arXiv research

Can certain shapes be drawn with a pencil and eraser?

Simple curves enclose two small disks if they're wide and bend moderately.

Study calculates first $p$-widths of unit disk.

In 1998, Gompf described a Stein domain structure on the disk cotangent bundle of any closed surface S, by a Legendrian handlebody diagram. We prove that Gompf's Stein domain is symplectomorphic to the disk cotangent bundle equipped with its canonical symplectic structure and the boundary of this domain is contactomorp…

Sharp curvature bounds for minimal graphs over unit disk.

This paper classifies planar-Rips complexes and their unit disk graphs up to homotopy.

We describe a Lefschetz fibration of genus one on the disk cotangent bundle of any closed orientable surface S. As a corollary, we obtain an explicit genus one open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of S.

In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory …

Method solves Calderón problem for surfaces near disks.

Study constructs disks with curved boundaries in a 3D ball.

A left orderable completely metrizable topological group is exhibited containing Artin's braid group on infinitely many strands. The group is the mapping class group (rel boundary) of the closed unit disk with a sequence of interior punctures converging to the boundary. This resolves an issue suggested by work of Dehor…

Bi-Lipschitz proof for 2-varifolds near critical Allard condition.

We first study holomorphic isometries from the Poincaré disk into the product of the unit disk and the complex unit $n$-ball for $n\ge 2$. On the other hand, we observe that there exists a holomorphic isometry from the product of the unit disk and the complex unit $n$-ball into any irreducible bounded symmetric domain …

We prove that if M is a closed, connected, oriented, rationally inessential manifold, then the Hofer-Zehnder capacity of the unit disk bundle of the cotangent bundle of M is finite.

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface $Σ_g$, where $g$ is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of $Σ_g$. For …

The paper improves estimates of Gaussian curvature for minimal graphs over a unit disk.

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show…

Let $Σ$ be a compact Riemann surface and $D_1,...,D_n$ a finite number of pairwise disjoint closed disks of $Σ$. We prove the existence of a proper harmonic map into the Euclidean plane from a hyperbolic domain $Ω$ containing $Σ\backslash\cup_{j=1}^n D_j$ and of its topological type. Here, $Ω$ can be chosen as close as…

The study confirms two cases of the convex body isoperimetric conjecture in the plane.

Study of parabolas in Funk metric on unit disk.

Paper solves long-standing Gaussian curvature conjecture for minimal graphs.

We generalize Meeks and Yau's embeddedness result on the solutions of the Plateau problem to the constant mean curvature disks. We show that any minimizing H-disk in an H_0-convex domain is embedded for any H in [0,H_0). In particular, for the unit ball B in R^3, this implies that for any H in [0,1], any Jordan curve i…

Study on disk configurations in strips shows stability patterns.

M Handel has proved in [Topology 38 (1999) 235--264] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that may be extended to the closed disk and that satisfies a linking property of orbits. We give here a new proof of Handel's fixed point theorem, based on Brouwer theory and som…

Study finds Scherk type surfaces as extremals for zero-curvature minimal graphs.

This paper shows that every totally-geodesic isometry from the unit disk to a finite-dimensional Teichmüller space for the intrinsic Kobayashi metric is either holomorphic or anti-holomorphic; in particular, it is a Teichmüller disk. Additionally, a similar result is proved for a large class of disk-rigid domains, whic…

Study on a metric for disk automorphisms with maximal modulus.

Surface parameterizations have been widely used in computer graphics and geometry processing. In particular, as simply-connected open surfaces are conformally equivalent to the unit disk, it is desirable to compute the disk conformal parameterizations of the surfaces. In this paper, we propose a novel algorithm for the…

Shorter proof for wave front length in Euclidean disk

We prove that the unique least-perimeter way of partitioning the unit 2-dimensional disk into three regions of prescribed areas is by means of the standard graph consisting in three balanced constant geodesic curvature curves meeting themselves at 120 degrees, and reaching orthogonally the boundary of the disk.

Constructs minimal surfaces in a 3-ball using PDE gluing.

Let $\varphi :\mathbb{D}\to\mathbb{C}$ be an integrable holomorphic function on the unit disk $\mathbb{D}$ and $D_{\varphi}:\mathbb{D}\to T(\mathbb{D})$ the Teichmüller disk in the universal Teichmüller space $T(\mathbb{D})$. For a positive $t$ it is known that $D_{\varphi}(t)\to [μ_{\varphi}]\in PML_b(\mathbb{D})$ as …

We give a parametrization to the asymptotic Teichmuller space of the open unit disk through equivalent classes of shear functions induced by quasisymmetric homeomorphisms on the Farey tesselation of the unit disk. Then using the parametrization, we define a new metric on the asymptotic Teichmuller space. Two other rela…

Geodesics on polygons in a unit disk are studied with unique metric properties.

Paper proves rigidity of minimal disks in 3-balls with non-negative Ricci curvature.

We give a short proof of the fact that bounded earthquakes of the unit disk induce quasisymmetric maps of the unit circle. By a similar method, we show that symmetric maps are induced by bounded earthquakes with asymptotically trivial measures.

For a smooth immersion $f$ from the punctured disk $D\backslash\{0\}$ into $\mathbb{R}^n$ extendable continuously at the puncture, if its mean curvature is square integrable and the measure of $f(D)\cap B_{r_k}=o(r_k)$ for a sequence $r_k\to 0$, we show that the Riemannian surface $(D_r\backslash\{0\},g)$ where $g$ is …

Minimal diffeomorphisms extend uniquely with $L^1$ Hopf differential.

Let $Σ$ be a hypersurface in an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We study the isometric extension problem for isometric immersions $f:Σ\to\mathbb R^n$, where $\mathbb R^n$ is equipped with the Euclidean standard metric. We prove a general curvature obstruction to the existence of merely differen…

In a recent paper A. Fraser and R. Schoen have proved the existence of free boundary minimal surfaces $Σ\_n$ in $B^3$ which have genus $0$ and $n$ boundary components, for all $n \geq 3$. For large $n$, we give an independent construction of $Σ\_n$ and prove the existence of free boundary minimal surfaces $\tilde Σ\_n…

This paper generalizes a result about bounded differentials to higher-order differentials and studies their geometric implications.

New bounds on slice genus from knot invariants.

The paper resolves a problem about metric inequivalence and characterizes proper holomorphic maps.

Let $Φ$ be a strictly plurisubharmonic and radial function on the unit disk ${\cal D}\subset {\complex}$ and let $g$ be the \K metric associated to the \K form $ω=\frac{i}{2}\partial\bar\partialΦ$. We prove that if $g$ is $g_{eucl}$-balanced of height 3 (where $g_{eucl}$ is the standard Euclidean metric on ${\complex}=…

Solves Christoffel problem for disk area measures on spheres.

We simplify proof of the theorem that close to any pseudoholomorphic disk there passes a pseudoholomorphic disk of arbitrary close size with any pre-described sufficiently close direction. We apply these results to the Kobayashi and Hanh pseudodistances. It is shown they coincide in dimensions higher than four. The res…

Let $D$ be a 2-dimensional closed unit disk and $\rm{Symp}(D,0)_{\rm{rel}}$ the group of symplectomorphisms preserving the origin and the boundary $\partial D$ pointwise. We consider the $\mathbb{R}$-valued flux homomorphism on $\rm{Symp}(D,0)_{\rm{rel}}$ and define the central $\mathbb{R}$-extension called the $\mathb…

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disk. The surfaces are constructed by singular perturbation methods and have three boundary components. They are …