arXiv research

We first construct a genus zero positive allowable Lefschetz fibration over the disk (a genus zero PALF for short) on the Akbulut cork and describe the monodromy as a positive factorization in the mapping class group of a surface of genus zero with five boundary components. We then construct genus zero PALFs on infinit…

We present a new method to compare the shapes of genus-zero surfaces. We introduce a measure of mutual stretching, the symmetric distortion energy, and establish the existence of a conformal diffeomorphism between any two genus-zero surfaces that minimizes this energy. We then prove that the energies of the minimizing …

Paper generalizes discrete uniformization for genus-zero surfaces.

The aim of this paper is to introduce a group containing the mapping class groups of all genus zero surfaces. Roughly speaking, such a group is intended to be a discrete analogue of the diffeomorphism group of the circle. One defines indeed a {\it universal mapping class group of genus zero}, denoted $\B$. The latter i…

Proves existence of maps with arbitrary ends and conditions for maxfaces.

Proves any three or more knots can form a genus-zero link in a 3-manifold.

Minimal surfaces in a ball have limited area.

In 1841, Delaunay constructed the embedded surfaces of revolution with constant mean curvature (CMC); these unduloids have genus zero and are now known to be the only embedded CMC surfaces with two ends and finite genus. Here, we construct the complete family of embedded CMC surfaces with three ends and genus zero; the…

Defines super stable maps and proves quotient superorbifolds for genus zero.

Smooth torus actions on moduli spaces of super stable curves and maps of genus zero.

Paper proves surfaces with specific symmetries have the first Steklov eigenvalue.

We show that the moduli space of genus zero stable maps is a real projective variety if the target space is a smooth convex real projective variety. We show that evaluation maps, forgetful maps are real morphisms. We analyze the real part of the moduli space.

Monoidal categorifies genus zero skein algebra using K-theory.

This paper explains the conjectured algebraic duality between genus zero Gromov-Witten theory and genus zero "Closed String topology". This duality in another perspective is discussed on page 87 of the book "Frobenius manifold, quantum cohomology, and moduli spaces" (by Yuri Manin). This paper also discusses Fulton Mac…

We construct a genus zero PALF structure on each of plugs introduced by Akbulut and Yasui and describe the monodromy as a positive factorization in the mapping class group of a fiber. We also examine the monodromies of PALFs on a certain pair of compact Stein surfaces such that one is obtained by applying a plug twist …

A Riemann surface $M$ is said to be $K$-quasiconformally homogeneous if for every two points $p,q \in M$, there exists a $K$-quasiconformal homeomorphism $f \colon M \rightarrow M$ such that $f(p) = q$. In this paper, we show there exists a universal constant $K_0 > 1$ such that if $M$ is a $K$-quasiconformally homogen…

We consider constant mean curvature surfaces of finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors in arXiv:math.DG/0102183. Here we extend the arguments to the case of an arbitrary number o…

A very interesting problem in the classical theory of minimal surfaces consists of the classification of such surfaces under some geometrical and topological constraints. In this short paper, we give a brief summary of the known classification results for properly embedded minimal surfaces with genus zero in $\mathbb{R…

We construct new examples of immersed minimal surfaces with catenoid ends and finite total curvature, of both genus zero and higher genus. In the genus zero case, we classify all such surfaces with at most $2n+1$ ends, and with symmetry group the natural $\bfZ_2$ extension of the dihedral group $D_n$. The surfaces are …

In 1988, Karcher generalized the family of singly periodic Scherk minimal surfaces by constructing, for each natural $n\geq 2$, a $(2n-3)$-parameter family of singly periodic minimal surfaces with genus zero and $2n$ Scherk-type ends in the quotient, called {\it saddle towers}. They have been recently classified by Pér…

We exhibit sufficient conditions for a finite collection of periodic orbits of a Reeb flow on a closed $3$-manifold to bound a positive global surface of section with genus zero. These conditions turn out to be $C^\infty$-generically necessary. Moreover, they involve linking assumptions on periodic orbits with Conley-Z…

Given k>=2, we construct a (2k-2)-parameter family of properly embedded minimal surfaces in H^2 x R invariant by a vertical translation T, called Saddle Towers, which have total intrinsic curvature 4 pi(1-k), genus zero and 2k vertical Scherk-type ends in the quotient by T. As limits of those Saddle Towers, we obtain J…

We construct a smooth Riemannian metric on any 3-manifold with the property that there are genus zero embedded minimal surfaces of arbitrarily high Morse index.

Using the DPW method, we construct genus zero Alexandrov-embedded constant mean curvature (greater than one) surfaces with any number of Delaunay ends in hyperbolic space.

We construct constant mean curvature surfaces in euclidean space with genus zero and n ends asymptotic to Delaunay surfaces using the DPW method.

New surfaces described that are symmetric and solve a specific equation.

Researchers solved a number-theoretic hypothesis to determine the spin parity of k-differentials.

Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.

We prove the existence of nonperiodic, properly embedded minimal surfaces in $\mathbb{R}^2\times\mathbb{S}^1$ with genus zero, infinitely many ends and one limit end (in particular, they have infinite total curvature).

We propose localization techniques for computing Gromov-Witten invariants of maps from Riemann surfaces with boundaries into a Calabi-Yau, with the boundaries mapped to a Lagrangian submanifold. The computations can be expressed in terms of Gromov-Witten invariants of one-pointed maps. In genus zero, an equivariant ver…

We show that any complete $f$-stable translating soliton $M$ admits no codimension one cycle which does not disconnect $M$. As a corollary, it follows that any two dimensional complete $f$-stable translating soliton has genus zero.

We announce the classification of complete, almost embedded surfaces of constant mean curvature, with three ends and genus zero: they are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends.

Constructs minimal surfaces near the boundary of a ball.

We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to Gelfand-Dickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdV_…

Proves Alexander and Markov theorems for higher genus virtual doodles.

Theory of point vortices extended to closed surfaces.

For each end of complete minimal surface in the Euclidean 3-space, the flux vector is defined. It is well-known that the sum of the flux vector over all ends are zero. Consider the following inverse problem: For each balanced n-vectors, find an n-end catenoid which realizes these vectors as flux. Here, an n-end catenoi…

This work generalizes a formula linking Seiberg-Witten prepotential and topological recursion.

Paper proves rigidity for self-similar solutions in 3D flows.

We compute the genus zero family Gromov-Witten invariants for K3 surfaces using the topological recursion formula and the symplectic sum formula for a degeneration of elliptic K3 surfaces. In particular we verify the Yau-Zaslow formula for non-primitive classes of index two.

For any m > 0, we construct properly embedded minimal surfaces in H^2 x R with genus zero, infinitely many vertical planar ends and m limit ends. We also provide examples with an infinite countable number of limit ends. All these examples are vertical bi-graphs.

In this paper, we use the conjugate surface construction to prove the existence of certain non-periodic symmetric immersed minimal surfaces. These surfaces have finite total curvature and embedded catenoid ends, and they have positive genus yet maintain the symmetry of their genus-zero counterparts constructed by Jorge…

This paper shows connections between two complex mathematical theories are equivalent.

Paper derives a formula for the determinant of Dirichlet-to-Neumann operator on Riemann surfaces.

Detects knots in thickened surfaces using instanton homology.

We study the modularity of the genus zero open Gromov-Witten potentials and its generating matrix factorizations for elliptic orbifolds. These objects constructed by Lagrangian Floer theory are a priori well-defined only around the large volume limit. It follows from modularity that they can be analytically continued o…

For a surface $S$ with $n$ marked points and fixed genus $g\geq2$, we prove that the logarithm of the minimal dilatation of a pseudo-Anosov homeomorphism of $S$ is on the order of $(\log n)/n$. This is in contrast with the cases of genus zero or one where the order is $1/n$.

We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to $\mathbb{Z}$ factors through a sphere with finitely many punctures. In fact we get an uncountable family of such maps.