Boundary branch points of least-area disks in the classical Plateau problem for smooth non-analytic boundary curves

The classical Plateau problem asks for a surface of least area spanning a given Jordan curve. Douglas and Radó proved existence of a least-area disk, represented by a conformal harmonic map from the closed unit disk, and Morrey extended the theory to homogeneously regular Riemannian manifolds; these are the Douglas–Morrey solutions. The parametrization may in principle fail to be an immersion at branch points. Interior branch points of minimizing disks were excluded around 1970 (Osserman; Gulliver; Alt; Gulliver–Osserman–Royden), and boundary branch points were excluded for real-analytic boundary curves (Gulliver–Lesley). For smooth but non-analytic curves the question has remained open since the 1930s: can a Douglas–Morrey least-area disk spanning a $C^k$ Jordan curve have a branch point on the boundary? Meeks and Yau (Topology, 1982) assume precisely this exclusion as a hypothesis in their embeddedness theorem for least-area disks in convex three-manifolds and list the question among the problems their paper leaves open; Tromba (2011) calls it a fundamental question 'open for some eighty years'. Partial exclusions exist — Wienholtz's inequality $2m-2<3n$, and Tromba's exclusion of nonexceptional boundary branch points for $C^k$ local minimizers spanning $C^\infty$ curves with nonzero curvature and torsion — but the exceptional cases and the general smooth category remain unsettled.

Background

The classical Plateau problem asks for a surface of least area spanning a given Jordan curve (a one-to-one continuous image of the circle) in $\mathbb{R}^3$. Douglas and Radó proved in the early 1930s that a least-area disk exists, realized as a conformal harmonic map $f:\bar D\to\mathbb{R}^3$ from the closed unit disk whose boundary restriction parametrizes the curve weakly monotonically (i.e. it wraps once around the curve preserving the cyclic order); Morrey later extended existence to maps into homogeneously regular Riemannian manifolds, meaning compact regions whose metric is uniformly comparable to a Euclidean metric in suitable charts. For conformal maps the Dirichlet energy $\int_{\bar D}\|df\|^2$ equals twice the parametrized area, so in the Douglas–Courant variational formulation one minimizes the Dirichlet energy over disk-type surfaces spanning the curve; such a minimizing conformal harmonic map is what is called a Douglas–Morrey solution of the Plateau problem.

A branch point is a point where the differential $df$ vanishes, so that the parametrization fails to be an immersion even though the image surface may still be geometrically regular. At an interior branch point the local expansion of a nonplanar minimal surface in a suitable normal form begins with a homogeneous term of order $n\ge 1$ (the order of the branch point), and the third coordinate function vanishes to an order described by an integer index $m$; the branch point is called exceptional when $m+1=2(n+1)$ or $m+1=3(n+1)$, which are exactly the index pairs at which the standard higher-derivative variation tests degenerate. False branch points, which arise only from a non-injective conformal covering of an unbranched surface, were excluded for minimizing disks by Gulliver and by Alt; Osserman and Gulliver–Osserman–Royden excluded true (geometric) interior branch points, so a least-area disk has no branch points in the open disk. Boundary branch points are subtler: Nitsche established Kellogg-type boundary regularity for the parametrization and showed that boundary branch points, if present, are finite in number, and Gulliver and Lesley proved that a least-area disk spanning a real-analytic Jordan curve has no boundary branch points. For smooth but non-analytic curves no general exclusion is known.

The question matters because of the embeddedness program of Meeks and Yau. They proved that least-area disks are embedded under convexity and barrier hypotheses — the key that turns minimal-surface theory into a tool for three-manifold topology, including minimal-surface proofs of Dehn's lemma and the loop theorem (the classical statements that a simple closed curve on the boundary of a three-manifold which is null-homotopic inside the manifold bounds a properly embedded disk). In their Topology 1982 paper the embedding theorem for a least-area disk in a convex three-manifold is proved under the hypothesis that the solution has no branch points on the boundary; for extremal curves (curves contained in the boundary of their convex hull) the paper excludes boundary branch points by a barrier argument, but for general smooth curves the hypothesis is unverified, and the paper's introduction poses the question explicitly. Subsequent partial exclusions in the smooth category use higher derivatives of the Dirichlet energy in the directions of forced Jacobi fields — variations of the surface generated by boundary reparametrizations that vanish at the branch point: Wienholtz excluded boundary branch points whose order and index satisfy $2m-2<3n$, and Tromba excluded nonexceptional boundary branch points for $C^k$ local minimizers spanning $C^\infty$ curves with nonzero curvature and torsion. For the embeddedness question itself, only special curve classes are understood (extremal curves; curves of total boundary curvature at most $4\pi$, by Ekholm–White–Wienholtz; weak-extreme curves, meaning curves that decompose into extreme pieces, by Coskunuzer), and examples of Coskunuzer show that mean convexity of a containing domain does not force the ambient least-area disk to be embedded. The cases left open by these partial exclusions — exceptional index pairs $m+1\in\{2(n+1),3(n+1)\}$ for $C^\infty$ curves, curves of finite smoothness class $C^k$ with $2\le k<\infty$, curves whose curvature or torsion vanishes, and the untreated index ranges — constitute the remaining gap in the smooth category.

Problem Statement

Let $\Gamma$ be a Jordan curve in $\mathbb{R}^3$, or more generally in a homogeneously regular Riemannian three-manifold $N$ (Morrey's setting), of smoothness class $C^\infty$ (or, more generally, $C^k$ with $k\ge 2$), not assumed to be real-analytic, extremal, or otherwise nondegenerate, and let $f:\bar D\to N$ be a Douglas–Morrey solution for $\Gamma$: a conformal harmonic map of the closed unit disk whose boundary restriction parametrizes $\Gamma$ weakly monotonically and which has least area among all disk-type surfaces spanning $\Gamma$ (equivalently, minimizes the Dirichlet energy in the Douglas–Courant formulation, conformality of the minimizer being part of the conclusion).

Question: can $f$ have a branch point on the boundary $\partial D$, i.e. a point $p\in\partial D$ with $df(p)=0$?

The question is answered negatively by proving that for every such $C^\infty$ Jordan curve $\Gamma$, every Douglas–Morrey least-area solution is immersed up to and including the boundary; the stronger form of the statement covers every $C^k$ local minimizer of area or of Dirichlet energy in the $C^k$ topology. It is answered affirmatively by constructing a $C^\infty$ (or $C^k$, $k\ge 2$) Jordan curve $\Gamma$ and a Douglas–Morrey least-area disk spanning $\Gamma$ that has a boundary branch point of order $n$ and index $m$.

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