We prove a general contractibility criterion for Riemannian metrics on a disc.
problem Contractibility of subsets of Riemannian metrics on a disc.
method General contractibility criterion for Riemannian metrics.
result The space of metrics with positive Gauss curvature and convex boundary is contractible.
The paper explores isometric models and Busemann functions for Funk and Hilbert discs.
problem Exploring isometric models and Busemann functions for Funk and Hilbert discs.
method Finding and describing isometric models and computing Busemann functions.
result Proving asymptotic harmonicity of the Funk disc and showing its dependence on measure.
New proofs confirm travel time data determine simple metrics on a disc.
problem Determining a simple Riemannian metric from travel time data.
method Proofs based on Myers-Steenrod theorem, Lipschitz-type stability estimate.
result Travel time data determine a simple Riemannian metric on a disc up to natural gauge.
The paper proves conditions for converting homotopies to monotone homotopies in Riemannian discs and spheres.
problem Conditions for converting homotopies to monotone homotopies in Riemannian discs and spheres.
method Analyzing the boundary of Riemannian discs and spheres to determine if they can be contracted monotonously.
result A monotone homotopy can be constructed for a Riemannian disc and sphere under certain length constraints.
The diameter of a disc filling a loop in the universal covering of a Riemannian manifold may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram filling a word that represents the i…
The article proves Randers Poincaré disc satisfies isoperimetric equality.
problem Extending Riemannian isoperimetric equality to Finslerian case.
method Analyzes Randers Poincaré disc with different volume forms.
result Osserman's result cannot be extended to Finslerian case.
Computes a new metric quantity Y(M) for Riemannian 2d-manifolds.
problem No simple metric quantity exists for Riemannian manifolds.
method Defines Y(M) and Y_disc(M) involving sectional curvatures and computes them for specific manifolds.
result Y(M) and Y_disc(M) differ from the Euler characteristic and can be positive or negative.
Defines a distance function on a manifold using symplectic embeddings and recovers the metric.
problem Recovering a Riemannian metric from symplectic embeddings in cotangent bundles.
method Defines a distance-like function ρ W ρ_W ρ W using symplectic embeddings and recovers the metric when W W W is the unit disc-cotangent bundle. result The distance function ρ W ρ_W ρ W recovers the Riemannian metric when W W W is the unit disc-cotangent bundle. We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal, is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this r…
The paper solves an embedding problem for discs with specific curvature properties.
problem Embedding discs with positive Gauss curvature into Euclidean 3-space.
method Analyzes the free boundary isometric embedding problem with specific curvature conditions.
result Discs with specified curvature can be isometrically embedded into R 3 \mathbb{R}^3 R 3 orthogonally to the unit sphere. We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadra…
We prove a "gluing" theorem for monotone homotopies; a monotone homotopy is a homotopy through simple contractible closed curves which themselves are pairwise disjoint. We show that two monotone homotopies which have appropriate overlap can be replaced by a single monotone homotopy. The ideas used to prove this theorem…
The paper sharpens a theorem about surfaces with zero Gaussian curvature.
problem Quantifying the isometric property of surfaces with zero Gaussian curvature.
method Asymptotically sharp quantitative version of a classical theorem using isothermal coordinates.
result An isothermal coordinate map from a Riemannian disc to an Euclidean disc is bi-Lipschitz with a constant of exp(4ε).
Let D D D be a Riemannian 2-disc of area A A A , diameter d d d and length of the boundary L L L . We prove that it is possible to contract the boundary of D D D through curves of length ≤ L + 200 d max { 1 , ln A d } \leq L + 200d\max\{1,\ln {\sqrt{A}\over d} \} ≤ L + 200 d max { 1 , ln d A } . This answers a twenty-year old question of S. Frankel and M. Katz, a version of which was asked …
We prove that if the unit codisc bundle of a closed Riemannian manifold embeds symplectically into a symplectic cylinder of radius one then the length of the shortest nontrivial closed geodesic is at most half the area of the unit disc.
We answer a question of Liokumovich-Nabutovsky-Rotman showing that if D is a Riemannian 2-disc with boundary length L, diameter d and area A << d then D can be filled by a homotopy where the lengths of the intermediate curves are bounded by L + 2 d + O ( A ) L+2d+O(\sqrt A) L + 2 d + O ( A ) .
Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
problem Smooth isotopy of 3-discs in 4-sphere.
method Pushing 3-discs into 5-dimensional space.
result Isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.
problem Existence of holomorphic discs for higher A ∞ A_\infty A ∞ operations. method Showing existence of minimal discs with specific properties implies existence of holomorphic discs.
result Minimal discs in Kähler manifolds with certain boundary conditions are holomorphic.
Study minimal discs in metric spaces with a quadratic isoperimetric inequality.
problem Understanding the geometry of minimal discs in metric spaces.
method Associate a compact metric space to each minimal disc, controlling its properties by the isoperimetric inequality.
result The geometry of the associated space can control the shapes of curves and the original space's topology.
Continuous functions on Riemannian manifolds with poles have fixed points.
problem Extending continuous functions on Riemannian manifolds with poles.
method Simple geometrical technique to generalize Brouwer fixed point theorem.
result Any continuous function on the boundary of a convex domain of a 2D Riemannian manifold with a pole has a fixed point that can be extended to the domain.
Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.
problem Topology of ordered disc configurations and their homotopy types.
method Analysis of ordered configuration spaces of hard discs, focusing on homotopy types and nontrivial classes.
result Exhibit nontrivial classes in π_{n-3} for all n, and their persistence in deformed ambient discs.
Classifies homotopy ribbon discs for certain slice knots.
problem Characterizing homotopy ribbon discs for specific slice knots.
method Classifies Γ \Gamma Γ -homotopy ribbon slice discs up to topological ambient isotopy. result In the infinite cyclic case, there is a unique equivalence class of such slice discs. For the Baumslag-Solitar group, there are at most two equivalence classes of Γ \Gamma Γ -homotopy ribbon discs. New knots found with tough, unsliceable discs.
problem Finding tough knots that can't be sliced smoothly.
method Constructed infinitely many knots with non-approximable slice discs.
result Smoothly sliceable knots have non-approximable slice discs.
Classifies ancient flows in a disc with boundary.
problem Ancient convex flows in a disc with boundary.
method Classifies flows using curve shortening.
result Ancient convex flows in a disc are classified.
New method constructs AdS 3-manifolds and applies to Higgs bundles and minimal immersions.
problem Understanding AdS 3-manifolds and their connections to Higgs bundles and minimal immersions.
method Developed a new construction method for AdS structures.
result Recovered Tholozan's formula for AdS 3-manifold volumes and characterized representations for minimal immersions.
Study of geodesics on circle using Moebius transformations.
problem Understanding geodesics on the circle with a special metric.
method Analyzing force-free Moebius motions of the circle and their geodesics.
result Geodesics on the circle are hypocycloids, matching the 'better than nice' metric.
Holomorphic discs cover a ball in complex space.
problem Covering a ball in complex space with holomorphic discs.
method Showed a nonsingular holomorphic foliation by complete discs.
result The open unit ball in complex space admits a foliation by complete discs.
Circular disc can be tiled with up to 3 congruent pieces, showing symmetry.
problem Tiling a circular disc with congruent pieces.
method Proving the existence of a k k k -fold rotational symmetry for k ≤ 3 k \leq 3 k ≤ 3 . result First nontrivial estimate on minimum number of tiles for certain tiling configurations.
Study geodesic discs with boundary length bounds, finding their closure in metric space.
problem Geodesic discs with boundary length constraints in metric spaces.
method Investigate closure in Gromov-Hausdorff space, relate to disc retracts.
result Closure of geodesic discs is related to disc retracts in metric spaces.
The rotation angle of a rolling disc is shown to be a geometric phase related to the Hopf fibration.
problem Understanding the geometric nature of rotation angles in kinematic models.
method Using the Hopf fibration and Gauss map, the geometric phase is decomposed into dynamical and geometric components.
result The geometric phase of rotation is described as the holonomy of the Hopf fibration.
Develops a new theory of width for embedded circles in Riemannian manifolds.
problem Defining and understanding the width of embedded circles in Riemannian manifolds.
method Morse-Lusternik-Schnirelmann theory applied to geodesics and minimising configurations.
result Classifies configurations of minimising geodesics intersecting embedded circles.
Study smooth manifolds using disc-presheaves.
problem Understanding smooth manifolds.
method Using presheaves on a category of discs.
result Disc-presheaves have desirable properties and strong applications.
This note characterizes monohedral tilings of regular polygons with up to three tiles.
problem Characterizing monohedral tilings of regular polygons with up to three tiles.
method Connecting the results for squares and circles to generalize for any regular n n n -gon. result Characterization of monohedral tilings of any regular n n n -gon with up to three tiles. Study horizontal discs in fat distributions, proving their existence.
problem Existence of embedded horizontal discs in fat distributions.
method Analyzing nonlinear PDEs and proving local invertibility.
result Existence of germs of embedded horizontal discs.
Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.
problem Investigating relationships between three invariants of complex hyperbolic disc orbibundles.
method Analyzing Euler characteristic, Euler number, and Toledo invariant of disc orbibundles over 2-orbifolds.
result Proved that -3|τ| = 2e + 2χ holds for certain complex hyperbolic disc orbibundles.
We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.
The paper classifies homotopy ribbon discs with specific groups.
problem Classifying homotopy ribbon discs with given fundamental groups.
method Using geometric and algebraic properties of groups, particularly Farrell-Jones conjecture.
result Classification of homotopy ribbon discs for specific knot groups and Baumslag-Solitar groups.
Study of rotation angles in a rotating disc model.
problem Understanding geometric phase in rotating systems.
method Analyzes a simple kinematic model of rotating discs.
result Explicit form of geometric phase Δ g Δ_g Δ g found using Baumkuchen lemma. Study calculates homotopy groups and derivatives for disc diffeomorphisms.
problem Understanding the homotopy groups of diffeomorphisms of discs.
method Computes rational homotopy groups and uses Weiss' orthogonal calculus.
result Determines optimal rational concordance stable range for high-dimensional discs.
It is well-known that Teichmuller discs that pass through "integer points'' of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmuller discs is mostly unexplored: their number, genus, area, cusps, etc. We prove that in genus two …
The study finds conditions for free boundary Hamiltonian stationary discs in complex 2-space.
problem Conditions for free boundary Hamiltonian stationary Lagrangian discs in complex 2-space.
method Established conditions for weakly conformal, branched Ω Ω Ω -free boundary Hamiltonian stationary Lagrangian immersions of discs. result If conditions are met, a disc is a free boundary minimal immersion.
Minimal perimeter polygons in punctured discs are found with inscribed horocycles.
problem Finding polygons with minimal perimeter in punctured discs.
method Proving minimal perimeter by inscribed horocycles, generalizing to cone points and geodesic boundaries.
result Minimum perimeter polygons found with inscribed horocycles.
Algebraic treatment of connection reduction over a special disc.
problem Reduction theory for connections over a specific geometric structure.
method Purely algebraic approach for arbitrary groups, with quantitative results.
result New quantitative results in reduction theory.
Holomorphic discs converge to maximal surfaces under specific flows.
problem Understanding the evolution of holomorphic discs under mean curvature flow.
method Mean curvature flow with boundary conditions in the space of oriented lines.
result Holomorphic discs converge to Bishop filling by holomorphic discs under certain conditions.
Modified Engel structures allow complete h-principle for overtwisted discs.
problem Engel structures and their overtwisted discs.
method Engel twist modification and h-principle proof.
result Complete h-principle for overtwisted Engel structures.
This work improves optic disc and cup segmentation for glaucoma detection.
problem Automatic segmentation of optic disc and cup on eye fundus images for glaucoma diagnosis.
method Modification of U-Net convolutional neural network.
result Our method achieves comparable quality to state-of-the-art methods, with faster prediction times.
Disc graphs are uniformly quasiconvex in curve graphs of surfaces.
problem Characterizing quasiconvexity in curve graphs of surfaces.
method Proof using a universal constant K without train tracks.
result Disc graphs are K-quasiconvex in curve graphs.
The paper finds the Finsler structure of Apollonian weak metric on unit disc.
problem Understanding the Finsler structure of Apollonian weak metric on the unit disc.
method Analyzing the deformation of hyperbolic Poincaré metric by a closed 1-form.
result The Apollonian weak-Finsler structure has bounded below S S S -curvature and flag curvature K K K satisfying − ∞ < K < − 1 -\infty < K < -1 − ∞ < K < − 1 .