Invariant for 3-manifolds with torus boundary defined.
problem Defining invariants for 3-manifolds with specific boundaries.
method Module over a weighted A-infinity algebra associated to a torus.
result Invariant constructed for bordered 3-manifolds with torus boundary.
A torus manifold M is a 2n-dimensional orientable manifold with an effective action of an n-dimensional torus such that MT=∅. In this paper we discuss the classification of torus manifolds which admit an invariant metric of non-negative curvature. If M is a simply connected torus manifold which a…
In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent tor…
The paper proves conditions for 2-torus manifolds to be equivariantly formal.
problem Characterizing 2-torus manifolds as equivariantly formal.
method Proving 2-torus manifolds are equivariantly formal under specific conditions.
result 2-torus manifolds are equivariantly formal if and only if the action is locally standard and all faces of the orbit space are mod 2 acyclic.
We give an affirmative answer to the Halperin-Carlsson conjecture for the homologically injective torus actions on closed manifolds. This class contains holomorphic torus actions on compact Kahler manifolds, torus actions on compact Riemannian flat manifolds.
Researchers found two types of graphs for 6D torus manifolds with Euler number 6.
problem Identifying and constructing 6D almost complex torus manifolds with specific Euler numbers.
method Examined labeled directed graphs associated with fixed points and isotropy spheres, used to construct manifolds and determine Chern numbers.
result Proved the existence of two types of 6D almost complex torus manifolds with Euler number 6.
The study determines Z2-Thurston norms in Sol manifolds and embeds non-orientable surfaces.
problem Determining Z2-Thurston norms in Sol manifolds and embedding non-orientable surfaces. method Analyzing the action of torus maps on curve complexes and constructing incompressible surfaces.
result Determination of Z2-Thurston norms and embeddability of non-orientable surfaces in Sol manifolds. Quantum trace map defined for 3-manifolds with torus boundaries.
problem Quantifying topological structures of 3-manifolds with torus boundaries.
method Defining a quantum trace map from skein module to a quantum torus module.
result Established a 3D quantum trace map for 3-manifolds with torus boundaries.
Extends Landweber-Stong result to torus actions on manifolds of generalized odd type.
problem Signature vanishing for manifolds with torus actions of generalized odd type.
method Extends Landweber-Stong result to torus actions.
result Signature vanishes for manifolds with torus actions of generalized odd type.
Study Dolbeault cohomology on complex manifolds with torus action.
problem Describe Dolbeault cohomology of complex manifolds with torus action.
method Describe Dolbeault cohomology algebra of canonical foliation, provide dga model, prove Hodge decomposition.
result Hodge decomposition for basic Dolbeault cohomology proved.
The paper classifies manifolds with free torus actions and positive Ricci curvature.
problem Classifying manifolds with specific geometric properties.
method Using circle bundles and free torus actions on connected sums of products of spheres.
result Infinitely-many manifolds with positive Ricci curvature and torus actions are classified.
A 2-torus manifold is a closed smooth manifold of dimension n with an effective action of a 2-torus group (Z2)n of rank n, and it is said to be locally standard if it is locally isomorphic to a faithful representation of (Z2)n on Rn. This paper studies the equivariant classification of locally standar…
Study on LCK metrics and their existence criteria on complex manifolds.
problem Existence of LCK metrics on complex manifolds.
method Investigation of holomorphic torus actions and biholomorphisms on LCK type manifolds.
result Existence of Vaisman and LCK metrics under specific conditions on the manifold's biholomorphism group.
Study shows non-negative curvature on 4-manifolds with torus symmetry.
problem Classifying 4-manifolds with torus symmetry under non-negative curvature.
method Investigated invariant metrics on 4-manifolds with circle and torus actions.
result Found that almost non-negative curvature implies non-negative curvature for 4-manifolds with torus symmetry.
Builds torus fibrations over singular manifolds with specific features.
problem Creating torus fibrations over manifolds with singularities.
method Constructs a topological space X as a torus fibration over an integral affine manifold B with singularities. result The fibration has a discriminant in codimension 2.
Study shows groups can act on torus without extending to 3-manifold.
problem Bordism problem for group actions on torus.
method Examples of groups acting on torus by diffeomorphisms isotopic to identity.
result Groups cannot be extended to action on bounding 3-manifold.
Combinatorial definition of bordered Floer theory for torus-boundary manifolds.
problem Defining Floer homology for manifolds with torus boundaries.
method Combinatorial definition with Z coefficients. result Recovery of combinatorial Heegaard Floer homology.
The n-dimensional torus is uniquely characterized by specific harmonic forms.
problem Characterizing the n-dimensional torus via harmonic forms.
method Analyzing closed 1-forms on the torus to determine unique properties.
result The n-dimensional torus is the unique manifold supporting a linearly independent set of (n-1) closed 1-forms whose product determines a non-zero cohomological class.
New classification for curved manifolds with specific symmetries.
problem Classifying non-negatively curved manifolds with specific symmetries.
method Extending equivariant classification results for manifolds with isotropy-maximal or strictly almost isotropy-maximal torus actions.
result Almost isotropy-maximal manifolds are diffeomorphic to products of spheres.
We associate a root system to a finite set in a free abelian group and prove that its irreducible subsystem is of type A, B or D. We apply this general result to a torus manifold, where a torus manifold is a 2n-dimensional connected closed smooth manifold with a smooth effective action of an n-dimensional compact t…
Framework extends sutured manifold invariants to pairs with torus boundaries.
problem Extending sutured manifold invariants to pairs with torus boundaries.
method Establishing a framework for extending invariants of sutured manifolds to pairs of sutured manifolds differing by attaching a basic slice along a torus boundary.
result Framework recovers bordered-sutured Floer homology for a corresponding bordered-sutured three-manifold.
We show that if a hyperbolic 3-manifold M with a single torus boundary admits two Dehn fillings at distance 5, each of which contains an essential torus, then M is a rational homology solid torus, which is not large in the sense of Wu. Moreover, one of the surgered manifold contains an essential torus which meets t…
The paper extends torus surgery results in 4-manifolds and shows diffeomorphisms.
problem Extending torus surgery results in 4-manifolds.
method Analyzing 4-manifolds diffeomorphic to S1 times exterior of fibered knots and T2imesD2 bundles over T2. result The result of torus surgery in S1imesY along S1imesK fibers over S1. We construct a new example of an A-manifold, i.e. a Riemannian manifold with a cyclic-parallel Ricci tensor, which can be viewed as a generalization of the Einstein condition. The underlying manifold for our construction is a principal torus bundle over Kähler-Einstein manifold a with fibre a torus of arbitrary dimensi…
Classifies symplectic torus actions up to equivariant symplectomorphism.
problem Classifying symplectic torus actions up to equivariant symplectomorphism.
method Classification theorems based on Duistermaat and Pelayo's work on symplectic torus actions with coisotropic orbits.
result Every almost isotropy-maximal symplectic torus action is equivariantly diffeomorphic to a product of a symplectic toric manifold and a torus.
Each closed oriented 3-manifold M is naturally associated with a set of integers D(M), the degrees of all self-maps on M. D(M) is determined for each torus bundle and torus semi-bundle M. The structure of torus semi-bundle is studied in detail. The paper is a part of a project to determine D(M) for all 3-ma…
Study shows certain knots can be surgically altered to create non-left-orderable 3-manifolds.
problem Understanding non-left-orderable fundamental groups in 3-manifolds.
method Analyzing surgeries on negatively twisted torus knots.
result Certain knots can be surgically altered to create 3-manifolds with non-left-orderable fundamental groups.
Classifies torus bundles bounding 4-manifolds with rational homology.
problem Classifying torus bundles over the circle that bound 4-manifolds with rational homology.
method Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
result Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
Study nearly Kähler 6-manifolds with 2-torus symmetry, proving geometric properties and constructing new manifolds.
problem Characterize nearly Kähler 6-manifolds with 2-torus symmetry.
method Use multi-moment map and Laplace operator eigenfunctions, analyze quotient geometry, and construct new manifolds.
result Prove T2-action is free on level sets and determine the geometry of quotients. The paper supports a conjecture about a vanishing identity for certain 3-manifolds.
problem The vanishing identity of adjoint Reidemeister torsions for hyperbolic 3-manifolds with torus boundary.
method Examined hyperbolic once-punctured torus bundles and torus knot exteriors.
result The vanishing identity holds for all hyperbolic once-punctured torus bundles with tunnel number one, but not for torus knot exteriors.
Study of rod packings in 3-torus using 3-manifold geometry.
problem Understanding crystal structures in crystallography through rod packings in 3-torus.
method Use of 3-manifold geometry and topology to analyze complements of rod packings.
result Find families of complements that are hyperbolic and Seifert fibred.
We show that the cone associated with a moment map for an action of a torus on a contact compact connected manifold is a convex polyhedral cone and that the moment map has connected fibers provided the dimension of the torus is bigger than 2 and that no orbit is tangent to the contact distribution. This may be consider…
Study mapping class groups on invariant incompressible surfaces in reducible 3-manifolds.
problem Understanding the effect of mapping class groups on incompressible surfaces in reducible 3-manifolds.
method Analyzing the mapping class group of reducible 3-manifolds and their incompressible surfaces.
result Characterize reducible 3-manifolds that admit Anosov tori.
Study nearly parallel G2-structures with torus symmetry using multi-moment maps.
problem Characterize and construct nearly parallel G2-structures with torus symmetry.
method Use multi-moment map techniques and analyze the geometry of the base spaces.
result Locally, the construction may produce examples with four-torus symmetry.
Study non-negatively curved GKM orbifolds and their cohomology.
problem Characterize non-negatively curved GKM orbifolds and their cohomology.
method Analyze rational cohomology rings and isometric actions of finite groups.
result Rational cohomology rings of GKM orbifolds are isomorphic to model orbifolds.
We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants…
We construct new examples of manifolds with cyclic-parallel Ricci tensor, so called A-manifolds, on a r-torus bundle over a product of almost Hodge A-manifolds.
Study shows cohomological limits on extending actions on 3-manifold boundaries.
problem Limits to extending group actions on 3-manifold boundaries.
method Cohomological obstructions for C0-actions on 3-manifolds. result Cohomological obstructions prevent extending actions on certain 3-manifold boundaries.
Let (G) be a connected compact non-abelian Lie-group and (T) a maximal torus of (G). A torus manifold with (G)-action is defined to be a smooth connected closed oriented manifold of dimension (2\dim T) with an almost effective action of (G) such that (M^T\neq \emptyset). We show that if there is a torus manifold (M) wi…
New equations on manifolds linked to torus actions, proving existence and uniqueness.
problem Existence and uniqueness of solutions for generalized Kazdan-Warner equations.
method Linear action of a torus on complex vector spaces, existence and uniqueness proof on compact manifolds.
result Existence and uniqueness of solutions for the generalized Kazdan-Warner equations.
Study on (λ,λ)-eigenfunctions on compact manifolds, showing manifold properties and eigenfamily dimensions.
problem Characterizing compact manifolds with (λ,λ)-eigenfunctions and understanding their eigenfamilies. method Analyzing (λ,λ)-eigenfamilies on compact Riemannian manifolds, showing that any such manifold is a mapping torus and any (λ,λ)-eigenfamily is one-dimensional. result Any compact manifold admitting a (λ,λ)-eigenfunction is a mapping torus and any (λ,λ)-eigenfamily is one-dimensional. The aim of this paper is to classify simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology. It is shown that there is a one-to-one correspondence between equivariant diffeomorphism types of these manifolds and 3-valent labelled graphs, called torus graphs introduced by Maeda-Masuda-Panov. …
Study shows closed manifolds close to flat tori under Kato Ricci curvature bounds.
problem Stability of closed Riemannian manifolds with small Kato Ricci curvature.
method Geometric and diffeomorphic stability results for manifolds with small Kato Ricci curvature.
result Closed manifolds with small Kato Ricci curvature are close to flat tori and diffeomorphic to tori.
Embeds cohomology of hyperkahler manifolds into torus cohomology.
problem Embedding cohomology of hyperkahler manifolds into torus cohomology.
method Kuga-Satake construction and embedding of graded cohomology spaces.
result Compatibility of embeddings with Hodge structures and Lie algebra actions.
Model financial dynamics using 2-manifold geometries, revealing the torus as best for cyclical data.
problem Financial forecasting using complex market data.
method Embedding market data onto 2-manifolds (S2, R2, H2, T) guided by uniformization theorem, inferring latent curvature.
result The torus geometry best predicts cyclical financial data, aligning with IS-LM theory.
Study Ricci flow on torus bundles and related manifolds.
problem Compute Ricci flow formulas for torus bundles.
method Use invariant metrics compatible with connections on principal G-bundles.
result Solutions to Ricci flows on Heisenberg groups and Berger 3-spheres.
The paper proves finiteness of cosmetic fillings on a specific type of 3-manifold.
problem Cosmetic fillings on specific 3-manifolds.
method Analysis of Dehn fillings on compact, orientable 3-manifolds with incompressible torus boundaries.
result There are only finitely many pairs of purely cosmetic fillings.
Formula calculates invariant for 3-manifolds with torus boundaries.
problem Calculating an invariant for 3-manifolds with torus boundaries.
method Generalized Chern-Simons invariant and provided a gluing formula.
result A gluing formula for the invariant d(M,ρ).