Simple sphere eversion with a unique point.
problem Sphere eversion with a unique quadruple point.
method Inspired by Morse theory, without homotopical tools.
result Unique quadruple point in sphere eversion.
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48 results for “sphere eversion”
For decades, the sphere eversion has been a classic subject for mathematical visualization. The 1998 video "The Optiverse" shows geometrically optimal eversions created by minimizing elastic bending energy. We contrast these minimax eversions with earlier ones, including those by Morin, Phillips, Max, and Thurston. The…
We develop a general Minmax procedure in Euclidian spaces for constructing Willmore surfaces of non zero indices. We implement this procedure to the Willmore Minmax Sphere Eversion in the 3 dimensional euclidian space. We compute the cost of the Sphere eversion in terms of Willmore energies of Willmore Spheres in ${\R}…
We give a short, simple and conceptual proof, based on spin structures, of sphere eversion: an embedded 2-sphere in can be turned inside out by regular homotopy. Ingredients of this eversion are seamlessly connected. We also give the mathematical origins of the proof: the Hopf fibration, and the topological struc…
Yet another sphere eversionmath.GT
We present a (possibly) new sphere eversion based on the contractibility* of a certain subset of the space of immersions of the circle in the plane. (*: by strong deformation retraction)
Sphere eversions have been described so far by either pictures with minimal topological complexity, numerical evolution or complex equations. We write down relatively simple explicit formulas for the whole eversion, both analytic and topologically simpler, including also Boy surface (real projective plane), using a fam…
This paper formalizes the h-principle and sphere eversion in differential topology.
problem Formalizing the h-principle and sphere eversion in differential topology.
method Lean formalization of the local h-principle for first-order partial differential relations, using convex integration.
result Reproves Smale's sphere eversion theorem and formalizes advanced mathematics.
For each diagram of a -knot, we provide a way to construct a new diagram of the same knot such that any sequence of Roseman moves between and necessarily involves branch points. The proof is done by developing the observation that no sphere eversion can be lifted to an isotopy in -space.
We consider properties of the total absolute geodesic curvature functional on circle immersions into a Riemann surface. In particular, we study its behavior under regular homotopies, its infima in regular homotopy classes, and the homotopy types of spaces of its local minima. We consider properties of the total curvatu…
This paper has been withdrawn by the author, due to an error in Proposition 2.2.
We show that the sum of the Morse indices of the Willmore spheres realising the width of Willmore type sweep-outs is bounded by the number of the parameters of the min-max. As an application, we deduce that among the true Willmore spheres realising the min-max sphere eversion, at most one of them one has index 1, while…
We extend the classification of Robert Bryant of Willmore spheres in to variational branched Willmore spheres and show that they are inverse stereographic projections of complete minimal surfaces with finite total curvature in and vanishing flux. We also obtain a classification of variational…
P. M. Akhmetiev used a controlled version of the stable Hopf invariant to show that any (continuous) map N -> M between stably parallelizable compact n-manifolds, n\ne 1,2,3,7, is realizable in R^{2n}, i.e. the composition of f with an embedding M\subset R^{2n} is C^0-approximable by embeddings. It has been long believ…
Paper defines a new invariant for surface immersions.
problem Detecting and classifying jumps in surface immersions.
method Defines an integer-valued function to classify jumps involving quadruple points and triple-line tangencies.
result Classifies quadruple point jumps into five geometrically distinct cases.
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.
Ribbonness of Kervaire's sphere-link in homotopy 4-sphere and its consequences to 2-complexesmath.GT
Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.
problem Understanding the structure of 2-complexes and their asphericity.
method Using Kervaire's sphere-link and ribbon sphere-link equivalence, analyzing the compact complement of ribbon disk-links.
result Every connected subcomplex of a contractible finite 2-complex is aspherical.
New theory proves infinite homology 3-spheres in homology 4-spheres.
problem Existence of homology 3-spheres in homology 4-spheres.
method Diagrammatics of surface cross sections, Taubes' work.
result Infinite number of homology 3-spheres in homology 4-spheres.
Infinitely many splitting spheres found for unlinked 2-spheres in 4-space.
problem Existence of pairwise non-isotopic splitting spheres for unlinked 2-spheres in 4-space.
method Analytical proof showing non-isotopic spheres.
result Infinitely many non-isotopic splitting spheres found.
Reduces weak reducing pairs to spheres in 3-sphere Heegaard surfaces.
problem Finding reducing spheres for weak reducing pairs in Heegaard surfaces.
method Proves existence of reducing spheres for weak reducing pairs in 3-sphere Heegaard surfaces.
result Reduction of weak reducing pairs to spheres if genus is at most 3.
The study shows how to construct -spheres from -spheres and -balls without additional vertices.
problem Constructing -spheres from -spheres and -balls without additional vertices.
method Examining specific types of spheres (flag, stacked, join of spheres) and -balls to determine if constructions can be made without extra vertices.
result Affirmative answers to constructing -spheres from -spheres and -balls without additional vertices for certain types of spheres and -balls.
Patterns and Tracksmath.GT
New proof for sphere recognition algorithm.
problem Sphere recognition algorithm proof.
method New proof of a lemma in Abigail Thompson's algorithm.
result New proof of a lemma in Abigail Thompson's proof of the Recognition Algorithm for 3-spheres.
Stability of Convex Spheresmath.DG
Proves stability of convex spheres with similar geodesic lengths.
problem Stability of convex spheres with specific geodesic properties.
method Proves C^0 Cheeger-Gromov closeness to the round sphere.
result Strictly convex 2-spheres are close to the round sphere.
Soliton Spheresmath.DG
Soliton spheres are immersed 2-spheres in the conformal 4-sphere S^4=HP^1 that allow rational, conformal parametrizations f:CP^1->HP^1 obtained via twistor projection and dualization from rational curves in CP^{2n+1}. Soliton spheres can be characterized as the case of equality in the quaternionic Pluecker estimate. A …
Study on sphere immersions and their stability indices.
problem Analyzing the stability of sphere immersions.
method Calculation of Morse indices and stability indices for specific sphere immersions.
result Bounds on stability index of associative cone in .
The paper constructs biharmonic maps between spheres using polynomial maps.
problem Creating biharmonic maps between spheres.
method Using harmonic homogeneous polynomial maps of different degrees to generate proper biharmonic maps.
result Established a method for constructing proper biharmonic product maps.
Reduces connectivity problem for genus-4 Heegaard surface in 3-sphere.
problem Connectivity problem in reducing sphere complex for genus-4 Heegaard surface.
method Presented a sufficient condition for a non-separating weak reducing pair to be separated by a reducing sphere.
result Reduced connectivity problem to showing disjointness of representative reducing spheres from a fixed disk.
Author provides an alternate proof of the free ribbon lemma.
problem Proving that every free sphere-link in the 4-sphere is a ribbon sphere-link.
method An alternate proof of the free ribbon lemma.
result Provides an alternate proof of the free ribbon lemma.
Paper studies inscribed sphere and lines through centers of Apollonius spheres in n dimensions.
problem Tangency of spheres and lines through their centers.
method Lie sphere geometry and two-step construction of Apollonius spheres.
result Center of inscribed sphere coincides with point .
Sharp convergence theorem for sphere submanifolds proved.
problem Sphere submanifolds in spheres.
method Proved a sharp convergence theorem.
result New differentiable sphere theorem for submanifolds in spheres.
In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is the round 3-sphere of radius 1) we classify the stable constant mean curvature spheres, showing that in some Berger spheres (a close to 0) there are unstable constant mean curvature spheres. Also, we classify the orientable compact stable constant mea…
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 , where or , which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a que…
In this paper we show that for m>n the set of cobordism classes of maps from m-sphere to n-sphere is trivial. The determination of the cobordism homotopy groups of spheres admits applications to the covers for spheres.
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.
We show that among Seifert fibered integer homology spheres, Poincare sphere (with either orientation) is the only non-trivial example which has trivial Heegaard Floer homology. Together with an earlier result, this shows that if an integer homology sphere has trivial Heegaard Floer homology, then it is a connected sum…
We prove a Reeb sphere theorem for finite simple graphs. The result bridges two different definitions of spheres in graph theory. We also reformulate Morse conditions in terms of the center manifolds, the level surface graphs {f=f(x)} in the unit sphere S(x). In the Morse case these graphs are either spheres, the empty…
Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.
problem Characterize biharmonic hypersurfaces in spheres.
method Prove CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres.
result Supports the conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.
On a homotopy -spheremath.GT
Standard proved to be diffeomorphic to a curious homotopy sphere.
problem Determining the diffeomorphism of a curious homotopy sphere to the standard .
method Proof based on properties of homotopy spheres and loose corks.
result The curious homotopy sphere is diffeomorphic to the standard .
We provide a computer-assisted proof of the holomorphy of the quartic and the octic meromorphic differentials arising in the main Theorem 4.11 of our paper 'The Classification of Branched Willmore spheres in the -Sphere and the -Sphere' (arXiv:1706.01405), using the free mathematical software Sage.
New actions found on exotic spheres using group theory.
problem Understanding smooth transformations on exotic spheres.
method Recent progress in stable homotopy groups of spheres and group theory.
result Smooth circle and cyclic group actions on exotic spheres produced.
Here we generalize the Gromoll-Meyer construction of an exotic 7-sphere by producing geometric models of exotic 8, 10 and Kervaire spheres as quotients of sphere bundles over spheres by free isometric actions. We give a geometric application at the end.
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.
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.
Survey of Dupin hypersurfaces in Lie sphere geometry.
problem Classifying Dupin hypersurfaces in Lie sphere geometry.
method Detailed description of Lie sphere geometry concepts and classification results.
result Many classification results relating Dupin hypersurfaces to isoparametric hypersurfaces in spheres.
Paper studies spaces of flattenings of simplicial spheres and their homotopy type.
problem Existence and uniqueness of differentiable structures on simplicial spheres.
method Analyzes spaces of flattenings of simplicial spheres and their homotopy type.
result Spaces of flattenings have the homotopy type of the orthogonal group.
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.
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.
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…
Surgery on knots can produce non-separating spheres, using Heegaard Floer homology.
problem Conditions for surgery on knots to produce non-separating spheres.
method Heegaard Floer homology
result Sufficient conditions for a knot to be unknotted.