The study proves the existence of free boundary minimal disks in convex regions.
problem Proving the existence of free boundary minimal disks in convex regions.
method Based on a multiplicity-one theorem for the free boundary Simon-Smith min-max theory.
result Existence of at least three embedded free boundary minimal disks in strictly convex domains with nonnegative Ricci curvature.
Study AFPP of unions of convex digital disks in 2D.
problem Conditions for AFPP of union of convex disks in digital plane.
method Use results from [6] to analyze AFPP.
result Conditions for AFPP of union of convex disks.
Paper improves proof of theorem on convex 2-disk homeomorphisms.
problem Space of SL homeomorphisms of a convex 2-disk. method Improved proof of main lemma.
result Major improvement in understanding homeomorphisms of convex 2-disk.
The study finds at least two short, simple geodesic chords on a disk with convex boundary.
problem Existence of short, simple geodesic chords on a 2-disk with convex boundary.
method Proof of existence using Riemannian geometry and bounds on lengths.
result Existence of at least two short, simple orthogonal geodesic chords on a 2-disk with convex boundary.
Study minimal freezing sets in convex digital disks.
problem Finding minimal freezing sets in convex digital disks.
method Showed how to find minimal freezing sets for convex disks in digital plane.
result Found minimal freezing sets for convex disks in digital plane.
Proves stability of convex disks close to round caps.
problem Stability of convex disks with positive curvature and strictly convex boundary.
method Compactness result for a Liouville-type PDE problem.
result Proves stability for a theorem of F. Hang and X. Wang.
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.
The Blaschke rolling disk theorem is extended to non-convex domains.
problem Classical inclusion principle for non-convex domains.
method Geometric conditions based on curvature, algorithm for decomposition.
result Necessary and sufficient conditions for rolling disks in non-convex domains.
Solves Christoffel problem for disk area measures on spheres.
problem Conditions for a measure to be a disk area measure of convex bodies.
method Integral representation and differential equation reformulation.
result Reconstructs support function from disk area measure.
Proves nonnegatively curved hypersurfaces on a sphere are convex disks.
problem Characterizing nonnegatively curved hypersurfaces with free boundary on a sphere.
method Analyzes hypersurfaces in Euclidean space with constant mth mean curvature. result Compact hypersurfaces are embedded convex disks.
The study confirms two cases of the convex body isoperimetric conjecture in the plane.
problem The least perimeter to enclose a given area inside a unit disk is greater than inside any other convex set.
method Examined symmetric domains and perturbations of the unit disk.
result Two cases of the convex body isoperimetric conjecture are confirmed.
The study finds at least 2 free-boundary minimal disks in convex 3-balls.
problem Finding minimal disks in convex 3-balls.
method Combining mean curvature flow, min-max theory, and degree theory.
result Existence of at least 2 free-boundary minimal disks in convex 3-balls for generic metrics.
Study rigidity of minimal disks in specific 3-manifolds.
problem Rigidity of free boundary minimal disks in mean convex three-manifolds.
method Assuming strict stability, prove isometric neighborhoods using modified Hawking mass.
result Prove rigidity of minimal disks in specific 3-manifolds.
Classifies convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
problem Classifying convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
method Contact isotopy classification, h-principle, fundamental groups, contact mapping class group.
result Establishes an h-principle for convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
The study proves inequalities for area and boundary length of disks in convex manifolds.
problem Proving inequalities for area and boundary length of disks in convex manifolds.
method Analyzing disks with homotopically non-trivial boundaries in manifolds with convex mean curvature boundary and positive scalar curvature.
result Proves an inequality involving area and boundary length of disks.
This paper extends a 3D result to higher dimensions for manifolds with positive curvature.
problem Proving higher-dimensional manifolds with positive curvature operator and strictly convex boundary are diffeomorphic to the Euclidean disk.
method Using the positive curvature operator and strictly convex boundary conditions to deduce the manifold's diffeomorphism to the Euclidean disk.
result Compact n-manifolds with positive curvature operator and strictly convex boundary are diffeomorphic to the standard n-dimensional Euclidean disk.
Asymmetric expansion preserves convexity in hyperbolic geometry.
problem Maintaining convexity in hyperbolic geometry under asymmetric expansions.
method Generalizing earlier results on radial expansion to asymmetric expansion.
result Asymmetric expansion of hyperbolic convex sets remains convex.
Geometric interpretation of Fock-Goncharov positivity and disk stabilization in symmetric space.
problem Understanding Fock-Goncharov positivity and its geometric implications.
method Geometric interpretation and bending deformations of Fuchsian representations.
result Stabilization of a uniform Finsler quasi-convex disk in the symmetric space.
Paper proves ellipticity of certain Reeb orbits and estimates ECH spectrum on lens spaces.
problem Proving ellipticity of Reeb orbits in lens spaces and estimating ECH spectrum.
method Using rational self-linking number, Conley-Zehnder index, and ECH computations.
result First ECH spectrum on dynamically convex L(3,1) is estimated and shown to be equal to contact area infimum.
We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic curve, there is a canonical mean convex hull containing all minimal planes span…
The paper develops a new min-max theory for minimal disks with free boundary.
problem Constructing minimal disks with free boundary in Riemannian manifolds.
method Develops a min-max theory using harmonic replacement and energy convexity.
result Established an effective version of partial Morse theory for minimal disks with free boundary.
Paper proves disks are holomorphic in Teichmüller spaces.
problem Understanding holomorphic disks in Teichmüller spaces.
method Analyzes totally-geodesic isometries and disk-rigid domains.
result Every isometry is holomorphic or anti-holomorphic.
Three configurations of two perpendicular disks in R^3 are examined, the first in which the disks share centers and the other two in which the disks touch at precisely one point. Volume, surface area and mean width calculations dominate the discussion. Integrated mean curvature also appears as an indirect way to comput…
New method proves existence of constant mean curvature disks on convex surfaces.
problem Proving existence of constant mean curvature disks on convex surfaces.
method Sacks-Uhlenbeck type perturbation instead of heat flow.
result Existence for all H∈(0,H0) when Σ is convex and has mean curvature bounded below by H0. A mean-convex set can be regarded as a barrier for the construction of minimal surfaces. Namely, if we are given a mean-convex set and a null-homotopic Jordan curve on its boundary, then there exists an embedded minimal disk with boundary the given curve contained in the starting mean-convex set. Does a mean-convex set…
Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.
problem Preserving convexity in hyperbolic and spherical geometries under radial transformations.
method Used Poincaré disk model for hyperbolic geometry and stereographic projection for spherical geometry to prove preservation of convexity under radial expansion and contraction.
result Radial expansion and contraction preserve hyperbolic and spherical convexity, respectively.
Maps from spheres and disks to convex shapes via curvature flow.
problem Constructing contractions from spheres and disks to convex shapes.
method Inverse mean curvature flow to create normalized-area-preserving contractions.
result Proves E. Milman's conjecture and gives spectral comparison results.
Curvature conditions distinguish Euclidean space and disks in contractible manifolds.
problem Distinguish Euclidean space and disks among contractible manifolds.
method Investigate curvature conditions on open and compact contractible manifolds with boundary.
result Stronger curvature conditions can distinguish disks from Euclidean spaces.
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).
We show that if C is a simple closed curve bounding an embedded disk in a closed 3-manifold M, then there exists a disk D in M with boundary C such that D minimizes the area among the embedded disks with boundary C. Moreover, D is smooth, minimal and embedded everywhere except where the boundary C meets the interior of…
Harmonic functions on compact symmetric spaces exhibit strong convexity properties.
problem Understanding the convexity of harmonic functions on compact symmetric spaces.
method Analyzing the nonnegativity of the Laplacian powers of harmonic functions.
result Harmonic functions on compact symmetric spaces have nonnegative Laplacian powers, demonstrating strong convexity.
Consider a convex domain B of space. We prove that there exist complete minimal surfaces which are properly immersed in B. We also demonstrate that if D and D' are convex domains with D bounded and the closure of D contained in D' then any minimal disk whose boundary lies in the boundary of D, can be approximated in an…
We generalize Meeks and Yau's embeddedness result on the solutions of the Plateau problem to the constant mean curvature disks. We show that any minimizing H-disk in an H_0-convex domain is embedded for any H in [0,H_0). In particular, for the unit ball B in R^3, this implies that for any H in [0,1], any Jordan curve i…
The paper proves the existence and properties of geodesics on convex surfaces.
problem Existence and properties of geodesics on convex surfaces with free boundaries.
method Free boundary curve shortening flow on closed surfaces with strictly convex boundary.
result Existence of two free boundary embedded geodesics and geodesics with Morse Index 1 and 2.
Study calculates first p-widths of unit disk.
problem Computing first p-widths of the unit disk. method Regularity result for integral 1-varifolds on compact 2-manifolds with convex boundary, applied to unit disk.
result Computed first p-widths for p=1,...,4. The paper finds at least N orthogonal Finsler geodesic chords in a disk-like manifold.
problem Existence and multiplicity of orthogonal Finsler geodesic chords in a disk-like manifold.
method Study of Finsler geodesic chords under reversibility assumption.
result At least N orthogonal Finsler geodesic chords found in a disk-like manifold.
Geodesics on polygons in a unit disk are studied with unique metric properties.
problem Characterizing geodesics on polygons within a unit disk.
method Defining a metric on polygons such that geodesics are curves in the family C.
result The constructed metric space is not isometric to any convex domain in R^2.
The paper proves the existence of constant mean curvature disks with capillary boundary conditions.
problem Existence of constant mean curvature disks with specific boundary conditions.
method Extending Struwe's result to a broader range of boundary angles.
result Existence of constant mean curvature disks with index at most 1.
We show that for a generic nullhomotopic simple closed curve C in the boundary of a compact, orientable, mean convex 3-manifold M with trivial second homology, there is a unique area minimizing disk D embedded in M where the boundary of D is C. We also show that the same is true for absolutely area minimizing surfaces.
Surface area and mean width of a cylinder (the convex hull of two parallel disks) in R^3 are computed. It is more difficult to obtain analogous results for a cone (the convex hull of a disk D and a point p). Oblique formulas for mean width, as well as those for mean curvature, are new. Let L denote the unique diameter …
The paper proves the existence of a center in disks within Riemannian manifolds.
problem Proving the existence of a center in disks within Riemannian manifolds.
method Analyzing the existence of a center in C1 embedded k-disks in Riemannian n-manifolds. result Existence and equivariance of centers in certain conditions, non-existence in others.
This paper finds a global surface of section in dynamically convex L(p,p-1) using ECH.
problem Finding a global surface of section in dynamically convex L(p,p-1).
method Using Embedded Contact Homology (ECH).
result Relates periods of the surface of section to the first ECH spectrum.
The paper proves a Fenchel theorem for Gauss maps and shows circles and disks minimize certain energies.
problem Finding minimizers of nonlocal curvature energies.
method Combining Fenchel-type theorems with geometric analysis techniques.
result Circles and disks minimize specific energy functionals.
The paper finds representations of surface groups in SO(4,1) with specific curvature properties.
problem Finding convex-cocompact representations of surface groups with minimal map properties.
method Complex variation of Hodge structures and embedded minimal maps.
result Examples of generalized almost-Fuchsian representations not deformations of Fuchsian representations.
The Weyl problem is extended to hyperbolic and anti-de Sitter spaces, connecting geometry, analysis, and group theory.
problem The classical Weyl problem for surfaces in hyperbolic and anti-de Sitter spaces.
method Generalizations of the Weyl problem to unbounded convex subsets and convex surfaces, focusing on thin and thick asymptotic boundaries.
result Connections to Kleinian groups, complex analysis, circle packings, and grafting on the hyperbolic disk.
Paper proves rigidity of 3-manifolds with boundary using modified Hawking mass.
problem Rigidity of 3-manifolds with boundary under specific geometric conditions.
method Area estimates for free boundary strictly stable two-disks, modified Hawking mass analysis.
result 3-manifolds with boundary are locally isometric to half anti-de Sitter-Schwarzschild manifold.
We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic disk in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi di…
Inverse mean curvature flow converges to a disk in hyperbolic space.
problem Understanding flow behavior in hyperbolic geometry.
method Inverse mean curvature flow with free boundary on geodesic spheres.
result Flow converges to a totally geodesic disk.