New minimal surfaces in round 3-sphere by doubling equatorial 2-sphere.
problem Constructing minimal surfaces in round 3-sphere by doubling equatorial 2-sphere.
method Linearized Doubling (LD) methodology.
result Minimal surfaces can concentrate along arbitrary parallel circles, including poles.
Analytic saddle spheres in S^3 are equators.
problem Characterizing saddle-shaped minimal surfaces in 3-sphere.
method Purely geometric approach, no PDE imposed.
result Analytic saddle spheres in S^3 are equators.
The equatorial disk in a ball has the smallest area.
problem Finding the minimal area of free boundary minimal surfaces in a ball.
method Comparing excess of surfaces with excess of cones over the boundary.
result Existence of a gap in the area of minimal surfaces.
New minimal surfaces found in a ball, connected and with high genus.
problem Finding minimal surfaces with connected boundaries in a ball.
method Gluing construction tripling the equatorial disc.
result First examples of compact free boundary minimal surfaces with connected boundary in a ball.
Paper studies equatorial concentration of measure in sphere immersions and submersions.
problem Equatorial concentration of measure in sphere immersions and submersions.
method Analyzes concentration of measure phenomena in the sphere.
result Describes an equatorial concentration of measure for minimal immersions and submersions.
We construct closed embedded minimal surfaces in the round three-sphere, resembling two parallel copies of the equatorial two-sphere, joined by small catenoidal bridges symmetrically arranged either along two parallel circles of the equator, or along the equatorial circle and the poles. To carry out these constructions…
Study on ball widths and minimal submanifolds in space forms.
problem Understanding widths of balls and minimal submanifolds.
method Analyzing the area of equatorial balls and related bounds for minimal submanifolds.
result Lower bounds for the area of free boundary minimal submanifolds.
New minimal surfaces in a ball created by merging catenoids and disks.
problem Creating minimal surfaces in a bounded space.
method Desingularizing a critical catenoid and equatorial disk using singular perturbation methods.
result Constructed high genus free boundary minimal surfaces with three boundary components.
Constructs minimal surfaces in a 3-ball using PDE gluing.
problem Finding minimal surfaces in a 3-ball with boundary constraints.
method PDE gluing construction of discrete free boundary minimal annuli.
result Discrete family of non-rotational free boundary minimal annuli in a unit 3-ball.
Study shows critical width for rigidity of equatorial zones on spheres.
problem Mean curvature rigidity of equatorial zones on spheres.
method Used tangency principle and trap-slice lemma for strong rigidity, and constructed nontrivial perturbations using Delaunay surfaces for non-rigidity.
result Critical width exists for rigidity, beyond which zones are non-rigid.
Study proves mean curvature flows on spheres in higher dimensions.
problem Existence of mean curvature flows on spheres.
method Generalized previous results to higher dimensions, proving existence of flows.
result Existence of infinitely many eternal weak mean curvature flows in Sn+1 connecting specific hypersurfaces. Study eternal solutions to Allen-Cahn equation on 3-sphere, connecting Clifford tori to equatorial spheres.
problem Understanding eternal solutions to the Allen-Cahn equation on the 3-sphere.
method Realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, using classifications and rigidity results.
result Construction of eternal integral Brakke flows connecting Clifford tori to equatorial spheres.
We prove rigidity for hypersurfaces with boundary in the unit (n+1)-sphere with scalar curvature bounded below by n(n−1). Under appropriate boundary conditions, the hypersurfaces are shown to be part of the equatorial spheres. The lower bound n(n−1) is critical in the sense that the hypersurface may contain geode…
Constructs minimal hypersurfaces in S^4(1) by doubling equatorial S^3.
problem Constructing minimal hypersurfaces in S^4(1).
method PDE gluing methods and Linearized Doubling (LD) methodology.
result Minimal hypersurfaces ${reve{M}_m}$ doubling the equatorial S^3 in S^4(1).
Minimal surface doublings have specific index and nullity values.
problem Analyzing the properties of minimal surface doublings.
method Analytical proof of index and nullity values.
result Minimal surface doublings have index 2γ+5=2m+7 and nullity 6. The study finds conditions for free boundary Hamiltonian stationary discs in complex 2-space.
problem Conditions for free boundary Hamiltonian stationary Lagrangian discs in complex 2-space.
method Established conditions for weakly conformal, branched Ω-free boundary Hamiltonian stationary Lagrangian immersions of discs. result If conditions are met, a disc is a free boundary minimal immersion.
The Pachner graph of 2-spheres is studied, focusing on subgraphs of flag and stacked 2-spheres.
problem Characterize subgraphs of the Pachner graph of 2-spheres.
method Analyzes various induced subgraphs of the Pachner graph of n-vertex triangulated 2-spheres, proving connectivity and providing bounds on the number of connected components. result The subgraph of n-vertex flag 2-spheres is connected, while the subgraph of n-vertex stacked 2-spheres has at least as many connected components as trees with specific properties. Proves existence and uniqueness of rotating fluid bodies in GR to second order.
problem Understanding rotating fluid bodies in GR, especially beyond Newtonian limits.
method Second order perturbation theory, derived from first principles, with rigidly rotating finite perfect fluid ball assumptions.
result Equatorially symmetric spacetime determined by central pressure and uniform angular velocity.
Closed and broken electromagnetic orbits in Kerr-Newman spacetime
problem Constructing closed and broken electromagnetic orbits in the Kerr-Newman spacetime
method Constructing smooth closed electromagnetic orbits tangent to the axial Killing field and proving the existence of spherical electromagnetic orbits
result Proving the existence of spherical electromagnetic orbits and constructing closed broken electromagnetic orbits
Unique simple spines of homotopy 2-spheres are shown to be ambiently isotopic.
problem Characterizing and comparing simple spines of homotopy 2-spheres.
method Proving ambient isotopy of simple spines representing the same homology class.
result Simple spines of homotopy 2-spheres are unique.
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.
Concrete example of 2-sphere with constant curvature 1 and closed geodesics.
problem Constructing a 2-sphere with constant curvature 1 and closed geodesics.
method Constructing a concrete example of a Finsler metric on the 2-sphere.
result A 2-sphere with constant Gauss curvature 1 and all geodesics closed.
The paper studies minimal 2-spheres of constant curvature in complex hyperquadrics using matrix theory.
problem Exploring minimal 2-spheres of constant curvature in complex hyperquadrics.
method Using singular-value decomposition of complex matrices to study the moduli space of noncongruent spheres.
result Uniqueness proven for totally real constantly curved 2-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.
We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standar…
Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then apply the equation to obtain a classification of biharmonic maps in a family of r…
Study pinches the length of surfaces in a 3D ball.
problem Characterizing minimal surfaces with boundary constraints.
method Pinching condition on second fundamental form.
result Flat equatorial disk and critical catenoid are unique.
Extends curvature gap characterization for minimal surfaces in a ball.
problem Characterization of minimal surfaces in a ball.
method Pinching condition on second fundamental form.
result Extension to higher codimension.
The degree of certain holomorphic 2-spheres is bounded.
problem Understanding the degree of holomorphic 2-spheres in complex Grassmannians.
method Analyzing the degree of linearly full constantly curved holomorphic 2-spheres in G(2,n+2;C).
result The degree is bounded between n and 2n for specific n values.
Study hyperbolic 2-spheres with cone points, describing spaces for n=3.
problem Characterize the space of hyperbolic 2-spheres with cone points.
method Analyzing the space C(a0,a1,…,an) for n=3 and n=4. result Detailed description of spaces for n=3 and examples for n=4. We study metrics on conic 2-spheres when no Einstein metrics exist. In particular, when the curvature of a conic metric is positive, we obtain the best curvature pinching constant. We also show that when this best pinching constant is approached, the conic 2-sphere has an explicit Gromov-Hausdorff limit. This is a gene…
Taut foliations map leaves to branched 2-sphere covers.
problem Understanding taut foliations in 3-manifolds.
method Map foliation leaves to branched 2-sphere covers.
result Taut foliations are characterized by such maps.
Kähler structures on products of 2-spheres have identical Chern classes.
problem Classifying Kähler structures on products of 2-spheres.
method Using complex Bott manifolds and iterated P1-bundle constructions, we classify Kähler structures up to biholomorphism via Bott diagrams. result Kähler structures on products of 2-spheres have identical Chern classes.
The study proves the existence of geodesics on reversible Finsler spheres.
problem Existence of closed geodesics on Finsler 2-spheres.
method Generalization of Grayson's curve shortening flow.
result Existence of three simple closed geodesics and infinitely many closed geodesics.
Proves nonnegatively curved hypersurfaces on a sphere are convex disks.
problem Characterizing nonnegatively curved hypersurfaces with free boundary on a sphere.
method Analyzes hypersurfaces in Euclidean space with constant mth mean curvature. result Compact hypersurfaces are embedded convex disks.
Complete invariant defined for doodles on a sphere.
problem Doodles on a 2-sphere.
method Coefficients in series of chord diagrams.
result Finite type invariants of order at most 2n.
Study finds a minimal surface in a ball with specific properties.
problem Finding minimal surfaces in bounded domains.
method 6-sweepout technique to prove existence and properties of minimal surfaces.
result Existence of a free boundary minimal surface with specified topological and geometric constraints.
Sharp volume bounds for conic 2-spheres with Gaussian curvature constraints.
problem Volume constraints for conic 2-spheres with curvature bounds.
method Level set analysis and iso-perimetric inequality with new analytical tools.
result Computed minimal volume of conic spheres with curvature bounded by 1.
Proves existence of curves with constant curvature in a sphere.
problem Existence of curves with constant geodesic curvature in a Riemannian 2-sphere.
method Develops a min-max scheme for a weighted length functional.
result Proves existence for almost every prescribed curvature.
Study describes bifurcations of gradient flows on 2-sphere with holes.
problem Analyzing gradient flows on a 2-sphere with up to six singular points.
method Using separatrix diagrams to specify saddle-node and saddle connections.
result Identified all possible topological structures of bifurcations.
In a previous paper, we showed that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to a smooth variation through harmonic maps). In this paper, in contrast, we show that there are (non-full) harmonic maps from the 2-sphere to the 3-sphere and 4-sp…
Proves Arnol'd's conjecture about Legendrian curves.
problem Front of Legendrian curves in projectivized cotangent bundle of 2-sphere.
method Microlocal theory of sheaves and derived category of sheaves.
result Proves Arnol'd's three cusps conjecture.
The paper classifies Willmore 2-spheres in Sn.
problem Classifying Willmore 2-spheres in Sn. method Construction of a harmonic sequence in the real Grassmannian over the Lorentz space, derived from the harmonic conformal Gauss map.
result Any Willmore 2-sphere is either totally isotropic or strictly k-isotropic. In 1987, Kalai proved that stacked spheres of dimension d≥3 are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension d=2. In this article, we give a characterisation of stacked 2-spheres using what we call the {\em separatio…
We extend our previous definition of quasi-local mass to 2-spheres whose Gauss curvature is negative and prove its positivity.
Study finds all conformal minimal immersions of 2-spheres in a complex Grassmann manifold with parallel second fundamental form.
problem Classifying conformal minimal immersions with parallel second fundamental form.
method Analyzing immersions in complex Grassmann manifold G(2,N;C). result Determined all conformal minimal immersions of 2-spheres with parallel second fundamental form.
We pursue a geometrical approach to gravitational lensing theory. We present a survey of the background theory of General Relativity, including particular properties of the Schwarzschild and Kerr solutions. Next we outline a proof of the Gauss Bonnet theorem and its applications to surfaces in optical geometry, as deve…
Let S be a triangulated 2-sphere with fixed triangulation T. We apply the methods of thin position from knot theory to obtain a simple version of the three geodesics theorem for the 2-sphere [5]. In general these three geodesics may be unstable, corresponding, for example, to the three equators of an ellipsoid. Using a…