The paper classifies biharmonic immersions and submersions in specific spheres.
problem Classifying biharmonic immersions and submersions in specific spheres.
method Analyzing biharmonic isometric immersions and Riemannian submersions from Berger 3-spheres.
result Complete classification of proper biharmonic Hopf tori in Berger 3-sphere.
3D spheres can't be swept by short curves, complicating geodesic length estimates.
problem Obstructing geodesic length estimates in 3D spheres.
method Constructing specific 3D spheres with controlled diameter and volume.
result Min-max methods for geodesic lengths fail for certain 3D spheres.
5 minimal tori found in 3-spheres with positive Ricci curvature.
problem Existence of at least 5 embedded minimal tori in 3-spheres with positive Ricci curvature.
method Combination of min-max theory and mean curvature flow heuristics.
result Confirms B. White's conjecture for positive Ricci curvature.
The 3-sphere has either 2 minimal 2-spheres or an optimal foliation by 2-spheres.
problem Proving existence of minimal 2-spheres or optimal foliations in arbitrary Riemannian 3-spheres.
method Analyzing the properties of arbitrary Riemannian metrics on 3-spheres.
result The existence of at least two minimal 2-spheres or an optimal foliation in 3-spheres with arbitrary metrics.
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 studies Pansu spheres in a sub-Riemannian 3-sphere and their area-minimizing properties.
problem The study of Pansu spheres and their area-minimizing properties in a sub-Riemannian 3-sphere.
method Calibration arguments.
result The closed half-spheres of S0 with boundary C0 minimize sub-Riemannian area among compact C1 surfaces with the same boundary. Proves existence of special 2-spheres in curved 3-spaces.
problem Existence of constant mean curvature 2-spheres in Riemannian 3-spheres.
method Develops a min-max scheme for a weighted Dirichlet energy functional, using bi-harmonic approximation, derivative estimates, and Morse index estimates.
result Proves existence for almost every mean curvature and all for positively curved 3-spheres.
Paper connects hypersurfaces in 4D to surfaces in 3-sphere.
problem Understanding conformally flat hypersurfaces in 4D space forms.
method Characterizes conformal structures and relates to surfaces in 3-sphere.
result Relates 2-metrics in 3-sphere to surfaces giving rise to conformally flat hypersurfaces.
Existence of non-trivial monopoles on 3-spheres proven.
problem Existence of non-trivial SU(2)-monopoles on 3-manifolds.
method Gluing construction to prove existence on rational homology 3-spheres.
result Existence of non-trivial irreducible SU(2)-monopoles with Dirac singularities.
3D spheres with certain properties approach the round sphere.
problem Flexibility of Llarull's Theorem in dimension 3.
method Proof based on spacetime harmonic functions.
result 3D spheres with bounded Cheeger isoperimetric constant and scalar curvatures tending to 6 approach the round sphere.
Extends Zeitlin's model to 3-D axisymmetric Euler equations.
problem Preserving geometric structure in 3-D Euler equations.
method Axisymmetric discretization of 3-D Euler equations on the 3-sphere.
result First discretization of 3-D Euler equations preserving geometric structure.
Unified study of surfaces using Clifford algebras.
problem Classifying immersed surfaces in various manifolds.
method Using Clifford algebras to construct formalism for immersed bilegendrian surfaces.
result Full classifications of immersed bilegendrian surfaces in the unit tangent bundle of the 3-sphere.
New closed non-CMC biconservative surfaces found in round 3-sphere.
problem Existence of closed biconservative surfaces in space forms.
method Characterization of profile curves and proof of existence using curvature energy.
result Existence of a discrete family of closed, non-CMC biconservative surfaces in S3(ρ). Study of higher spin Killing spinors on 3D manifolds, proving rigidity and providing explicit expressions.
problem Understanding higher spin Killing spinors on 3D manifolds.
method Definition and detailed study of higher spin Killing spinors in arbitrary dimension, focusing on 3D manifolds. Rigidity result and explicit expressions for 3-sphere and 3-hyperbolic space.
result Proved a rigidity result for 3D manifolds admitting higher spin Killing spinors and provided explicit expressions for these spinors.
To an integral homology 3-sphere Y, we assign a well-defined Z-graded (monopole) homology $MH_*(Y, I_{\e}(\T; \e_0))$ whose construction in principle follows from the instanton Floer theory with the dependence of the spectral flow $I_{\e}(\T; \e_0)$, where $\T$ is the unique U(1)-reducible monopole of the Seiberg-…
We construct sequences of `expander manifolds' and we use them to show that there is a complete connected 2-dimensional Riemannian manifold with discontinuous isoperimetric profile, answering a question of Nardulli and Pansu. Using expander manifolds in dimension 3 we show that for any ε,M>0 there is a Riemannian 3-…
Sharp estimate for genus of embedded surfaces in 3-sphere.
problem Estimating the genus of embedded surfaces in the 3-sphere.
method Refined volume estimate and pinching method on the norm of traceless second fundamental form.
result Sharp pinching estimate for the genus of a surface in S3. A new geometric flow K-flow on 3-manifolds shrinks or preserves homogeneous spheres.
problem Analyzing the behavior of Thurston's model geometries under the K-flow. method Defining and studying the K-flow on 3-dimensional Riemannian manifolds, using a DeTurck-type argument for short-time existence. result The K-flow shrinks or preserves homogeneous spheres, showing short-time existence. We study the (massless) Dirac operator on a 3-sphere equipped with Riemannian metric. For the standard metric the spectrum is known. In particular, the eigenvalues closest to zero are the two double eigenvalues +3/2 and -3/2. Our aim is to analyse the behaviour of eigenvalues when the metric is perturbed in an arbitrar…
Let M be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric ≥0. We suppose that M is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let Σ be a compact connected and orientable surface immersed in M which is a stable constan…
Meeks, Pérez and Ros conjectured that a closed Riemannian 3-manifold which does not admit any closed embedded minimal surface whose two-sided covering is stable, must be diffeomorphic to a quotient of the 3-sphere. We give an counterexample to this conjecture. Also, we show that if we consider immersed surfaces ins…
Renormalized volume invariant for knots in 3-sphere computed.
problem Computing renormalized volume for knot embeddings in 3-sphere.
method Renormalizing volume associated to singular Yamabe metric.
result Renormalized volume is a global conformal invariant for knots in 3-sphere.
For constant mean curvature surfaces of class C2 immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classi…
This paper concerns the problem of existence of taut foliations among 3-manifolds. Since the contribution of David Gabai, we know that closed 3-manifolds with non-trivial second homology group admit a taut foliations. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite di…
The study finds nondegenerate harmonic 1-forms using symmetry conditions.
problem Existence of nondegenerate harmonic 1-forms over Riemannian manifolds.
method Utilizing Z3 symmetry to establish topological conditions. result Found nondegenerate Z2 harmonic 1-forms over branched coverings of links. Characterizes a specific type of convex curves on a 3-sphere.
problem Understanding convex curves on a 3-sphere.
method Decomposes curves on 3-sphere into 2-sphere curves, characterizes locally convex ones.
result Completely characterized a class of convex curves on the 3-sphere.
Real algebraic structures help classify overtwisted contact 3-spheres.
problem Classifying overtwisted contact structures on 3-spheres.
method Using real algebraic functions and open book decompositions.
result Most overtwisted contact structures are real algebraic.
Disk complexes show 3-sphere surfaces are topologically minimal.
problem Understanding minimal surfaces in 3-sphere topology.
method Analyzing disk complexes of genus >1 Heegaard surfaces.
result Genus >1 Heegaard surfaces have minimal index 2g-1.
We prove that any steady solution to the real analytic Euler equations on a Riemannian 3-sphere must possess a periodic orbit bounding an embedded disc. One key ingredient is an extension of Fomenko's work on the topology of integrable Hamiltonian systems to a degenerate case involving stratified integrals. The result …
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-…
The paper examines conditions for contact surgeries on rational homology 3-spheres.
problem Conditions for contact surgeries on rational homology 3-spheres.
method Analyzes sufficient conditions for contact surgeries using Legendrian knots and links.
result Provides sufficient conditions for surgeries to have vanishing contact invariants or to be overtwisted.
For any knot, a 3-sphere triangulation exists with a knotted edge.
problem Finding triangulations of the 3-sphere with specific knot configurations.
method Constructive proof using fully augmented links.
result Constructs one-vertex triangulations of the 3-sphere with knotted edges.
The paper studies invariants of surfaces in the 3-sphere using handlebody-links.
problem Understanding invariants of surfaces in the 3-sphere.
method Using Heegaard splittings and G-families of quandles to construct invariants. result Invariants can distinguish certain surfaces in the 3-sphere.
Given a genus-g Heegaard splitting of a 3-sphere, the genus-g Goeritz group is defined to be the group of the isotopy classes of orientation preserving homeomorphism of the 3-sphere that preserve the splitting. In this paper, we determine the twisted first (co)homology group of the genus-2 Goeritz group of 3-sphere.
We prove Mayberry-Murasugi's formula for links in homology 3-spheres, which was proved before only for links in the 3-sphere. Our proof uses Franz-Reidemeister torsions.
The paper classifies decompositions of 3-sphere and lens spaces with handlebodies.
problem Classifying decompositions of 3-manifolds with handlebodies.
method Studied decompositions of 3-sphere and lens spaces with three handlebodies, using stabilizations.
result Determined whether decompositions are stabilized.
Let M be a complete Sasakian sub-Riemannian 3-manifold of constant Webster scalar curvature κ. For any point p∈M and any number λ∈R with λ2+κ>0, we show existence of a C2 spherical surface Sλ(p) immersed in M with constant mean curvature λ. Our construction recovers in par…
Study properties of hypersurfaces in spacetimes with conformal transformations.
problem Properties of embedded hypersurfaces in spacetimes with a preferred spatial direction.
method Analysis of hypersurfaces with conformal transformations, scalar curvature conditions, and Riemannian manifold properties.
result Hypersurfaces are either Einstein or have vanishing twist, and under certain conditions, they are isomorphic to the 3-sphere.
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…
The original Smale Conjecture asserted that the inclusion of the group O(4) of isometries of the round 3-sphere S into the full diffeomorphism group Diff(S) is a homotopy equivalence. The (Generalized) Smale Conjecture asserts that the inclusion of Isom(M) into Diff(M) is a homotopy equivalence whenever M is an ellipti…
Perelman's proof confirmed, new method uses 4D topology.
problem Confirming the classical Poincaré conjecture.
method 4D topology, spun torus-knots, ribbonness, disk-chord system, Bing's result.
result Homotopy 3-sphere is diffeomorphic to the 3-sphere.
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.
Constructs fat, shellable 3-spheres with specific f-vectors.
problem Defines and constructs fat 3-spheres.
method Constructs strongly regular CW 3-spheres that are both shellable and dual shellable.
result Constructs arbitrarily fat, shellable and dual shellable 3-spheres with specific f-vectors. Study invariants of Z/p-homology 3-spheres from abelianization of mapping class groups.
problem Deciding and constructing invariants of Z/p-homology 3-spheres. method Formulating a criterion and using families of trivial 2-cocycles on the abelianization of the level-p mapping class group. result Disproved a conjectured extension of the Casson invariant for rational homology 3-spheres.
Study biharmonic conformal immersions into a 3D flat space, finding new examples and classifications.
problem Characterize and classify biharmonic conformal immersions into a conformally flat 3-space.
method Characterization of totally umbilical surfaces, method to produce biharmonic immersions, classification of maps, construction of examples.
result Construct many examples of biharmonic conformal immersions, including proper immersions and isometric ones.
New rack and multiple group rack cohomology for surfaces in 3-sphere.
problem Categorizing compact oriented surfaces in 3-sphere based on symmetry.
method Developed cohomology theory for racks and multiple group racks, constructed cocycle invariants.
result Identified new symmetry types of surfaces in 3-sphere.
New invariant fully describes finite type invariants of knots in homology 3-spheres.
problem Constructing a universal finite type invariant for knots in homology 3-spheres.
method Refined construction of a new invariant that is strictly stronger and universal.
result New invariant fully describes the graded space of finite type invariants of knots in homology 3-spheres.
We give a short proof of a theorem of Guth relating volume of balls and Uryson width. The same approach applies to Hausdorff content implying a recent result of Liokumovich-Lishak-Nabutovsky-Rotman. We show also that for any C>0 there is a Riemannian metric g on a 3-sphere such that vol(S3,g)=1 and for an…