arXiv research

Unique simple spines of homotopy 2-spheres are shown to be ambiently isotopic.

Concrete example of 2-sphere with constant curvature 1 and closed geodesics.

The 3-sphere has either 2 minimal 2-spheres or an optimal foliation by 2-spheres.

Complete obstruction found for 2-spheres in 5-manifolds to be isotopic.

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…

The degree of certain holomorphic 2-spheres is bounded.

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…

In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both $\mathbb{C}P^n$ and the hyperquadric of $\mathbb{C}P^n$. The moduli space of all those noncongruent ones is introduced, which can be described by certain complex symmetric matrices…

It is well-known that the Pachner graph of $n$-vertex triangulated $2$-spheres is connected, i.e., each pair of $n$-vertex triangulated $2$-spheres can be turned into each other by a sequence of edge flips for each $n\geq 4$. In this article, we study various induced subgraphs of this graph. In particular, we prove tha…

Taut foliations map leaves to branched 2-sphere covers.

The study proves the existence of geodesics on reversible Finsler spheres.

Complete invariant defined for doodles on a sphere.

We study topological properties of the Gromov-Hausdorff metric on the set of isometry classes of nonnegatively curved $2$-spheres.

Proves existence of curves with constant curvature in a sphere.

Study describes bifurcations of gradient flows on 2-sphere with holes.

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…

We extend our previous definition of quasi-local mass to 2-spheres whose Gauss curvature is negative and prove its positivity.

The paper classifies Willmore 2-spheres in $S^n$.

In 1987, Kalai proved that stacked spheres of dimension $d\geq 3$ 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…

Study finds all conformal minimal immersions of 2-spheres in a complex Grassmann manifold with parallel second fundamental form.

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…

The paper finds 3-colorings of 2-sphere triangulations.

Study classifies Morse functions with 4 critical points on immersed 2-spheres.

New Finsler metric on sphere disproves systolic ratio conjecture.

Shortest geodesic on curved spheres is no longer than 3 times the diameter.

Generalized Thurston's characterization for branched coverings of the 2-sphere.

We prove that a riemannian metric on the 2-sphere or the projective plane can be C2-approximated by a smooth metric whose geodesic flow has an elliptic closed geodesic.

We study the space $C(a_0,a_1,\dots,a_n)$ of hyperbolic 2-spheres with cone points of prescribed apex curvatures $2a_0,2a_1,\dots,2a_n\in]0,2π[$ and some related spaces. For $n=3$, we get a detailed description of such spaces. The euclidean 2-spheres were considered by W. P. Thurston: for $n=4$, the corresponding space…

The study finds resonance points in polarised curves with polynomial conserved quantities.

Synthetic construction of Hopf fibration in 4D space.

Here we discuss an example of topologically isotopic but smoothly non-isotopic pair of 2-spheres in a simply connected 4-manifold, which become smoothly isotopic after stabilizing by connected summing with S^2 x S^2.

A necessary and sufficient condition for the existence and uniqueness of a conformal metric on 2-sphere of constant curvature 1 and with three conical singularities of prescribed order is given.

As discussed in the paper, in a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inquality. Namely, its area must be bounded above by $4π/c$, where $c > 0$ is a lower bound on a natural energy momentum term. In this note we cons…

Study confirms a 2-sphere metric with three geodesics of minimal length.

We show that any number of disjointly embedded 2-spheres in 4-space can be pulled apart by a link homotopy, ie, by a motion in which the 2-spheres stay disjoint but are allowed to self-intersect.

Characterizes geodesics on spheres with Morse index bounds and inequalities.

In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by $4π/c$, where $c > 0$ is a lower bound on a natural energy-momentum term. We then consider th…

New criterion for branched covers between 2-spheres.

We give an exposition of a formula of Daskalopoulos, Hamilton and Sesum for solutions to the Ricci flow on the 2-sphere. This is one of several estimates used by them to classify ancient solutions on the 2-sphere.

We prove that an integral Cauchy-Riemann inequality holds for any pair of smooth functions $(f,h)$ on the 2-sphere $\mathbb{S}^2$, and equality holds iff $f$ and $h$ are related $λ_1$-eigenfunctions. We extend such inequality to 4-tuples of functions, only valid on the $L^2$-orthogonal complement of a suitable nonzero …

We prove effective uniformization for nearly round 2-spheres and investigate their stability.

We give a simple procedure to estimate the smallest Lipshitz constant of a degree 1 map from a Riemannian 2-sphere to the unit 2-sphere, up to a factor of 10. Using this procedure, we are able to prove several inequalities involving this Lipshitz constant. For instance, if the smallest Lipshitz constant is at least 1, …

Let M be a smooth 4-manifold which admits a relatively minimal hyperelliptic genus h Lefschetz fibration over the 2-sphere. If all of the vanishing cycles for this fibration are nonseparating curves, then we show that M is a 2-fold cover of a 2-sphere bundle over the 2-sphere, branched over an embedded surface. If the …

Proves existence of special 2-spheres in curved 3-spaces.

Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on hyperbolic 3-space.

The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.

This paper explores Brunnian twin groups and their properties.