The article explores the fundamental gap in Bakry-Emery geometry.
problem The fundamental gap in Bakry-Emery geometry.
method Recalled Bakry-Emery geometry and connected eigenvalues with boundary conditions. Showed a connection between fundamental gap and Bakry-Emery geometry.
result Presented key ideas in Andrews's and Clutterbuck's proof of the fundamental gap conjecture.
Our goal is to better understand the relationship between the polyhedron and the group associated with a fundamental domain in H^3. In this paper, we will study torsion-free groups and determine a formula for how many edge classes a given abstract polyhedron must have. We will use that result to classify all fundamenta…
The study finds conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.
problem Conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.
method Analyzes conditions and examples to generalize results on topological complexity and Lusternik-Schnirelmann category.
result Generalizes results on topological complexity and Lusternik-Schnirelmann category for manifolds with abelian fundamental groups.
The fundamental group of a Riemannian manifold with δ-pinched negative curvature, δ>1/4, cannot be the fundamental group of a quasicompact Kähler manifold. The proof also implies that a non-uniform lattice in F4(−20) cannot be the fundamental group of a quasicompact Kähler manifold. We also construct examples …
We show that the fundamental quandle defines a functor from the oriented tangle category to a suitably defined quandle category. Given a tangle decomposition of a link L, the fundamental quandle of L may be obtained from the fundamental quandles of tangles. We apply this result to derive a presentation of the funda…
In this article, we obtain some further estimates of fundamental solutions comparing to the result of Chau-Tam-Yu and give some applications of the estimates on asymptotic behaviors of fundamental solutions.
Homotopy classification for certain 4-manifolds with dihedral fundamental groups.
problem Classifying the homotopy types of specific 4-manifolds with dihedral fundamental groups.
method Using quadratic 2-type and combining with results from Hambleton-Kreck and Bauer.
result Homotopy types of finite oriented Poincaré 4-complexes are determined by their quadratic 2-type when fundamental group is dihedral.
Fundamental groups of certain Kähler orbifolds have polynomial growth.
problem Understanding the fundamental groups of specific types of orbifolds.
method Analyzing the orbifold fundamental group with respect to the nef anticanonical bundle.
result The orbifold fundamental group has polynomial growth.
We show that the resulting manifold by r-surgery on a large class of two-bridge knots has left-orderable fundamental group if the slope r satisfies certain conditions. This result gives a supporting evidence to a conjecture of Boyer, Gordon and Watson that relates L-spaces and the left-orderability of their funda…
Study surfaces with free product fundamental groups, proving existence and properties.
problem Understanding fundamental groups of quasi-projective surfaces.
method Prove existence of admissible maps and use addition-deletion Lemmas.
result Existence of admissible maps and properties of fundamental groups.
Study on fundamental groups of framed circle embeddings in 4-manifolds.
problem Understanding the fundamental groups of spaces of framed embeddings of a circle in 4-manifolds.
method Investigates the fundamental groups of spaces of embeddings of S1imesD3 in 4-manifolds, focusing on framed immersed circles. result Investigates the fundamental groups of spaces of framed embeddings of a circle in 4-manifolds, providing insights into the structure of these groups.
The fundamental n-quandles of links are residually finite for n ≥ 2.
problem Residual finiteness of fundamental n-quandles of oriented links.
method Investigation of residual finiteness and subquandle separability of quandles; use of Winker's work on 3-sphere branched covers.
result Fundamental n-quandles of oriented links are residually finite for each n ≥ 2.
Study shows fundamental gap of horoconvex domains in hyperbolic space has no positive lower bound.
problem Understanding the fundamental gap of horoconvex domains in hyperbolic space.
method Analysis of fundamental gap of geodesic balls as radius goes to infinity.
result Product of fundamental gap and square of diameter has no positive lower bound for horoconvex domains.
In this note we prove a decomposition related to the affine fundamental group and the projective fundamental group of a line arrangement and a reducible curve with a line component. We give some applications to this result.
Open manifolds with nonnegative Ricci curvature and Euclidean volume growth have finitely generated fundamental groups.
problem Understanding fundamental groups of open manifolds with specific curvature properties.
method Analyzing universal covers with Euclidean volume growth and applying topological group theory.
result Fundamental groups are finitely generated and virtually abelian under given conditions.
Hedlund constructed Riemannian metrics on n-tori, n≥3 for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert's existence results of minimal geodesics are optimal for nilpotent fundamental groups.
New convex domains in hyperbolic space can have lower fundamental gap than constant potentials.
problem Finding convex domains with lower fundamental gap than constant potentials.
method Constructing specific convex domains and potentials with controlled eigenfunctions.
result Fundamental gap of −Δ+V can be strictly smaller than −Δ for convex domains. Survey on open manifolds with nonnegative Ricci curvature and open questions.
problem Understanding fundamental groups of open manifolds with nonnegative Ricci curvature.
method Survey and open questions.
result Discussion of open questions in the field.
Study connects lens spaces' fundamental group to their symplectic fillings' second Betti numbers.
problem Relationship between lens spaces' fundamental group and symplectic fillings' second Betti numbers.
method Exploration of minimal symplectic fillings of lens spaces.
result Unified and generalized results on lens spaces' fundamental group and symplectic fillings' second Betti numbers.
Proves Singer conjecture for graph manifolds with residually finite groups.
problem Proving the Singer conjecture for graph manifolds with specific properties.
method Used residual finiteness and graph manifold properties to prove the conjecture.
result Proved the Singer conjecture for extended graph manifolds and pure complex-hyperbolic higher graph manifolds.
The aim of this paper is to obtain the fundamental tone for minimal submanifolds of the Euclidean or hyperbolic space under certain restrictions on the extrinsic curvature. We show some sufficient conditions on the norm of the second fundamental form that allow us to obtain the same upper and lower bound for the fundam…
Study homomorphisms from groups to 3-manifold fundamental groups.
problem Understanding homomorphisms between groups and 3-manifold fundamental groups.
method Proves foundational results and answers specific questions.
result Answers questions posed by Reid-Wang-Zhou and Agol-Liu.
New exotic structures found on 4-manifolds with specific groups.
problem Finding exotic smooth structures on 4-manifolds with certain fundamental groups.
method Constructing new smooth structures using definite intersection forms and fundamental groups.
result Infinitely many exotic smooth structures on 4-manifolds with fundamental group Z/2Z.
Study finds bounds for fundamental tone on special Riemannian manifolds.
problem Finding bounds for fundamental tones on specific Riemannian manifolds.
method Developed a general lower bound for the fundamental tone of the p-Laplacian.
result Results applied to negatively curved manifolds, warped products, and Riemannian submersions.
Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.
problem Understanding eigenvalues and gaps in spherical triangles.
method Explicit computation of Dirichlet eigenvalues and eigenfunctions for spherical lunes and triangles.
result Fundamental gap of spherical triangles increases as the angle of the lune decreases.
We obtain a description of Poisson--Furstenberg boundaries for (random walks on) fundamental groups of compact graph-manifolds. Together with previously known results due to V.A. Kaimanovich and others, this allows one to obtain descriptions of Poisson--Furstenberg boundaries for fundamental groups of all closed 3-mani…
Study reveals fundamental group properties of manifolds with specific curvature and growth.
problem Understanding the fundamental groups of manifolds with nonnegative Ricci curvature and linear volume growth.
method Analysis of covering spaces and rigidity results for RCD spaces.
result Fundamental groups of manifolds contain subgroups of finite index or are finite.
The study finds smooth structures on specific 4-manifolds with even fundamental groups.
problem Investigating smooth structures on 4-manifolds with even fundamental groups.
method Analyzing topological, closed, oriented, non-spin 4-manifolds with given constraints.
result Existence of either none or infinitely many distinct smooth structures on specified manifolds.
Study virtual fundamental classes of derived manifolds, proving invariant vanishes.
problem Computing virtual fundamental classes for derived manifolds.
method Combining derived differential geometry and cosection localization.
result Stable pair invariants of hyperkähler fourfolds are zero.
We present two results about the relationship between fundamental groups of quasiprojective manifolds and linear systems on a projectivization. We prove the existence of a plane curve with non-abelian fundamental group of the complement which does not admit a mapping onto an orbifold with non-abelian fundamental group.…
Characterizes fundamental groups of disjointly tree-graded spaces.
problem Understanding fundamental groups of complex geometric structures.
method Defines and analyzes disjointly tree-graded spaces, characterizing their fundamental groups.
result Fundamental groups of disjointly tree-graded spaces embed into inverse limits of free products of fundamental groups of pieces.
There are various results that frame left-orderability of a group as a geometric property. Indeed, the fundamental group of a 3-manifold is left-orderable whenever the first Betti number is positive; in the case that the first Betti number is zero this property is closely tied to the existence of certain nice foliation…
Generalizes O'Neill's equations to pseudo-Finsler submersions.
problem Extending Riemannian submersion equations to pseudo-Finsler geometry.
method Generalization of O'Neill's fundamental equations to pseudo-Finsler submersions and exploration of O'Neill tensors.
result Generalized fundamental equations for pseudo-Finsler submersions.
Classifies 4-manifolds with specific properties and fundamental group.
problem Classifying 4-manifolds with fundamental group Z and boundary. method Topological and smooth classification methods, including Hermitian forms and 2-handlebody constructions. result Every Hermitian form over Z[t±1] arises as the equivariant intersection form of exotic smooth 4-manifolds. Study on the topology of leaves in singular Riemannian foliations.
problem Characterizing the topology of leaves in singular Riemannian foliations.
method Analyzing the fundamental groups and using nilpotent spaces.
result Leaves of singular Riemannian foliations are finitely covered by nilpotent spaces when M is simply connected. We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and do…
Study simplicial volume for fixed fundamental groups, finding gaps.
problem Understanding simplicial volume for manifolds with fixed fundamental group.
method Relate gap problem to rationality questions in bounded (co)homology.
result Show existence of gaps in simplicial volume spectrum at zero.
In this paper we show that the fibered isomorphism conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the fundamental groups of a large class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the neg…
For complete shrinking (gradient) Ricci solitons, we observe a quantification of Wylie's result that the fundamental group is finite.
Study satellite knots and their quandles related to incompressible tori.
problem Understanding quandles of satellite knots and their components.
method Algebraic approach to augmented fundamental quandles, presentations of fundamental quandles, and analysis of Alexander modules.
result Relationships between satellite knots, companion and pattern knots, and their fundamental quandles.
New criteria for non-isometric group actions in metric spaces.
problem Understanding group actions on non-isometric spaces.
method Generalizing results from isometric to continuous group actions.
result Criterion for cocompact cyclic groups to be inessential.
Simplified proof of a theorem about 3D shapes.
problem Proving a theorem about foliations on 3-manifolds.
method Using foliated branched covers.
result A simple proof of Novikov's theorem.
Paper finds conditions for two geodesics on complex manifolds.
problem Existence of two distinct closed geodesics on manifolds with infinite fundamental group.
method Topological and metric conditions for existence of geodesics in Riemannian and Finsler metrics.
result Generic Finsler metrics have two distinct closed geodesics.
Closed oriented 4-manifolds with the same geometrically 2-dimensional fundamental group (satisfying certain properties) are classified up to s-cobordism by their w2-type, equivariant intersection form and the Kirby-Siebenmann invariant. As an application, we obtain a complete homeomorphism classification of closed…
Study shows a central limit theorem for random coverings of manifolds with nilpotent groups.
problem Understanding the distribution of connected components in random coverings of manifolds with nilpotent fundamental groups.
method Used sampling homomorphisms from the fundamental group into the symmetric group and subgroup growth zeta functions of nilpotent groups.
result Proved a central limit theorem for the number of connected components of these random coverings.
Study on 4-manifolds restricts negatively curved metrics and fundamental groups.
problem Understanding negatively curved metrics on 4-manifolds.
method Analysis of transnormal and isoparametric functions.
result Closed 4-manifolds cannot be negatively curved.
The study shows simplicial volume finiteness for certain manifolds with amenable fundamental groups.
problem Determining the simplicial volume of manifolds with specific properties.
method Analyzing the fundamental group and using amenability properties.
result Simplicial volume is finite for certain manifolds with amenable fundamental groups.
We study the topology of small covers from their fundamental groups. We find a way to obtain explicit presentations of the fundamental group of a small cover. Then we use these presentations to study the relations between the fundamental groups of a small cover and its facial submanifolds. In particular, we can determi…