New research finds 145 infinite families of CS spheres are standard.
problem Determining which Cappell-Shaneson spheres are diffeomorphic to the standard 4-sphere.
method Using Kirby calculus and new families of CS spheres.
result Proves 145 new infinite families of CS spheres are standard.
Solves four problems related to sphere families in 3D space.
problem Four basic problems of sphere families in Euclidean 3-space.
method Solves all four basic problems of sphere families in Euclidean 3-space.
result All four basic problems are solved.
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.
It is known by the author that there exist 20 families of Dehn surgeries in the Poincaré homology sphere yielding lens spaces. In this paper, we give the concrete knot diagrams of the families and extend them to families of lens space surgeries in Brieskorn homology spheres. We illustrate families of lens space surgeri…
We construct a new infinite family of models of exotic 7-spheres. These models are direct generalizations of the Gromoll-Meyer sphere. From their symmetries, geodesics and submanifolds half of them are closer to the standard 7-sphere than any other known model for an exotic 7-sphere.
Extends exotic embeddings of RP^2 to a larger family and produces homotopy spheres.
problem Constructing exotic embeddings of RP^2 and homotopy spheres.
method Using Montesinos knots and roll-spun knots to prove the existence of homotopy spheres.
result An infinite family of homotopy spheres and homotopy CP^2s are produced.
The paper constructs metrics on spheres with families of minimal hypersurfaces.
problem Finding Riemannian metrics on spheres with specific families of minimal hypersurfaces.
method Used Nash-Moser Inverse Function Theorem in the tame maps setting.
result Generalized Guillemin's theorem for Zoll families of minimal hypersurfaces.
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 explores the topology and curvature of isoparametric families in spheres.
problem Investigating the topology and curvature of isoparametric families in spheres.
method The paper investigates the topology and curvature of isoparametric families in spheres using homotopy, homeomorphism, diffeomorphism types, parallelizability, and Lusternik-Schnirelmann category.
result The paper determines conditions for non-negative sectional curvatures and positive Ricci curvatures in isoparametric families.
The paper shows how certain circle families in S1imesD3 relate to sphere families in S2imesD2 and induces nontrivial barbell diffeomorphisms.
problem Understanding the relationship between circle and sphere families in specific 3-manifolds.
method Analyzing the fundamental groups and ambient extensions of circle and sphere families.
result Induces nontrivial barbell diffeomorphisms of S1imesS2imesI. New Zoll families of minimal spheres found in spheres and projective spaces.
problem Finding new Zoll families of minimal spheres in various spaces.
method Equivariant constructions using Nash-Moser-Hamilton implicit function theorem.
result First examples of metrics on real projective spaces with Zoll families of minimal projective hyperplanes.
We compute the Heegaard-Floer homology for the family Sigma(2,3,6n+1) of Brieskorn spheres using the algorithm by P. Ozsvath and Z. Szabo
For all positive integers n we construct a 1-parameter family of conformal tori of revolution in the 3-sphere with n bulges. These tori arise by Darboux transformations of constant mean curvature tori but do not have constant mean curvature in the 3-sphere.
New families of Brieskorn spheres bound rational homology balls.
problem Identifying new Brieskorn spheres that bound rational homology balls.
method Using techniques from Akbulut and Larson's work, we present new families of Brieskorn spheres.
result We discover new infinite families of Brieskorn spheres that non-trivially bound rational homology balls.
We discuss two kinds of almost contact metric structures on a one-parameter family of totally umbilical hyperspheres in the nearly Kaehler unit 6-sphere.
The study characterizes and constructs polynomial harmonic morphisms on spheres.
problem Characterizing and constructing polynomial harmonic morphisms on spheres.
method Characterization and construction of polynomial harmonic morphisms using eigenfamilies.
result Strong restrictions and classification of polynomial harmonic morphisms in low dimensions.
The paper finds infinite families of exotic spheres with free actions.
problem Detecting smooth free S1 and S3 actions on exotic spheres. method Topological modular forms.
result Infinite families of very exotic spheres with free actions.
Infinite Klein bottles with 4-fold meridians found.
problem Finding indecomposable Klein bottles with specific meridian orders.
method Exhibited an infinite family of Klein bottles in 4-sphere.
result Infinite family of indecomposable Klein bottles with order-4 meridians.
We prove that the infinite family of homotopy 4-spheres constructed by Daniel Nash are all diffeomorphic to 4-sphere.
The paper studies geodesics and isoparametric functions on Finsler spheres.
problem Analyzing geodesics and isoparametric functions on Finsler spheres.
method Global expressions of geodesics and isoparametric functions derived using navigation and Cartan-Münzner polynomials.
result Construction of isoparametric families and focal submanifolds.
The author, in her previous paper, constructed an infinite family of 3-bridge links each of which admits infinitely many 3-bridge spheres up to isotopy. In this paper, we prove that if a prime, unsplittable link L in S3 admits infinitely many 3-bridge spheres up to isotopy then L belongs to the family.
New links split by integer homology spheres but not by others.
problem Characterizing links split by integer homology spheres.
method Constructing specific links and homology spheres.
result Infinite families of links and homology spheres split by specific ones but not by others.
The paper proves geometric inequalities in sphere using locally constrained flows.
problem Deriving geometric inequalities in sphere.
method Established the longtime existence and convergence of a locally constrained flow.
result Proved new families of three-term geometric inequalities in sphere.
Hildebrand classified all semi-homogeneous cones in R3 and computed their corresponding complete hyperbolic affine spheres. We compute isothermal parametrizations for Hildebrand's new examples. After giving their affine metrics and affine cubic forms, we construct the whole associated family for each of Hil…
In this paper we construct families of homology spheres which bound 4-manifolds with intersection forms isomorphic to −E8. We show that these families have arbitrary large correction terms. This result says that among homology spheres, the difference of the maximal rank of minimal sub-lattice of definite filling and…
For each rational homology 3-sphere Y which bounds simply connected definite 4-manifolds of both signs, we construct an infinite family of irreducible rational homology 3-spheres which are homology cobordant to Y but cannot bound any simply connected definite 4-manifold. As a corollary, for any coprime integers $p,…
New minimal surfaces found in 3D sphere with low genus.
problem Finding minimal surfaces in 3D sphere with low genus.
method Equivariant min-max procedure applied to a novel sweepout.
result Discovered new minimal surfaces with low genus.
New families of embeddings in 4-manifolds, topologically trivial but smoothly non-trivial.
problem Constructing non-trivial smooth embeddings of 3-manifolds in 4-manifolds.
method Parameterized families of embeddings, using high-dimensional spheres.
result Embeddings of homology spheres and any 3-manifold in blown-up K3 surfaces.
Study spectral gaps in hyperbolic rational homology spheres.
problem Finding spectral gaps in hyperbolic rational homology spheres.
method Construction of families of hyperbolic rational homology spheres with coexact 1-form spectral gaps.
result Provided intervals containing limit points of spectral gaps, with the rightmost interval being [0.8196, 0.8277].
New tools prove smooth actions on exotic spheres.
problem Existence of smooth actions on exotic spheres.
method Homotopy-theoretic tools, complex and quaternionic Mahowald invariants.
result Existence of smooth U(1)- and Sp(1)-actions on exotic spheres. New bounds and examples for sphere unknotting numbers.
problem Comparing unknotting numbers for 2-spheres in 4-space.
method Algebraic and geometric techniques.
result Stabilization number is bounded above by one more than Casson-Whitney number.
An infinite family of generalized pseudo-Anosov homeomorphisms of the sphere S is constructed, and their invariant foliations and singular orbits are described explicitly by means of generalized train tracks. The complex strucure induced by the invariant foliations is described, and is shown to make S into a complex sp…
New contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
problem Existence of contractible domains with specific boundary conditions.
method Local bifurcation argument around geodesic disks, anisotropic Hölder spaces, computer-assisted techniques.
result Existence of nontrivial contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
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.
Researchers create a family of solitons connecting a cigar to a sphere.
problem Classifying steady gradient Ricci-Yang-Mills solitons on surfaces.
method Constructed a 1-parameter family of solitons on surfaces.
result Any complete steady gradient Ricci-Yang-Mills soliton on a surface must come from this family.
Curved flats linked to pairs of Lie applicable surfaces.
problem Understanding curved flats in Lie sphere geometry.
method One-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces via Darboux transformation.
result Curved flats correspond to specific Lie applicable surface pairs.
This article sets out to serve a dual purpose. On the one hand, we give an explicit description of the Lagrangian submanifolds of the nearly Kaehler 6-sphere which are ruled by circles of constant radius using Weierstrass formulae. On the other, we recognise all previous known examples of these Lagrangians as being rul…
Study of Randers metrics on spheres with simple cut loci.
problem Understanding Randers metrics on spheres and their cut loci.
method Analyzing geodesics, conjugate, and cut loci of Finsler metrics of Randers type.
result Found new families of Randers metrics with simple cut loci.
New family of triangulated 3-spheres identified from trees.
problem Challenges in enumerating triangulations of 3-spheres.
method Identifying a restricted family of triangulations and proving bijections with triples of trees.
result These triangulations are in bijection with a combinatorial family of triples of plane trees.
The paper explores biconservative surfaces in a 4D sphere, finding a unique family of non-isometric surfaces.
problem Characterizing biconservative surfaces with a parallel normalized mean curvature vector field in a 4D sphere.
method Analyzes existence and uniqueness, derives local parametrization.
result A 2-parameter family of non-isometric biconservative surfaces in a 4D sphere.
New normalizing flows for sphere distributions improve complexity and scale handling.
problem No straightforward normalizing flows for Fisher-Bingham distributions in higher dimensions.
method Zoom-linear-project (ZLP)-Fisher flows that gradually add complexity and handle varying scales.
result Generalizes Fisher-Bingham distributions to normalizing flows in any dimension.
This is a preliminary note on a family of minimal surfaces in the 3-sphere defined by a compatible fourth order equation. The minimal surfaces are geometrically characterized either by having a surface of revolution like induced metric, or by having a flat structure 3-web. We observe that the structure equation un-coup…
There are known infinite families of Brieskorn homology 3-spheres which can be realized as boundaries of smooth contractible 4-manifolds. In this paper we show that free periodic actions on these Brieskorn spheres do not extend smoothly over a contractible 4-manifold. We give a new infinite family of examples in which …
A metric with positive sectional curvature on the Gromoll-Meyer exotic 7-sphere is constructed explicitly. The proof relies on a 2-parameter family of left invariant metrics on Sp(2) and a one-parameter family of conformal deformations via an isoparametric function F on it. One byproduct is a metric with positive secti…
We investigate families of Legendrian submanifolds of 1-jet spaces by developing and applying a theory of families of generating family homologies. This theory allows us to detect an infinite family of loops of Legendrian n-spheres embedded in the standard contact (2n+1)-space (for n>1) that are contractible in the smo…
We show that there exists an infinite family of knots, each of which has, for each integer k>=0, a destabilized (2k+5)-bridge sphere. We also show that, for each integer n>=4, there exists a knot with a destabilized 3-bridge sphere and a destabilized n-bridge sphere.
New surfaces described that are symmetric and solve a specific equation.
problem Understanding symmetric shapes of membranes.
method Characterized and described axially symmetric Helfrich spheres using the reduced membrane equation.
result These surfaces are symmetric and belong to a specific family.
Authors create stable proper biharmonic maps from unit ball to spheres.
problem Constructing stable proper biharmonic maps from compact domains.
method Established second variation formula of bienergy, examined stability of previously constructed maps.
result Existence of an explicit family of stable proper biharmonic maps to Euclidean spheres.