arXiv research

We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots $D^{n-2}\into D^n$. Cobordisms of disk knots that do not fix the boundary sphere knots are easily classified by the cobordism properties of these boundaries, and any two even-dimensional disk knots with isotopic boundary knots are cobordant r…

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

The study proves the existence of free boundary minimal disks in convex regions.

Proves existence of non-planar minimal disks in ellipsoids.

The study finds at least two short, simple geodesic chords on a disk with convex boundary.

Rare Teichmüller disks converge to small limit sets.

The paper proves the existence of constant mean curvature disks with capillary boundary conditions.

Compact Special Weingarten surfaces with planar convex boundaries are disks.

Study p-Willmore disks with boundary energies, finding equilibrium configurations.

Classifies convex disks with Legendrian boundary in overtwisted contact 3-manifolds.

The study proves inequalities for area and boundary length of disks in convex manifolds.

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 …

The study finds at least 2 free-boundary minimal disks in convex 3-balls.

We show that if C is a simple closed curve bounding an embedded disk in a closed 3-manifold M, then there exists a disk D in M with boundary C such that D minimizes the area among the embedded disks with boundary C. Moreover, D is smooth, minimal and embedded everywhere except where the boundary C meets the interior of…

Study rigidity of minimal disks in specific 3-manifolds.

Paper proves rigidity of 3-manifolds with boundary using modified Hawking mass.

Curvature conditions distinguish Euclidean space and disks in contractible manifolds.

Study of disks in complex projective space with specific fundamental groups.

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

The Blaschke rolling disk theorem is extended to non-convex domains.

For a 3-manifold M and a subsurface $X$ of the boundary of M with empty or incompressible boundary we use surgery to identify a graph whose vertices are disks with boundary in X and which is quasi-isometrically embedded in the curve graph of X.

If M is a manifold with compressible boundary, we analyze essential disks in M, as well as incompressible, but not necessarily boundary incompressible, surfaces in M. We are most interested in the case where M is a handlebody or compression body. The analysis depends on a new normal surface theory. We hope the normal s…

The paper proves the existence of capillary geodesics on Riemannian 2-disks.

We perform a replacement procedure in order to produce a free boundary minimal surface whose area achieves the min-max value over all disk sweepouts of a manifold whose boundary lie in a submanifold. Our result is based on a proof of the convexity of the energy for free boundary harmonic maps and a generalization of Co…

This paper extends a 3D result to higher dimensions for manifolds with positive curvature.

Paper studies minimal surfaces in curved spaces, proving existence and properties.

Proves stability of convex disks close to round caps.

We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a three dimensional constant curvature submanifold and is totally umbilic. These res…

It is proved by Brendle in [4] that the equatorial disk $D^k$ has least area among $k$-dimensional free boundary minimal surfaces in the Euclidean ball $B^n$. By comparing the excess of free boundary minimal surfaces with the excess of the associated cones over the boundary, we prove the existence of a gap for the area…

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

Let $D_n$ be the $n$-punctured disk. We prove that a family of essential simple arcs starting and ending at the boundary and pairwise intersecting at most twice is of size at most $\binom{n+1}{3}$. On the way, we also show that any nontrivial square complex homeomorphic to a disk whose hyperplanes are simple arcs inter…

The study finds disks for certain constant mean curvature surfaces in a specific 3D space.

New ribbon disks in 4D space, non-isotopic to each other.

Inverse mean curvature flow converges to a disk in hyperbolic space.

Moduli spaces of holomorphic disks in a complex manifold Z, with boundaries constrained to lie in a maximal totally real submanifold P, have recently been found to underlie a number of geometrically rich twistor correspondences. The purpose of this paper is to develop a general Fredholm regularity criterion for holomor…

Holomorphic disks on compact Lagrangian surfaces are shown to exist.

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…

We study the classification of slice disks of knots up to isotopy and diffeomorphism using an invariant in knot Floer homology. We compute the invariant of a slice disk obtained by deform-spinning, and show that it can be effectively used to distinguish non-isotopic slice disks with diffeomorphic complements. Given a s…

It is investigated Hurwitz numbers, that correspond to covering of disk with single non-simple boundary critical value. It is found differential equations, that describe a generating function for these numbers.

The paper classifies CMC free boundary hypersurfaces in rotational domains.

We prove that in Euclidean space $R^{n+1}$ any compact immersed nonnegatively curved hypersurface $M$ with free boundary on the sphere $S^n$ is an embedded convex topological disk. In particular, when the $m^{th}$ mean curvature of $M$ is constant, for any $1\leq m\leq n$, $M$ is a spherical cap or an equatorial disk.

We show that for a generic nullhomotopic simple closed curve C in the boundary of a compact, orientable, mean convex 3-manifold M with trivial second homology, there is a unique area minimizing disk D embedded in M where the boundary of D is C. We also show that the same is true for absolutely area minimizing surfaces.

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.

The n-strand braid group can be defined as the fundamental group of the configuration space of n unlabeled points in a closed disk based at a configuration where all n points lie in the boundary of the disk. Using this definition, the subset of braids that have a representative where a specified subset of these points …

A mean-convex set can be regarded as a barrier for the construction of minimal surfaces. Namely, if we are given a mean-convex set and a null-homotopic Jordan curve on its boundary, then there exists an embedded minimal disk with boundary the given curve contained in the starting mean-convex set. Does a mean-convex set…

The study explores deep and shallow slice knots in 4-manifolds, linking them to conjectures and proving existence and nonexistence results.

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

New method proves existence of constant mean curvature disks on convex surfaces.