New findings on isospectral tori and harmonic maps between flat tori.
problem Determining if isospectral tori are isometric using harmonic maps.
method Examined harmonic maps between flat tori, focusing on Milnor's isospectral tori.
result Milnor's isospectral tori cannot be distinguished by harmonic maps from lower-dimensional tori, but can be distinguished by higher-dimensional ones.
The paper finds rotationally symmetric critical metrics for Laplace eigenvalues on tori.
problem Maximizing the first normalized Laplace-Beltrami eigenvalue on tori.
method Constructing equivariant harmonic maps to spheres and analyzing their properties.
result Rotationally symmetric critical metrics for the first eigenvalue are found and characterized.
Compact metric f-K-contact manifolds constructed via specific transformations.
problem Constructing all metric f-K-contact manifolds.
method Iteration of constructions of mapping tori, rotations, and type II deformations.
result Compact metric f-K-contact manifolds are derived from compact K-contact manifolds.
Researchers compute zeta-determinants and analytic torsion for metric mapping tori.
problem Computing zeta-determinants and analytic torsion for metric mapping tori.
method Using the BFK-gluing formula for zeta-determinants.
result Computed zeta-determinants and analytic torsion for metric mapping tori.
We study Thurston's Lipschitz and curve metrics, as well as the arc metric on the Teichmueller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston'…
Study minimal Lagrangian tori on Kähler manifolds, answering questions about their existence and stability.
problem Characterize minimal Lagrangian tori on Kähler manifolds.
method Investigate orbits of torus actions, analyze stability, and relate to ambient geometry.
result Partial answers to questions about minimal Lagrangian tori existence and stability.
Classifies mapping tori of specific groups, generalizing known results.
problem Classifying mapping tori of specific groups.
method Using Hopf-type properties and Poincaré Duality groups.
result Generalizes and provides new proofs for fibered 3-manifolds.
Paper explains dynamics of homeomorphisms to mapping tori geometry.
problem Understanding dynamics of end-periodic homeomorphisms.
method Illustration-driven overview of recent results.
result Analogue of Brock's theorem for infinite-type surfaces.
New non-Kähler examples of generalized Kähler manifolds constructed via mapping tori.
problem Constructing new non-Kähler generalized Kähler manifolds.
method Starting from a 3-torus and a compact Kähler manifold, constructing via mapping tori.
result Obtained new non-Kähler examples and recovered known examples of generalized Kähler solvmanifolds.
The study characterizes subgroups of mapping tori of free groups.
problem Characterizing subgroups of mapping tori of free groups.
method Using canonical maximal sub-mapping tori and relative hyperbolicity.
result Characterizes locally quasi-convex hyperbolic groups.
Minimal volume entropy vanishes for mapping tori over 3-manifolds.
problem Volume entropy of mapping tori over 3-manifolds.
method A variation of amenable category and minimal volume entropy of a homology class.
result Minimal volume entropy vanishes.
Free maps exist on low-dimensional tori and closed surfaces.
problem Embedding closed surfaces in high-dimensional spaces.
method Factorization trick for constructing free immersions.
result Every closed surface embeds freely in \(\mathbb{R}^5\).
We exhibit families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds.
New compactification of Teichmüller space via renormalized volume.
problem Compactify Teichmüller space with new distance.
method Horocompactification with renormalized volume.
result Translation length of pseudo-Anosov mapping classes equals hyperbolic volume of mapping tori.
We extend to metric compact mapping tori a splitting result for coKähler manifolds. In particular, we prove that a compact Vaisman manifold is finitely covered by the product of a Sasakian manifold and a circle.
Lower bound on volumes of special mapping tori.
problem Calculating the minimum volume of compactified mapping tori.
method Using strongly irreducible end-periodic homeomorphisms and properties of pants graphs.
result Volume of compactified mapping tori is comparable to the translation length of the homeomorphism on pants graphs.
This article studies the harmonicity of vector fields on Riemannian manifolds, viewed as maps into the tangent bundle equipped with a family of Riemannian metrics. Geometric and topological rigidity conditions are obtained, especially for surfaces and vector fields of constant norm, and existence is proved on two-tori.…
Integral filling volume of mapping tori grows sublinearly with complexity.
problem Characterizing mapping classes with vanishing integral filling volume.
method Analyzing Dehn twists and mapping tori, using simplicial volume and complexity.
result Integral simplicial volume of mapping tori grows sublinearly with respect to the monodromy power.
Homology growth of specific mapping tori vanishes for certain groups.
problem Homology growth of polynomially growing mapping tori in various groups.
method Proof of the cheap rebuilding property for specific groups.
result Torsion homology growth vanishes for Farber sequences in every degree.
Biharmonic maps are generalizations of harmonic maps. A well-known result of Eells and Wood on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere (whatever the metrics chosen) in the homotopy class of maps of Brower degree ±1. It would be interesting to know if there …
We observe that the class of metric f-K-contact manifolds, which naturally contains that of K-contact manifolds, is closed under forming mapping tori of automorphisms of the structure. We show that the de Rham cohomology of compact metric f-K-contact manifolds naturally splits off an exterior algebra, and rel…
New construction method for special geometric structures.
problem Creating strong HKT and generalized hyperkähler structures.
method Mapping tori construction of skew-symmetric torsion.
result Non-product structures with parallel skew-symmetric torsion.
Study higher rank inner products and their tilings to describe tori degenerations.
problem Understanding metric degenerations of tori.
method Introduce higher rank inner products and their tilings, use to describe degenerations.
result Describe metric degenerations of polarized tori and Hausdorff limits of tilings.
Researchers calculate dimensions of skein modules for 2-torus mapping tori.
problem Determining dimensions of Kauffman bracket skein modules for specific cases.
method Using generic q and decomposing twisted Hochschild homology of G-skein algebras. result Dimensions of skein modules for G=SL2 and G=GL1 are calculated. Researchers compute dimensions of GLN-skein modules for genus-one mapping tori.
problem Computing dimensions of GLN-skein modules for mapping tori.
method Explicit Euler product expansion of the skein partition function.
result Explicit computation of dimensions and generating function.
Proves conjecture for hyperbolic-by-cyclic groups using geometric methods.
problem Proving Farrell-Jones Conjecture for specific group structures.
method Geometric methods and structure theory of mapping tori.
result Proved Farrell-Jones Conjecture for mapping tori of automorphisms of hyperbolic-by-cyclic groups.
New theorem on flat tori stability using harmonic maps and Ricci flow.
problem Stability of flat tori under Ricci and scalar curvature bounds.
method Harmonic map heat flow, Ricci flow, and RCD theories.
result Gromov-Hausdorff stability theorem for flat 3-tori.
Counts minimal tori in Riemannian manifolds with 6 or more dimensions.
problem Counting minimal tori in Riemannian manifolds.
method Introduces a function to count minimal tori and shows invariance under metric perturbations.
result The count function is invariant under metric perturbations.
Using Takahashi theorem we propose an approach to extend known families of minimal tori in spheres. As an example, the well-known two-parametric family of Lawson tau-surfaces including tori and Klein bottles is extended to a three-parametric family of tori and Klein bottles minimally immersed in spheres. Extremal spect…
Otsuki tori form a countable family of immersed minimal two-dimensional tori in the unitary three-dimensional sphere. According to El Soufi-Ilias theorem, the metrics on the Otsuki tori are extremal for some unknown eigenvalues of the Laplace-Beltrami operator. Despite the fact that the Otsuki tori are defined in quite…
Study bounds topological entropy of maps on surfaces with punctures based on mapping torus homology.
problem Relating topological entropy of pseudo-Anosov maps to homology of mapping tori.
method Analyzing the topological entropy of pseudo-Anosov maps on surfaces with punctures and relating it to the rank of the first homology of their mapping tori.
result Entropy of a pseudo-Anosov map is bounded by a formula involving the genus, number of punctures, and homology rank.
This research studies end-periodic mapping tori and their hyperbolic structures.
problem Understanding the geometry and dynamics of end-periodic mapping tori.
method Analyzes invariant laminations and hyperbolic structures of mapping tori.
result Establishes a relationship between the geodesic length of boundary components and the infimum of geodesic lengths in hyperbolic structures.
New length functions on mapping class groups linked to simplicial volumes of mapping tori.
problem Understanding the relationship between mapping class groups and simplicial volumes of mapping tori.
method Introducing filling volumes as length functions and proving their properties.
result Real filling volumes equal the simplicial volume of mapping tori, while integral filling volumes are not smaller than the stable integral simplicial volume.
Maps from rational homology solid tori yield rank inequalities in Heegaard Floer homology.
problem Rank inequalities in Heegaard Floer homology.
method Using Hanselman-Rasmussen-Watson's bordered Floer homology, we extend their proof to rational homology solid tori.
result We provide rank inequalities for Heegaard Floer homology.
Study exotic tori and their SL_d(Z) actions, proving many do not admit nontrivial actions.
problem Characterize exotic tori admitting SL_d(Z) actions.
method Compute mapping class groups, analyze homology actions, and prove non-existence of nontrivial actions.
result Many exotic tori do not admit nontrivial SL_d(Z) actions.
Study shows tori metrics converging to flat under specific conditions.
problem Understanding convergence of metrics on tori with non-negative scalar curvature.
method Uniformly conformal metrics and controlled geometry sequences.
result Sequence of metrics converges to flat metric in multiple senses.
The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. …
Embeddings of mapping tori for end-periodic graph maps are proven.
problem Embedding mapping tori of end-periodic graph maps into finite complexes.
method Flowline-preserving homotopy equivalence and π1-injective map. result Every mapping class of Γ arising from an end-periodic homotopy equivalence contains a representative whose mapping torus realizes such an embedding.
New contact structures on folded sums of contact mapping tori are tight under certain conditions.
problem Understanding tight contact structures on folded sums of contact mapping tori.
method Alternative bundle-theoretical construction and gluing process near the fold.
result Folded contact structures on folded sums of contact mapping tori are tight under specific conditions.
Proves Schoen's conjecture on tori with specific conditions.
problem Proving Schoen's conjecture on tori with non-negative scalar curvature.
method Uses weighted scalar curvature and the relative index theorem.
result If the fundamental group of the singular set is not surjective, the metric extends to a smooth flat metric.
Study Kähler metrics on complex tori with almost non-negative scalar curvature.
problem Stability of Kähler metrics on complex tori.
method Proved convergence of non-collapsing subsequence of Kähler metrics to flat torus.
result Kähler metrics with almost non-negative scalar curvature on complex tori converge to flat torus.
We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational, integer, and finite field coefficients, the hyperbolic volume …
Non-coherence proven for certain groups with specific mapping tori.
problem Proving non-coherence for groups with specific mapping tori.
method Analyzing the structure of groups as extensions and properties of mapping tori.
result Groups with certain mapping tori are not coherent.
We show that the graph TQFT for Heegaard Floer homology satisfies a strong version of Atiyah's duality axiom for a TQFT. As an application, we compute some Heegaard Floer mixed invariants of 4-dimensional mapping tori in terms of Lefschetz numbers on HF+.
We study a generalization of the familiar Poincaré map, first implicitely introduced by N.N. Nekhoroshev in his study of persistence of invariant tori in hamiltonian systems, and discuss some of its properties and applications. In particular, we apply it to study persistence and bifurcation of invariant tori.
We consider a random walk on the mapping class group of a surface of finite type. We assume that the random walk is determined by a probability measure whose support is finite and generates a non-elementary subgroup H. We further assume that H is not consisting only of lifts with respect to any one covering. Then w…
Authors create non-isometric 3-orbifolds with identical topology and volume.
problem Finding non-isometric hyperbolic 3-orbifolds with the same topological type and volume.
method Constructing pairs of non-isometric hyperbolic 3-orbifolds with the same topological type and volume.
result Demonstrated the existence of non-isometric hyperbolic 3-orbifolds with the same topological type and volume.
We prove that for every P there is a bound B depending only on P so that the mapping torus of every P--small irreducible train-track map can be obtained by surgery from one of B mapping tori. We show that given an integer P>0 there is a bound M depending only on P, so that there exists a presentation of the fundament…