We study the structure of the branch set of solutions to Plateau's problem in metric spaces satisfying a quadratic isoperimetric inequality. In our first result, we give examples of spaces with isoperimetric constant arbitrarily close to the Euclidean isoperimetric constant $(4π)^{-1}$ for which solutions have large branch set. This complements recent results of Lytchak--Wenger and Stadler stating, respectively, that any space with Euclidean isoperimetric constant is a CAT($0$) space and solutions to Plateau's problem in a CAT($0$) space have only isolated branch points. We also show that any planar cell-like set can appear as the branch set of a solution to Plateau's problem. These results answer two questions posed by Lytchak and Wenger. Moreover, we investigate several related questions about energy-minimizing parametrizations of metric disks: when such a map is quasisymmetric, when its branch set is empty, and when it is unique up to a conformal diffeomorphism.
We strengthen our previous results regarding the moduli spaces of Zoll metrics and Zoll projective structures on S^2. In particular, we describe a concrete, open condition which suffices to guarantee that a totally real embedding of RP^2 in CP_2 arises from a unique Zoll projective structure on the 2-sphere. Our methods ultimately reflect the special role such structures play in the initial value problem for the 3-dimensional Lorentzian Einstein-Weyl equations.
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, annuli or Moebius bands. This answers a question posed by Hoffman and White.
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.
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.
We desingularize a branch point $p$ of a minimal disk $F_0(\mathbb{D})$ in $\mathbb{R}^4$ through immersions $F_t$'s which have only transverse double points and are branched covers of the plane tangent to $F_0(\mathbb{D})$ at $p$. If $F_0$ is a topological embedding and thus defines a knot in a sphere/cylinder around the branch point, the data of the double points of the $F_t$'s give us a braid representation of this knot as a product of bands.
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 due to Meeks, we prove that such a blow-up set K must be contained in a C^1 embedded curve.
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 forms part of our ongoing works on the existence of complete non-compact free boundary minimal planes in an asymptotically flat three-dimensional Riemannian manifold with boundary. We set up the degree theory for the space of properly embedded partially free boundary minimal half disks in a three-dimensional half ball. We prove that the space of such surfaces is a Banach manifold. The projection map which projects partially free boundary minimal half disks into their Dirichlet boundaries is a Fredholm map of index zero and has a well-defined mod-2 degree under suitable assumptions. …
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 results extend to higher dimensions earlier three dimensional work of J. C. C. Nitsche and R. Souam.
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 for Plateau problem of minimal disks, which can be used to generalize the famous work by Morse-Thompkins and Shiffman on minimal surfaces in $\mathbf{R}^n$ to the Riemannian setting. More precisely, we generalize the min-max construction of minimal surfaces using harmonic replace…
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 the free boundary analogue of the Costa-Hoffman-Meeks surfaces and the surfaces constructed by Kapouleas by desingularizing coaxial catenoids and planes. It is plausible that the minimal surfaces we constructed here are the same as the ones obtained recently by Ketover using the …
Motivated by a classical result of Finn and Osserman (1964), who proved that the Scherk surface over the square inscribed in the unit disk is extremal for the Gaussian curvature of the point $O$ (so-called \emph{centre}) of the minimal graphs above the center $0$ of unit unit disk, provided the tangent plane is horizontal, we ask and answer to the question concerned the extremal of "second derivative" of the Gaussian curvature of such graphs provided that its curvature at $O$ is zero. We prove that the extremals are certain Scherk type minimal surfaces over the regular hexagon inscribed in the…
Stability and uniqueness of minimal disks in non-constant curvature
Universidad de la República（乌拉圭共和国大学）；CNRS（法国国家科学研究中心）；Université Paris-Saclay（巴黎萨克雷大学）；Laboratoire de mathématiques d’Orsay（奥赛数学实验室）；University of Chicago（芝加哥大学）；PUC-Rio（天主教里约热内卢大学）
We consider the Gaussian curvature conjecture of a minimal graph $S$ over the unit disk. First of all we reduce the general conjecture to the estimating the Gaussian curvature of some Scherk's type minimal surfaces over a quadrilateral inscribed in the unit disk containing the origin inside. As an application we improve so far the obtained upper estimates of Gaussian curvature at the point above the center. Further we obtain an optimal estimate of the Gaussian curvature at the point $\mathbf{w}$ over the center of the disk, provided $\mathbf{w}$ satisfies certain "symmetric" conditions. The re…
We show that any 2-valued C^{1, α} (α\in (0, 1)) function u = {u_{1}, u_{2}} on an open ball B in {\mathbb R}^{n} with values u_{1}, u_{2} \in {\mathbb R}^{k} whose graph, viewed as a varifold with multiplicity 2 at points where u_{1} = u_{2} and with multiplicity 1 at points where u_{1}, u_{2} are distinct, is stationary in the cylinder B \times {\mathbb R}^{k} must be a C^{1, 1/2} function, and the set of its branch points, if non-empty, must have Hausdorff dimension (n-2) and locally positive (n-2)-dimensional Hausdorff measure. The C^{1, 1/2} regularity is optimal.
Minimal surfaces in a Riemannian manifold $M^n$ are surfaces which are stationary for area: the first variation of area vanishes. In this paper we focus on surfaces of the topological type of the real projective plane $\R P^2$. We show that a minimal surface $f:\R P^2\to M^3$ which has the smallest area, among those mappings which are not homotopic to a constant mapping, is an immersion. That is, $f$ is free of branch points. As a major step toward treating minimal surfaces of the type of the projective plane, we extend the fundamental theorem of branched immersions to the nonorientable case. …
In this paper, we consider minimal graphs in the three-dimensional Riemannian manifold $M\times\mathbb{R}$. We mainly estimate the Gaussian curvature of such surfaces. We consider the minimal disks and minimal graphs bounded by two Jordan curves in parallel planes. The key to the proofs is the Weierstrass representation of those surfaces via $\wp-$harmonic mappings. We also prove some Schwarz lemma type results and some Heinz type results for harmonic mappings between geodesic disks in Riemannian surfaces.
It was proved by Montiel and Ros that for each conformal structure on a compact surface there is at most one metric which admits a minimal immersion into some unit sphere by first eigenfunctions. We generalize this theorem to the setting of metrics with conical singularities induced from branched minimal immersions by first eigenfunctions into spheres. Our primary motivation is the fact that metrics realizing maxima of the first non-zero Laplace eigenvalue are induced by minimal branched immersions into spheres. In particular, we show that the properties of such metrics induced from $\mathbb{S…
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 by gluing together two oppositely--oriented double spiral staircases. The second gives a pair of pants decomposition of any such surface when there are small necks, cutting the surface along a collection of short curves. After the cutting, we are left with graphical pieces that ar…
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 fundamental tool for studying complete constant mean curvature surfaces embedded in $\mathbb{R}^3$ with finite topology or with positive injectivity radius.
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 contain all minimal disks with boundary on it? Does it contain the solutions of Plateau's problem? We answer this question negatively and characterize the least barrier enclosing all the minimal hypersurfaces with boundary on a given set.
We prove that the area of a free boundary minimal surface $Σ^2 \subset B^n$, where $B^n$ is a geodesic ball contained in a round hemisphere $\mathbb{S}^n_+$, is at least as big as that of a geodesic disk with the same radius as $B^n$; equality is attained only if $Σ$ coincides with such a disk. More generally, we prove analogous results for a class of conformally euclidean ambient spaces. This follows work of Brendle and Fraser-Schoen in the euclidean setting.
In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or exists for all time and converges to a minimal disk. We further classify the type of the singularity. We additionally present applications of these results to the uniqueness problem for minimal hypersurfaces with free boundary on such suppport hypersurfaces; the results obtained t…
An equivariant minimal surface in $\mathbb{CH}^n$ is a minimal map of the Poincaré disc into $\mathbb{CH}^n$ which intertwines two actions of the fundamental group of a closed surface $Σ$: a Fuchsian representation on the disc and an irreducible action by isometries on $\mathbb{CH}^n$. The moduli space of these can been studied by relating it to the nilpotent cone in each moduli space of $PU(n,1)$-Higgs bundles over the conformal surface corresponding to the map. By providing a necessary condition for points on this nilpotent cone to be smooth this article shows that away from the points corre…
In this note we use the strong maximum principle and integral estimates prove two results on minimal hypersurfaces $F:M^n\rightarrow\mathbb{R}^{n+1}$ with free boundary on the standard unit sphere. First we show that if $F$ is graphical with respect to any Killing field, then $F(M^n)$ is a flat disk. This result is independent of the topology or number or boundaries. Second, if $M^n = \mathbb{D}^n$ is a disk, we show the supremum of the curvature squared on the interior is bounded below by $n$ times the infimum of the curvature squared on the boundary. These may be combined the give an impress…
