces with at least two boundary components.
result First result in this setting for surfaces with conical singularities of both positive and negative orders.
Proves existence of mean curvature flow with surgery for free boundary surfaces.
problem Existence of mean curvature flow with free boundary.
method New approach for flows with surgery in free boundary setting.
result Flow converges to stable minimal surfaces without surgeries for large t.