arXiv research

We deform a minimal disk in $\mathbb{R}^4$ with a branch point into symplectic minimally immersed disks with only transverse double points.

Proves existence of non-planar minimal disks in ellipsoids.

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

Curvature bounds preserved in length-minimizing disks.

This is a survey of our work on embedded minimal disks.

Study rigidity of minimal disks in specific 3-manifolds.

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

The paper proves a disk's energy minimizer is holomorphic and calculates its Morse index.

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

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 …

We consider two natural classes of minimal laminations in three-manifolds. Both classes may be thought of as limits - in different senses - of embedded minimal disks. In both cases, we prove that, under a natural geometric assumption on the three-manifold, the leaves of these laminations are topologically either disks,…

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

The minimal area of Finsler disks with minimizing geodesics is at least 6/π r^2.

Optimal Liouville theorem for minimal disks in any codimension.

In this paper we describe a new deformation that connects minimal disks with planar ends with minimal disks with helicoidal ends. In this way, we are able to construct a family of complete minimal surfaces with helicoidal ends that contains the singly periodic genus one helicoid of Hoffman, Karcher and Wei.

We show that an embedded minimal disk in R^3 with large curvature is bilipschitz with a piece of a helicoid. Additionally, a simplified proof of the uniqueness of the helicoid is provided.

The paper proves a Fenchel theorem for Gauss maps and shows circles and disks minimize certain energies.

Study minimal freezing sets in convex digital disks.

We construct a sequence of embedded minimal disks in a ball where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper.

A theorem simplifies mass-minimizing flat chains' regularity.

Any sequence of properly embedded minimal disks in an open subset U of Euclidean 3-space has a subsequence such that the curvatures blow up on a relatively closed subset K of U and such that the disks converge in the complement of K to a minimal lamination of U\K. Assuming results of Colding-Minicozzi and an extension …

We show that a topologically minimal disk in a tetrahedron with index $n$ is either a normal triangle, a normal quadrilateral, or a normal helicoid with boundary length 4(n+1). This mirrors geometric results of Colding and Minicozzi.

Constructs minimal immersions with singularities.

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…

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…

Minimal surfaces in hyperbolic space have a renormalized area criterion.

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.

A Seifert surface F for a knot K is disk decomposable if there is a taut sutured manifold heirarchy for the complement of F, whose decomposing surfaces are all disks. It follows that F has minimal genus for the knot K, and has handlebody complement, i.e., F is free. We show that these necessary conditions for disk deco…

Let $α$ be a polygonal Jordan curve in $\bfR^3$. We show that if $α$ satisfies certain conditions, then the least-area Douglas-Radó disk in $\bfR^3$ with boundary $α$ is unique and is a smooth graph. As our conditions on $α$ are not included amongst previously known conditions for embeddedness, we are enlarging the set…

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

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

We prove a chord arc bound for disks embedded in $\mathbb{R}^3$ with constant mean curvature. This bound does not depend on the value of the mean curvature. It is inspired by and generalizes the work of Colding and Minicozzi in [2] for embedded minimal disks. Like in the minimal case, this chord arc bound is a fundamen…

This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for non-simply connected embedded minimal surfaces of any given fixed genus. The first of these asserts that any such surface without small necks can be obtained b…

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

We show that the disk complex of a genus $g>1$ Heegaard surface for the 3-sphere is homotopy equivalent to a wedge of $(2g-2)$-dimensional spheres. This implies that genus $g>1$ Heegaard surfaces for the 3-sphere are topologically minimal with index $2g-1$.

Sharp curvature bounds for minimal graphs over unit disk.

Following Riemann's idea, we prove the existence of a minimal disk in Euclidean space bounded by three lines in generic position and with three helicoidal ends of angles less than $π$. In the case of general angles, we prove that there exist at most four such minimal disks, we give a sufficient condition of existence i…

This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.

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

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

For any prescribed closed subset of a line segment in Euclidean 3-space, we construct a sequence of minimal disks that are properly embedded in an open solid cylinder around the segment and that have curvatures blowing up precisely at the points of the closed set.

This paper is the second in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an emb…

The study examines uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.

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…

In this paper we prove a theorem concerning lamination limits of sequences of compact disks $M_n$ embedded in $\mathbb{R}^3$ with constant mean curvature $H_n$, when the boundaries of these disks tend to infinity. This theorem generalizes to the non-zero constant mean curvature case Theorem 0.1 by Colding and Minicozzi…

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…

Study on minimal disks in metric spaces, focusing on branch set structure.

The paper finds representations of surface groups in SO(4,1) with specific curvature properties.