Study on spheres in simply-connected 4-manifolds with abelian complements.
problem Classifying locally flat spheres in simply-connected 4-manifolds.
method Focus on complements with abelian fundamental groups.
result New classifications of spheres in specific 4-manifolds.
Smooth actions on certain 3-spheres can't extend to acyclic 4-manifolds.
problem Smooth actions on Brieskorn homology 3-spheres that bound acyclic 4-manifolds.
method Equivariant Yang-Mills moduli spaces
result No smooth extensions of actions to acyclic 4-manifolds.
Classifies smooth manifolds homotopy equivalent to sphere products
problem Classifying smooth manifolds homotopy equivalent to sphere products
method Using normal-invariant map and explicit families of manifolds
result Classifies smooth manifolds up to almost diffeomorphism
Sphere bundles over 4-manifolds are trivial after looping, except for two cases.
problem Understanding the triviality of sphere bundles over 4-manifolds.
method Analyzing the splitting of sphere bundles after looping.
result The loop spaces of total manifolds of sphere bundles are homotopy equivalent, except for two special cases.
New examples of Kirby-Ramanujam spheres found, leading to contractible 4-manifolds.
problem Finding new examples of Kirby-Ramanujam spheres.
method Tracing the initial steps of Kirby's example and providing additional examples, showing diffeomorphisms and linear independence.
result Found three infinite families of Kirby-Ramanujam spheres that bound contractible 4-manifolds.
The paper establishes a new sphere theorem for certain types of manifolds.
problem Finding conditions under which compact manifolds are spheres.
method Developed a generalized sphere theorem for manifolds with radial Ricci curvature.
result Established conditions for compact manifolds to be topologically spheres.
2-spheres in 4-manifolds have complete concordance obstructions if they have immersed dual spheres.
problem Determining concordance of 2-spheres in 4-manifolds.
method Adapting Stong's methods to 4-manifolds, using self-intersection sets and concordance obstructions.
result Complete concordance obstructions for 2-spheres with immersed dual spheres in 4-manifolds.
Complete obstruction found for 2-spheres in 5-manifolds to be isotopic.
problem Obstructing homotopic embeddings of 2-spheres into 5-manifolds from being isotopic.
method Using a level preserving Whitney move in codimension 3 to eliminate double points, and classical methods.
result New results for simply-connected 5-manifolds and 2-spheres with algebraic dual 3-spheres.
The paper studies rigid sphere packings on 3D manifolds with boundary.
problem Investigating rigid sphere packings on 3D manifolds with boundary.
method Introducing generalized sphere packings, proving rigidity, introducing combinatorial curvature flows.
result Generalized sphere packing metrics are determined by combinatorial scalar curvature.
Manifolds can be dominated by hypersurfaces in a sphere.
problem Dominating manifolds with hypersurfaces.
method Proving any smooth, closed, oriented manifold can be dominated by a codimension 1 submanifold of the sphere.
result Any smooth, closed, oriented manifold can be dominated by a codimension 1 submanifold of the sphere.
Approximates compact and non-compact Sasakian manifolds in spheres.
problem Approximating Sasakian structures in spheres.
method CR immersions in standard spheres.
result Compact and non-compact Sasakian manifolds can be approximated.
The paper studies rigidity of sphere packings on 3D manifolds with boundary.
problem Rigidity of sphere packings on 3D manifolds with boundary.
method Introduced generalized Thurston's sphere packings and proved their rigidity properties.
result Generalized Thurston's sphere packings are locally determined by combinatorial scalar curvatures and cannot be deformed while keeping combinatorial Ricci curvatures fixed.
We introduce and study a new class of homotopy spheres called Farrell-Jones spheres. Using Farrell-Jones sphere we construct examples of closed negatively curved manifolds M2n, where n=7 or 8, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a que…
The paper defines a new structure on tangent sphere bundles and characterizes their properties.
problem Characterizing properties of tangent sphere bundles with contact pseudo-metric structures.
method Introduced a contact pseudo-metric structure on TεM and proved manifold properties based on constant sectional curvature. result The tangent sphere bundle TεM is (κ,μ)-contact pseudo-metric manifold if and only if the manifold M has constant sectional curvature. Sphere theorems for specific manifolds with curvature constraints.
problem Sphere theorems for Riemannian manifolds with scalar curvature bounds and non-collapsed RCD(n−1,n) spaces. method Analysis of scalar curvature and mean distance constraints.
result Established sphere theorems for the specified manifolds.
The Price twist creates three 4-manifolds from a 4-sphere.
problem Understanding the properties of a non-simply connected 4-manifold created from a 4-sphere.
method Cutting and pasting operation on a P2-knot S in a 4-manifold. result The non-simply connected 4-manifold τS(S4) is studied for Kinoshita type P2-knots. Modified proof constructs dual spheres for 4-manifolds.
problem Prove dual spheres for 4-manifolds.
method Modified proof of disc embedding theorem, geometric construction.
result Constructs geometrically dual spheres.
The paper studies geometric structures on tangent and sphere bundles over statistical manifolds.
problem Defining geometric structures on tangent and sphere bundles over statistical manifolds.
method Using a statistical structure (g,abla), the paper defines a Riemannian structure on the tangent bundle and derives expressions for various curvatures. result Basic formulas for the geometry of sphere bundles are established, and rigidity results are proved for these structures.
Study of 3-manifolds in 5-sphere using bridge decompositions.
problem Understanding embeddings of 3-manifolds in 5-sphere.
method Introduce and study bridge decompositions, use multisections of 5-manifolds.
result Every embedded 3-manifold admits a bridge decomposition.
Study finds bound on energy of minimal spheres on complex manifolds.
problem Finding bounds on energy of minimal spheres on complex manifolds.
method Proving existence of harmonic spheres with Morse index bound one.
result Sum of energies of minimal spheres realizes a geometric invariant width.
A manifold has infinitely many sphere fibrations over a sphere.
problem Infinitely many fibrations over a sphere.
method Examining X=S2imesS3 and finding fibrations over B=S2 for lens spaces. result Found infinitely many fibrations from L(p,1) to X over B. We prove that two spheres of the same constant mean curvature in an arbitrary homogeneous three-manifold only differ by an ambient isometry, and we determine the values of the mean curvature for which such spheres exist. This gives a complete classification of immersed constant mean curvature spheres in three-dimension…
We show that a graph manifold which is a Z-homology 3-sphere not homeomorphic to either the 3-sphere or the Poincaré homology 3-sphere admits a horizontal foliation. This combines with known results to show that the conditions of not being an L-space, of having a left-orderable fundamental group, and of admitting a co-…
Study finds infinite families of Sasaki-Einstein metrics on spheres.
problem Finding Sasaki-Einstein metrics on spheres and exotic spheres.
method Analyzing odd-dimensional spheres and exotic spheres that bound parallelizable manifolds.
result Infinitely many families of Sasaki-Einstein metrics on spheres and exotic spheres.
In this paper, we prove some differentiable sphere theorems and topological sphere theorems for submanifolds in Kähler manifold, especially in complex space forms.
The study classifies transverse spheres in flag manifolds and finds new examples.
problem Classifying transverse spheres in flag manifolds.
method Using topological K-theory and constructions of transverse spheres.
result Classification of transverse spheres in various flag manifolds.
We show that Brieskorn manifolds with their standard contact structures are contact branched coverings of spheres. This covering maps a contact open book decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.
Study of spheres and circles on a manifold with a specific metric structure.
problem Understanding geometric objects on a manifold with a skew-circulant structure.
method Analyzing hyper-spheres, spheres, and circles in a tangent space of a 4D manifold with a skew-circulant tensor structure.
result Characterization of geometric objects under an indefinite metric.
Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
problem Characterizing manifolds that are both rational homology spheres and double disk bundles.
method Analyzing the structure of manifolds as unions of disk bundles and using properties of rational homology and cohomology.
result Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
A filling Dehn sphere Σ in a closed 3-manifold M is a sphere transversely immersed in M that defines a cell decomposition of M. Every closed 3-manifold has a filling Dehn sphere. The Montesinos complexity of a 3-manifold M is defined as the minimal number of triple points among all the filling Dehn spheres …
A strategy for constructing an embedded sphere in a 4-manifold realizing a given homology class which has been successfully applied in the past is to represent the class as a first step stably by an embedded sphere, i.e. after adding products of 2-spheres, and to move that sphere back into the original manifold. In thi…
In this paper, we introduce a non linear ODE method to construct CMC surfaces in Riemannian manifolds with symmetry. As an application we construct unstable CMC spheres and outlying CMC spheres in asymptotically Schwarzschild manifolds with metrics like gij=(1+l1)2δij+O(l−2). The existence of uns…
A taut contact sphere on a 3-manifold is a linear 2-sphere of contact forms, all defining the same volume form. In the present paper we completely determine the moduli of taut contact spheres on compact left-quotients of SU(2) (the only closed manifolds admitting such structures). We also show that the moduli space of …
Proves existence of sphere foliations with prescribed mean curvature on Riemannian manifolds.
problem Finding sphere foliations with prescribed mean curvature on Riemannian manifolds.
method Proves existence of foliations by spheres with mean curvature proportional to a given function on non-degenerate critical points.
result Essentially unique foliation of spheres with prescribed mean curvature exists in a neighborhood of a non-degenerate critical point.
Curved 10-manifolds with torus symmetry are spheres or complex projective spaces.
problem Characterizing positively curved manifolds with torus symmetry.
method Analyzing actions of 3-dimensional tori on closed, simply connected 10-manifolds. result Closed, simply connected, positively curved 10-manifolds with T3-symmetry are homotopy spheres or complex projective spaces. The paper explores conditions for homology spheres to bound acyclic smooth manifolds and symplectic fillings.
problem Conditions for integral homology 3-spheres to bound acyclic smooth 4-manifolds and their symplectic fillings.
method Structural results and analysis of smooth embeddings of lens spaces in C2. result Smooth embeddings of connected sums of lens spaces in C2 cannot be upgraded to Stein embeddings. We present a systematic calculation of the volumes of compact manifolds which appear in physics: spheres, projective spaces, group manifolds and generalized flag manifolds. In each case we state what we believe is the most natural scale or normalization of the manifold, that is, the generalization of the unit radius co…
The chromatic number of sphere graphs in 3-manifolds is bounded.
problem Understanding the chromatic number of sphere graphs in 3-manifolds.
method Analogous to curve graphs of surfaces, bounds are provided using the prime decomposition of 3-manifolds.
result Upper and lower bounds for the chromatic number of sphere graphs in 3-manifolds are derived.
The study of symmetries in manifolds derived from colored polytopes.
problem Existence and types of symmetries in rational homology 3-spheres.
method Analysis of hyperbolic manifolds and right-angled polytopes.
result Described how to create colorings with specific symmetries.
Study calculates indices and nullities of focal manifolds in spheres.
problem Calculating indices and nullities of focal manifolds in spheres.
method Analyzes isoparametric hypersurfaces in spheres with three distinct principal curvatures.
result Index equals the ambient space dimension, nullity determined by Killing vector fields.
The study shows manifolds with special generic maps also have nice multisections.
problem Characterizing manifolds with special generic maps and multisections.
method Analyzing manifolds with special generic maps and their properties, and showing how these maps restrict the differentiable structures of spheres and manifolds.
result Manifolds admitting special generic maps also admit nice generalized multisections.
New constraints on embedded spheres and projective planes in 4-manifolds from Seiberg-Witten theory.
problem Constraints on configurations of embedded spheres and real projective planes in 4-manifolds.
method Equivariant Seiberg-Witten invariants and gluing formula for relative Seiberg-Witten invariants.
result Existence of certain configurations of surfaces leads to 4-manifolds of non-simple type.
LVM and LVMB manifolds are a large family of examples of non kähler manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be seen as LVMB manifolds. The LVM manifolds have a very natural action of the real torus and the quotient of this action is a simple polytope. This quotient allows us to relate c…
LVM and LVMB manifolds are a large family of examples of non kahler manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be seen as LVMB manifolds. The LVM manifolds have a very natural action of the real torus and the quotient of this action is a simple polytope. This quotient allows us to relate c…
The paper classifies 3D spherical Sasakian manifolds using geometric and algebraic methods.
problem Classifying 3D spherical Sasakian manifolds with specific properties.
method Establishing correspondence between different sets of parameters and geometrically describing the moduli space.
result Determination of Sasakian automorphism groups and detection of homogeneous Sasakian manifolds.
New steady Euler flows found on 3-sphere and Sasakian manifolds.
problem Finding new steady Euler solutions on specific manifolds.
method Bifurcating from an existing ansatz and extending it to Sasakian 3-manifolds.
result Previously known solutions are not isolated, and new solutions found on 3-sphere and other Sasakian manifolds.
We show how the periodicity of a homology sphere is reflected in the Reshetikhin-Turaev-Witten invariants of the manifold. These yield a criterion for the periodicity of a homology sphere.
Alexander trick applied to homology spheres for manifold homeomorphisms.
problem Group of homeomorphisms of contractible manifolds.
method Strong uniqueness statement for one-sided h-cobordisms.
result Group of homeomorphisms is contractible for d≥6.