Study on embedding hyperbolic 2-orbifolds in Bianchi orbifolds.
problem Embedding closed totally geodesic hyperbolic 2-orbifolds in Bianchi orbifolds.
method Analyzing Bianchi orbifolds H3/PSL(2,Od) for large d. result Existence of at least cd closed embedded totally geodesic hyperbolic 2-orbifolds for large d. We prove the existence of Alexandrov embedded closed magnetic geodesics on closed hyperbolic surfaces. Closed magnetic geodesics correspond to closed curves with prescribed geodesic curvature.
We prove the existence of two Alexandrov embedded closed magnetic geodesics on any two dimensional sphere with nonnegative Gauss curvature.
Study on sphere widths and geodesic multiplicity.
problem Understanding the width of spheres and geodesic multiplicity.
method Computed sphere widths for k=1 to 8 and used min-max critical varifolds.
result Unstable geodesics can arise with multiplicity.
Holomorphic deformations of weighted projective spaces yield Finsler spheres with closed geodesics.
problem Finding Finsler 2-spheres with constant curvature and closed geodesics.
method Establishing a correspondence between Finsler structures and Weyl connections on orbifolds.
result Holomorphic deformations of Veronese embeddings provide examples of Finsler 2-spheres with constant curvature and closed geodesics.
Geodesic surfaces embed into hyperbolic 3-manifolds for all finite group actions.
problem Embedding geodesic surfaces into hyperbolic 3-manifolds.
method Analyzing finite group actions on surfaces and proving geodesic embeddings for all irreducible cases.
result All quasiplatonic surfaces embed geodesically into hyperbolic 3-manifolds.
Proved contractibility of geodesic triangulation space on hyperbolic surfaces.
problem Open problem on contractibility of geodesic triangulations on hyperbolic surfaces.
method Generalized Tutte's embedding theorem for negative curvature surfaces.
result Contractibility of geodesic triangulation space proved.
The abstract finds conditions for creating curves of constant curvature.
problem Finding conditions for closed embedded curves of constant curvature.
method Using Almgren-Pitts min-max method for geodesic curvature.
result Closed embedded curves of any prescribed constant curvature found in S2. Totally geodesically embeddings of infinitely many closed 7-manifolds into 13-dimensional positively curved closed Riemannian manifolds are constructed. The problems of computing pinching constants and existence of other totally geodesical embeddings are discussed.
The study measures how sparse geodesics are on hyperbolic surfaces.
problem Measuring the density of simple geodesics on hyperbolic surfaces.
method Analyzes embedded metric disks and their relation to geodesics.
result Disks of specific radius are disjoint from simple geodesics.
Paper proves geodesics are evenly distributed on surfaces.
problem Existence of equidistributed closed geodesics on surfaces.
method Volume property of embedded contact homology, local variational constructions, and transversality arguments.
result Equidistribution of nondegenerate closed geodesics for generic metrics on closed surfaces.
Study on embedding surfaces into 3-manifolds, focusing on equivariant cases.
problem Embedding hyperbolic surfaces into hyperbolic 3-manifolds with specific symmetries.
method Examined orientation-preserving and orientation-reversing actions on surfaces, including nonorientable ones.
result Found conditions for equivariant embeddings of hyperbolic surfaces into hyperbolic 3-manifolds.
Proves min-max theory for constant geodesic curvature curves on closed surfaces.
problem Prescribing mean curvature on surfaces with constant geodesic curvature.
method Min-max theory applied to classify blowups and ensure almost embedded solutions.
result Produces a solution with constant geodesic curvature c on closed surfaces. We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used.
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.
In this paper we construct quasiconformal embeddings from Y-pieces that contain a short boundary geodesic into degenerate ones. These results are used in a companion paper to study the Jacobian tori of Riemann surfaces that contain small simple closed geodesics.
The study finds minimal volume hyperbolic 4-manifolds with embedded 3-manifolds.
problem Existence of small volume hyperbolic 4-manifolds with embedded 3-manifolds.
method Analysis of hyperbolic manifolds and their submanifolds.
result Minimal volume hyperbolic 4-manifolds with embedded 3-manifolds exist.
We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.
In a paper of Menasco and Reid, it is conjectured that there exist no hyperbolic knots in S^3 for which the complement contains a closed embedded totally geodesic surface. In this note, we show that one can get "as close as possible" to a counter-example. Specifically, we construct a sequence of hyperbolic knots {K_n} …
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).
Luo and Tan gave a new identity for hyperbolic surfaces with/without geodesic boundary in terms of dilogarithms of the lengths of simple closed geodesics on embedded three-holed spheres or one-holed tori. However, the identity was trivial for a hyperbolic one-holed torus with geodesic boundary. In this paper we adapt t…
Random simple closed curves map Teichmüller space to geodesic currents.
problem Mapping Teichmüller space to geodesic currents.
method Using a formula for intersection numbers of multicurves and Dehn coordinates.
result Proper embedding of Teichmüller space into the space of geodesic currents.
Theorem shows generic metrics yield non-degenerate geodesic nets.
problem Characterizing geodesic nets on generic metrics.
method Proving all connected embedded nets are non-degenerate for Baire-generic metrics.
result All stationary geodesic nets are non-degenerate for generic metrics.
Study of curves in hyperbolic plane with variable curvature.
problem Finding curves with prescribed almost constant curvature in hyperbolic plane.
method Analyzing closed and embedded curves with geodesic curvature.
result Existence of curves with specified curvature variations.
Study shows certain 4D hyperbolic links don't contain geodesic 3-manifolds.
problem Proving certain hyperbolic link complements don't contain geodesic 3-manifolds.
method Analyzing hyperbolic link complements of 2-tori in S^4.
result Proves certain hyperbolic link complements do not contain closed embedded totally geodesic hyperbolic 3-manifolds.
The paper shows equidistribution of geodesics and nets on manifolds.
problem Equidistribution of geodesics and nets on manifolds.
method Generic metrics, Weyl Law for volume spectrum, and equidistribution of geodesic nets.
result Equidistribution of geodesic nets on 3-manifolds.
Study on manifolds with flat sections and their geodesics.
problem Characterizing geodesics in manifolds with flat sections.
method Analyzing closed non-positively curved Riemannian manifolds with fat k-flats.
result Existence of uncountably many closed geodesics in manifolds with fat k-flats.
The paper finds an upper limit for the length of geodesic chords on Riemannian manifolds.
problem Finding the maximum length of geodesic chords on Riemannian manifolds.
method Establishing an upper bound for geodesic chord length using geometric bounds on the manifold.
result An upper bound for the length of geodesic chords is derived, with a specific example for 2-dimensional spheres.
Proves existence and embeddability of least area minimal hypersurfaces in higher dimensions.
problem Existence and embeddability of least area minimal hypersurfaces in higher dimensions.
method Min-max theory applied to closed (n+1)-manifolds with 2≤n≤6. result Existence and embeddability of least area minimal hypersurfaces in higher dimensions.
New constructions show stable geodesics and figure-eights in convex hypersurfaces.
problem Constructing stable geodesics and figure-eights in convex hypersurfaces.
method Explicit billiard trajectories with controlled parallel transport in convex polytopes.
result Construction of stable figure-eights and index-zero geodesics in convex hypersurfaces.
Given a hyperbolic 3-manifold M containing an embedded closed geodesic, we estimate the volume of a complete hyperbolic metric on the complement of the geodesic in terms of the geometry of M. As a corollary, we show that the smallest volume orientable hyperbolic 3-manifold has volume >.32 .
Maximal representations are studied using tree embeddings and geodesic currents.
problem Maximal representations of surface groups in symplectic groups.
method Metric properties, geodesic currents, and tree embeddings.
result Translation length can be computed as intersection with a geodesic current.
Paper proves existence of magnetic geodesics on sphere.
problem Existence of closed K-magnetic geodesics on S2. method Lyapunov-Schmidt reduction and local variational formulation.
result Existence and multiplicity of closed K-magnetic geodesics. We embed directed acyclic graphs using hyperbolic spaces and geodesic cones.
problem Learning graph representations that preserve hierarchical structure.
method Use hyperbolic spaces and geodesic cones to define embeddings of directed acyclic graphs.
result Our method significantly outperforms existing approaches in graph representation learning.
We use pinched smooth hyperbolization to show that every closed, nonpositively curved n-dimensional manifold M can be embedded as a totally geodesic submanifold of a closed, nonpositively curved (n+1)-dimensional manifold M^ of geometric rank one.
Let M be a closed hyperbolic three manifold. We construct closed surfaces which map by immersions into M so that for each one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection.
A new Riemannian metric on curve spaces is complete and smooth.
problem Defining a complete metric on the space of embedded curves.
method Proposed a new Riemannian metric and proved its completeness.
result The proposed metric is complete in multiple senses.
New formulas for geodesics on Stiefel and flag manifolds using trust-region method.
problem Computing geodesics and logarithms on Stiefel and flag manifolds.
method Closed-form geodesic formulas, trust-region solver, Fréchet derivatives.
result Efficient computation of geodesic distance and logarithm map.
We give a positive answer to M. Traizet's open question about the existence of complete embedded minimal surfaces with Scherk-ends without planar geodesics. In the singly periodic case, these examples get close to an extension of Traizet's result concerning asymmetric complete minimal submanifolds of Euclidean space wi…
We correct and complete a conjecture of D. Gabai, R. Meyerhoff and N. Thurston on the classification and properties of thin tubed closed hyperbolic 3-manifolds. We additionally show that if N is a closed hyperbolic 3-manifold, then either N=Vol3 or N contains a closed geodesic that is the core of an embedded tube of ra…
Smooth Riemannian manifolds can be embedded without boundary.
problem Embedding smooth Riemannian manifolds with boundary into complete manifolds without boundary.
method General gluing and conformal-deformation construction.
result Any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannian manifold without boundary.
Study Zoll manifolds with boundary, showing unique geodesic properties.
problem Characterize Zoll manifolds with boundary.
method Analyzing geodesics, boundary conditions, and manifold structure.
result All free boundary geodesics have the same length and Morse index.
New non-cobordant hyperbolic manifolds found in certain dimensions.
problem Identifying closed hyperbolic manifolds that are not cobordant.
method Using the cobordism class and fixed point set of an involution, combined with a geodesic embedding.
result Existence of non-cobordant closed hyperbolic manifolds in dimensions not of the form 4m+3. The paper shows how to recover true node positions from a graph or similarity matrix.
problem Recovering true distances and positions from a graph or similarity matrix.
method Two steps: matrix factorisation followed by nonlinear dimension reduction.
result Nonlinear dimension reduction can recover latent positions close to a manifold where geodesic distance is encoded.
Thurston's boundary to the universal Teichmüller space T(H) is the set of asymptotic rays to the embedding of T(H) in the space of geodesic currents; the boundary is identified with the projective bounded measured laminations PMLbdd(H) of H. We prove that each Teichmüller …
The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature.
This paper studies certain embedded spheres in closed affine manifolds. For n≥3, we investigate the dome bodies in a closed affine n-manifold M with its boundary homeomorphic to a sphere under the assumption that a developing map restricted to a component of ∂M^ is an embedding onto a strictly …
The study proves nearly geodesic surfaces are filling in hyperbolic 3-manifolds.
problem Proving nearly geodesic surfaces are filling in hyperbolic 3-manifolds.
method Using quasi-Fuchsian surfaces and totally geodesic surfaces, the study proves filling properties with rigorous mathematical proofs.
result Proves nearly geodesic surfaces are filling in hyperbolic 3-manifolds.