For smooth embeddings of an integral homology 3-sphere in the 6-sphere, we define an integer invariant in terms of their Seifert surfaces. Our invariant gives a bijection between the set of smooth isotopy classes of such embeddings and the integers. It also gives rise to a complete invariant for homology bordism classe…
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48 results for “embedded spheres”
A surface in the 4-sphere is trivially embedded, if it bounds a 3-dimensional handle body in the 4-sphere. For a surface trivially embedded in the 4-sphere, a diffeomorphism over this surface is extensible if and only if this preserves the Rokhlin quadratic form of this embedded surface.
The paper proves the existence of at least 4 embedded minimal tori in a three-sphere with positive Ricci curvature.
problem Proving the existence of embedded minimal tori in three-spheres with positive Ricci curvature.
method The proof relies on a multiplicity one theorem for the Simon-Smith min-max theory.
result There exist at least 4 distinct embedded minimal tori in the three-sphere with positive Ricci curvature.
New compact mean convex hypersurfaces found for positive λ.
problem Finding compact embedded hypersurfaces for positive λ.
method Constructing compact mean convex hypersurfaces diffeomorphic to spheres.
result No compact convex embedded λ-hypersurfaces except a round sphere for λ > 0.
Authors construct hypertori with constant negative mean curvature in a sphere.
problem Constructing constant mean curvature hypertori in a sphere.
method Constructing two different constant mean curvature -dimensional hypertori in a -dimensional sphere.
result Two different constant mean curvature -dimensional hypertori with negative mean curvature in a -dimensional sphere.
A graph embedded in the 3-sphere is called irreducible if it is non-splittable and for any 2-sphere embedded in the 3-sphere that intersects the graph at one point the graph is contained in one of the 3-balls bounded by the 2-sphere. We show that irreducibility is preserved under certain deformations of embedded graphs…
Smoothly knotted 5RP^2 found in 4-sphere.
problem Finding knotted embeddings in higher dimensions.
method Topological and smooth knotting analysis in 4-sphere.
result Smoothly knotted 5RP^2 found in 4-sphere.
Proves existence of at least two minimal spheres in any 3D space.
problem Existence of minimal spheres in arbitrary 3D spaces.
method Iterative relative min-max constructions.
result Proves existence of at least two embedded minimal spheres.
In this paper, we show that, under arbitrary bounded Willmore energy assumption, embedded Willmore spheres (or more generally, embedded Willmore spheres under area constraint) with small diameter in a given -dimensional Riemannian manifold necessarily concentrate at a critical point of the scalar curvature …
By the Fox's re-embedding theorem, any compact submanifold of the 3-sphere can be re-embedded in the 3-sphere so that it is unknotted. It is unknown whether the Fox's re-embedding can be replaced with twistings. In this paper, we will show that any closed 2-manifold embedded in the 3-sphere can be unknotted by twisting…
Constraints on embedded spheres and real projective planes in 4-manifolds from Seiberg-Witten theorymath.GT
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.
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…
The paper extends isometric embedding results to null cones and spheres.
problem Isometric embedding of metrics on spheres in null cones.
method Extending Li-Wang's result to compact manifolds, specializing to 2D, developing existence and uniqueness theorems, and proving foliations.
result Existence and uniqueness of isometric embeddings in null cones.
Minimal spheres found in ellipsoids with large axes.
problem Finding non-planar minimal spheres in ellipsoids.
method Quantitative proof of minimal spheres existence with constraints.
result Ellipsoids with large axes contain at least three non-planar minimal spheres.
New formulas for minimal surfaces with specific end conditions.
problem Existence and explicit formulas for minimal surfaces with embedded planar ends.
method Provided new explicit formulas for genus 0 minimal surfaces in R^3 with 2k+1 embedded planar ends.
result Existence and explicit formulas for minimal surfaces with 2k+1 embedded planar ends for all k ≥ 4.
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.
The minimal dimension of a sphere with an equivariant embedding of the bouquet of circles is math.GT
Minimal sphere dimension for equivariant embedding of circles.
problem Embedding a bouquet of circles into a sphere.
method Finding the minimal dimension of the sphere for equivariant embedding.
result The minimal dimension is .
Similar simplices can be inscribed in most smoothly embedded spheres.
problem Inscribing families of similar simplices in spheres.
method Diffeomorphic mapping and techniques from previous work on inscribing triangles.
result A dense family of spheres allows inscribing similar simplices of every pose.
New methods detect nontrivial loops of spheres in 4-manifolds.
problem Detecting nontrivial loops of spheres in 4-manifolds.
method Pseudo-isotopy and embedding space theory.
result Invariant detects nontrivial loops of spheres in and connected sums.
Improved lower bound for the first eigenvalue of embedded minimal hypersurfaces in the unit sphere
problem First eigenvalue of embedded minimal hypersurfaces
method Establishing an improved lower bound
result Better than Duncan-Sire-Spruck's bound
In this paper, we characterize non-hyperbolic 3-component links in the 3-sphere whose exteriors contain essential 3-punctured spheres with non-integral boundary slopes. We also show the existence of embeddings of some multibranched surfaces in the 3-sphere which satisfy some homological conditions to be embedded in the…
Study embeddings of manifolds via acyclic maps and surgery.
problem Relationship between embeddings of manifolds in spheres.
method Solve Poincaré duality variant, apply surgery machine, focus on homology spheres.
result Deduce results about homotopy type of embedding spaces.
Any Sasakian structure can be closely mimicked by embeddings into weighted spheres.
problem Approximating Sasakian structures on closed manifolds.
method Using CR embeddings into weighted Sasakian spheres and strengthening previous approximation results.
result Sasakian structures can be approximated in the -norm by embeddings into weighted Sasakian spheres.
The paper explores CMC hypersurfaces in spheres, verifying Yau's conjecture.
problem Exploring the space of CMC hypersurfaces in spheres.
method Description and verification of CMC hypersurfaces, focusing on cases.
result Verification of Yau's conjecture for minimal hypersurfaces in spheres.
New -hypersurfaces not isometric to standard spheres.
problem No Alexandrov theorem for -hypersurfaces.
method Constructing compact embedded -hypersurfaces diffeomorphic to a sphere.
result Found -hypersurfaces not isometric to standard spheres.
The paper bounds the min-max width of embedded circles on spheres and manifolds.
problem Bounding the min-max width of embedded circles on spheres and manifolds.
method Inducing a sweepout by pairs of points in embedded circles from a given sweepout of the sphere by closed curves.
result Lower bounds for the Birkhoff min-max invariant of a Riemannian sphere in terms of the min-max width of its embedded circles.
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.
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 constructs homotopy 4-spheres using pochette surgery.
problem Creating homotopy 4-spheres from pochette surgeries.
method Pochette surgery generalizes Gluck surgery to construct embeddings of pochettes into the 4-sphere and proves homotopy 4-spheres are diffeomorphic to the 4-sphere.
result Homotopy 4-spheres obtained from pochette surgeries are all diffeomorphic to the 4-sphere.
We describe the (complex) quaternionic geometry encoded by the embeddings of the Riemann sphere, with nonnegative normal bundles.
For a non-orientable closed surface standardly embedded in the 4-sphere, a diffeomorphism over this surface is extendable if and only if this diffeomorphism preserves the Guillou-Marin quadratic form of this embedded surface.
Anti-self-dual blowupsmath.DG
New metric found for 4-manifolds with specific properties.
problem Finding metrics on 4-manifolds with embedded spheres.
method Constructing a Riemannian metric with anti-self-dual harmonic forms.
result Existence of a metric representing a cohomology class of a sphere.
We show that a compact embedded minimal or constant mean curvature annulus with non-vanishing Gaussian curvature which is tangent to two spheres of same radius or tangent to a sphere and meeting a plane in constant contact angle is rotational.
The paper calculates Morse indices and nullities for embedded networks on spheres.
problem Computing Morse indices and nullities for embedded networks on spheres.
method Using the Dirichlet-to-Neumann map and properties of eigenvalues and eigenfunctions.
result For all stationary triple junction networks in , there is only one eigenvalue -1.
It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in . In this note we show that distortion minimisers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that is…
Minimal surfaces in spheres found for any genus.
problem Finding minimal surfaces with arbitrary genus in 3-spheres.
method Topological structure analysis and embedding theorem.
result Every positive Ricci curvature 3-sphere contains a genus g surface.
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 .
result Smooth embeddings of connected sums of lens spaces in cannot be upgraded to Stein embeddings.
New types of Delaunay hypersurfaces found in spheres.
problem Characterizing Delaunay hypersurfaces in spheres.
method Analyzing hypersurfaces in with .
result Found new types of Delaunay hypersurfaces, including embedded ones.
Two spheres found with specific curvature constraints.
problem Existence of spheres with prescribed mean curvature.
method Proved existence of at least two embedded spheres with curvature satisfying pinching condition.
result Existence of at least two embedded spheres with prescribed mean curvature .
Maps on surfaces can be embedded into spheres with minimal dimensions.
problem Embedding periodic maps of surfaces into spheres with the smallest possible dimensions.
method Determining the minimal dimensions for embeddings of periodic maps of order on surfaces of genus into spheres .
result For each integer , there exist infinitely many periodic maps such that the smallest possible is equal to .
The study finds conditions for minimal spheres into ellipsoids using eigenfunctions.
problem Conditions for branched minimal immersions of spheres into ellipsoids to be embedded.
method Using eigenfunctions with respect to a critical metric, the study provides sufficient conditions for embeddings and constructions of non-planar minimal spheres.
result Conditions for embeddings and constructions of non-planar minimal spheres using eigenfunctions.
Spheres in curve complexes are almost simply connected.
problem Understanding connectivity of spheres in curve complexes.
method Defining spheres as induced subgraphs and showing almost simple connectivity.
result Spheres in high-complexity surfaces are almost simply connected.
3-balls in 4-sphere become isotopic in 5-ball.
problem Whether 3-balls in 4-sphere become isotopic in 5-ball.
method Analyzing the embedding of 3-balls in 4-sphere and 5-ball.
result Affirmative answer to Gay, Hughes, Kim, and Miller's question.
Veronese minimizes normal curvatures to sphere.
problem Bounding normal curvatures of submanifolds.
method Veronese embeddings of projective planes.
result Optimal bound on normal curvatures guarantees sphere.
The paper connects diffeomorphism groups and sphere embeddings, proving a group structure.
problem Understanding the homotopy types of diffeomorphism groups and sphere embeddings.
method Cerf's upgraded proof, scanning maps, canceling handles, Embedding Calculus.
result The monoid of Schoenflies spheres forms a group under connect-sum.
We show that one-sided Alexandrov embedded constant mean curvature cylinders of finite type in the 3-sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3-sphere are rotational.
Triple torsion, triple cup products, and embedding obstructions for rational homology 3-spheresmath.GT
Identifies a mod- triple cup product for rational homology 3-spheres with specific first homology.
problem Locally flat embeddings in for rational homology 3-spheres
method Using triple torsion linking form and torsion-linking duality
result Identifies the mod- triple cup product for specific rational homology 3-spheres
In this note we study whether specific elements in the second homology of specific simply connected closed -manifolds can be represented by smooth or topologically flat embedded spheres.