Totally geodesic surfaces found in knots and links.
problem Finding totally geodesic surfaces in knots and links.
method Constructing infinite families of knots and links with totally geodesic spanning surfaces in various 3-manifolds.
result Infinite families of knots and links with totally geodesic spanning surfaces in multiple 3-manifolds.
Homoclinic orbits found in geodesic flows on surfaces.
problem Existence of homoclinic orbits in geodesic flows.
method Kupka-Smale metric on closed surfaces.
result Homoclinic orbits for all hyperbolic geodesics.
Study on complexity of systolic geodesics on Bolza surface.
problem Complexity of systolic geodesics on the Bolza surface.
method Analysis of intersections and triangulation of geodesics.
result There are 12 second systolic geodesics forming a triangulation of the surface.
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.
Self-crossing geodesics on convex surfaces are studied.
problem Understanding patterns of geodesics crossing themselves.
method Analyzing closed geodesics on convex surfaces.
result Self-crossing geodesics exist on convex surfaces.
Paper finds shortest geodesic paths on hyperbolic surfaces.
problem Finding the shortest geodesic paths on hyperbolic surfaces.
method Analyzes genus g hyperbolic surfaces to find minimal length geodesics.
result Minimal geodesic length is realized by a specific polygon.
Study counts geodesic surfaces in knot complements, finding unique ones for small knots.
problem Counting totally geodesic surfaces in knot complements.
method Adapting boundary slope and intersection techniques, extending obstructions.
result Uniqueness of geodesic surfaces for specific knots, no geodesic surfaces for 47 knots.
The study finds totally geodesic surfaces in hyperbolic 3-manifolds and verifies a conjecture.
problem Finding totally geodesic surfaces in hyperbolic 3-manifolds.
method Developed algorithms to determine and verify the existence of totally geodesic surfaces.
result Discovered nine 3-manifolds with totally geodesic surfaces and verified Menasco-Reid's conjecture for knots up to 12 crossings.
The paper proves the existence of surfaces of section for geodesic flows on closed surfaces.
problem Existence of surfaces of section for geodesic flows on closed surfaces.
method Study of configurations of simple closed geodesics and use of the curve shortening flow.
result Construction of surfaces of section that intersect or have hyperbolic components in their boundary.
Type A surfaces are the locally homogeneous affine surfaces which can be locally described by constant Christoffel symbols. We address the issue of the geodesic completeness of these surfaces: we show that some models for Type A surfaces are geodesically complete, that some others admit an incomplete geodesic but model…
Study on geodesics on high genus expander surfaces, proving filling and non-simple properties.
problem Properties of geodesics on expander surfaces of high genus.
method Adapting Margulis' counting strategy to low length scales.
result Almost every geodesic of certain lengths is filling or non-simple.
Veering branched surfaces help construct geodesic flows on curved surfaces.
problem Constructing geodesic flows on negatively curved surfaces.
method Introduce veering branched surfaces and surgeries, then use them to construct veering triangulations that correspond to geodesic flows.
result Explicit constructions of veering branched surfaces corresponding to geodesic flows on negatively curved surfaces.
Minimal surfaces can't have certain epitrochoid geodesics.
problem Minimal surfaces and geodesics.
method Analyzing minimal immersions and epitrochoid properties.
result Minimal surfaces cannot contain certain epitrochoid geodesics.
Shortest non-simple closed geodesics on hyperbolic surfaces found.
problem Finding the shortest non-simple closed geodesics on hyperbolic surfaces.
method Analyzing closed geodesics with at least k self-intersections on hyperbolic surfaces.
result The shortest non-simple closed geodesics lie on an ideal pair of pants and have length $2\arccosh(2k+1)$ .
Proves stability of geodesic flows on closed surfaces.
problem Stability of geodesic flows on closed surfaces.
method Generic Riemannian metrics and Reeb flows.
result Proves C 2 C^2 C 2 -stability conjecture for geodesic flows. 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.
In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…
Almost Zoll affine surface found on cylinder.
problem Finding surfaces with special geodesic properties.
method Exhibited an affine structure on a cylinder.
result Affine structure on cylinder is almost Zoll.
Minimal geodesics on hyperbolic surfaces are long.
problem Finding the shortest closed geodesics on hyperbolic surfaces.
method Analyzing the self-intersection number to estimate geodesic lengths.
result The minimal length of geodesics grows logarithmically with the self-intersection number.
Closed geodesics densely cover a circle in dilation surfaces.
problem Density of closed geodesics in dilation surfaces.
method Study of Teichmüller flow and Delaunay triangulation.
result Directions of closed geodesics are dense in the circle.
The article constructs Bolza-like surfaces for infinitely many genera and studies their properties.
problem Maximizing systole functions in Teichmüller spaces for genus two and higher.
method Defining and constructing Bolza-like surfaces with specific triangulations and properties.
result Global maximal surfaces can be constructed using Bolza-like surfaces, and systolic geodesics intersect at even points.
The paper counts geodesic loops on surfaces without conjugate points.
problem Counting geodesic loops on surfaces of genus at least 2 without conjugate points.
method Proves asymptotic estimates for closed geodesic loops on compact surfaces with no conjugate points.
result Generalizes classical counting results and sector theorems for surfaces of strictly negative curvature.
Geodesics on hyperbolic surfaces become evenly spread over time.
problem Distribution of geodesics on hyperbolic surfaces.
method Equidistribution analysis of geodesics with length less than T as T approaches infinity.
result Closed geodesics on hyperbolic surfaces of finite area become evenly spread over time.
Study geodesics on K3 surfaces near orbifold limit.
problem Understanding geodesics on K3 surfaces near the orbifold limit.
method Improves metric estimates for K3 surfaces, uses hyperkähler identities.
result Restrictions and existence conditions for stable geodesics.
Estimates geodesics on surfaces without conjugate points.
problem Counting geodesics on surfaces without conjugate points.
method Margulis-type asymptotic estimates.
result Asymptotic estimates for geodesics on surfaces.
Study geodesic curvature of logarithmic spirals on curved surfaces.
problem Understanding geodesic curvature on curved surfaces.
method Computed geodesic curvature of logarithmic spirals on surfaces of constant Gaussian curvature.
result Asymptotic behavior of geodesic curvature is independent of the ambient surface's curvature.
Lower bounds on geodesic length with few intersections on hyperbolic surfaces.
problem Finding the minimum length of geodesics with at least 2 intersections.
method Analyzing geodesics on hyperbolic surfaces with at least 2 self-intersections.
result The minimum length of such geodesics is 2 log ( 5 + 2 6 ) 2\log(5+2\sqrt6) 2 log ( 5 + 2 6 ) , and this bound is sharp. We study the geodesics on an invariant surface of a three dimensional Riemannian manifold. The main results are: the characterization of geodesic orbits; a Clairaut's relation and its geometric interpretation in some remarkable three dimensional spaces; the local description of the geodesics; the explicit description o…
Currents on cusped hyperbolic surfaces have a denseness property similar to compact surfaces.
problem Proving denseness of rational currents on cusped hyperbolic surfaces.
method Using geodesic currents and subset currents, proving denseness through examples and continuous extension.
result Denseness of rational currents on cusped hyperbolic surfaces, including geodesics connecting cusps.
The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.
problem Finding the asymptotic behavior of shortest closed multi-geodesics on hyperbolic surfaces.
method Analyzing the length of shortest filling closed multi-geodesics using hyperbolic geometry and asymptotic analysis.
result The length of a shortest filling closed multi-geodesic is uniformly comparable to a specific formula involving the genus and lengths of closed geodesics.
Develops a method to define and characterize geodesics on hyperbolic surfaces.
problem Characterizing closed geodesics on hyperbolic surfaces without self-intersection.
method Constructive definition of the Goldman bracket using closed geodesics.
result Algebraic characterization of geodesics on hyperbolic surfaces.
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.
Study geodesic paths on flat surfaces, comparing length and singularity counts.
problem Comparing geometric length and singularity counts on geodesic paths.
method Apply counting limit laws to infinite graphs and then to flat surfaces.
result Statistical comparison of geometric length and singularity counts on geodesic paths.
This paper discovers new identities linking geodesic and orthogeodesic lengths on hyperbolic surfaces.
problem Understanding relationships between geodesic and orthogeodesic lengths on hyperbolic surfaces.
method Investigates a broad family of identities involving lengths of all closed geodesics and orthogeodesics.
result Introduces new identities that include lengths of all closed geodesics, contrasting with previous identities.
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.
The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces.
The paper connects Schrödinger equations to geodesics on a 2-surface.
problem Understanding the relationship between Schrödinger equations and geodesics.
method Analyzes the geodesic equation of a specific metric on a 2-surface.
result Explicit solutions for the metric and geodesics in terms of the Baker--Akhiezer function for finite-gap potentials.
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.
Improved bounds on geodesic intersections on hyperbolic surfaces.
problem Finding the shortest geodesic with a specific number of intersections.
method Proved a new formula for minimal length of geodesics with self-intersection number k.
result Improved the threshold for the existence of geodesics with self-intersection number k.
Polynomial decay of correlations shown for curved surfaces.
problem Analyzing geodesic flows on curved surfaces.
method Proving polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
result Polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
Researchers find geodesics on K3 surfaces using electrostatics.
problem Locating closed geodesics on K3 surfaces.
method Using Foscolo's construction of Ricci-flat Kahler metrics.
result Computed indices and lengths of geodesics with high precision.
New metric on geodesic currents connects different surface genera.
problem Understanding geodesic currents on surfaces of varying genera.
method Introducing a new asymmetric metric on the space of projective filling geodesic currents.
result Metric spaces of projective filling geodesic currents for surfaces of different genera are not isometric.
New results on geodesic flows using curve shortening flow.
problem Stability and existence of geodesics in Riemannian surfaces.
method Curve shortening flow approach.
result Existence of infinitely many geodesics intersecting a given one in every primitive class.
The paper calculates the volume growth of hyperbolic surfaces with short geodesics.
problem Understanding the volume growth of hyperbolic surfaces with short geodesics.
method Introduced a function L(g) to measure the length of geodesics and computed the volume growth rate.
result The volume of surfaces with short geodesics is equal to V_g almost surely as g approaches infinity.
Geodesic flows on surfaces have specific fractional-linear integrals related to constant cross-ratios.
problem Characterizing geodesic flows on surfaces with fractional-linear integrals.
method Proving the dimension of fractional-linear integrals and giving a geometric criterion.
result The dimension of fractional-linear integrals is either 3 or 5, corresponding to constant curvature.
Study proves uniqueness for ray transform on surfaces with obstacles.
problem Uniqueness of functions and 1-forms on surfaces with reflecting obstacles.
method Broken ray transform on twisted geodesics with nonpositive curvature and reflecting boundary.
result Proves uniqueness result for sums of functions and 1-forms.
This paper simplifies Anosov geodesic flows on surfaces.
problem Understanding Anosov geodesic flows on surfaces.
method Exposition of Eberlein's work, focusing on surface case.
result Provides a more accessible introduction to Anosov geodesic flows.
In this paper we study geodesic mappings of n n n -dimensional surfaces of revolution. From the general theory of geodesic mappings of equidistant spaces we specialize to surfaces of revolution and apply the obtained formulas to the case of rotational ellipsoids. We prove that such n n n -dimensional ellipsoids admit non tri…