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$ \newcommand\D{{\mathbb{D}}} \newcommand\Cak{{\mathbb{C}}} \newcommand\Chat{{\hat{\Cak}}} \renewcommand\P{{\mathbb P}} \renewcommand\R{{\mathbb R}} \newcommand\del{\partial} \newcommand\delbar{\bar{\del}} \newcommand\eps{\varepsilon}$

Received: 4 February 2016 / Revised: 30 November 2016 / Accepted: 23 December 2016

Convex Shapes
and Harmonic Caps

Laura DeMarco demarco@math.northwestern.edu

Department of Mathematics,

Northwestern University,

Evanston IL 60208 USA and Kathryn Lindsey klindsey@math.uchicago.edu

Department of Mathematics,

University of Chicago,

Chicago IL 60637 USA

Department of Mathematics,

Northwestern University,

Evanston IL 60208 USA and Kathryn Lindsey klindsey@math.uchicago.edu

Department of Mathematics,

University of Chicago,

Chicago IL 60637 USA

Any planar shape $ P\subset \Cak$ can be embedded isometrically as part of the
boundary surface $ S$ of a convex subset of $ \mathbb{R}^3$ such that
$ \partial P$ supports the positive curvature of $ S$. The complement
$ Q = S {\setminus} P$ is the associated cap
. We study the cap construction when the curvature is harmonic measure on
the boundary of $ (\Chat{\setminus} P, \infty)$. Of particular interest is the case when
$ P$ is a filled polynomial Julia set and the curvature is proportional to
the measure of maximal entropy.

A planar shape is a compact, connected subset of the Euclidean plane that contains at least two points and has connected complement. Given a probability measure $ \mu$ supported on the boundary of a planar shape $ P$, we investigate the existence of a conformal metric $ \rho = \rho(z) |dz|$ on the Riemann sphere $ \Chat$ so that

- (i) $ P$, with its Euclidean metric from $ \R^2$, embeds locally-isometrically into $ (\Chat, \rho)$; and
- (ii) the curvature distribution $ \omega_\rho = -\Delta \log \rho(z)$ on $ \Chat$ is equal to the push-forward of $ 4\pi \mu$ under the embedding.

Alexandrov's theorems on convex surfaces ([Alexandroff1942], [Aleksandrov1948], [Alexandrov2005]) assert that any abstract metrized sphere with non-negative curvature is isometric to the boundary surface of a convex body in $ \mathbb{R}^3$ with its induced metric (unique up to rigid motions of $ \mathbb{R}^3$). In particular, the metrized sphere $ (\Chat, \rho(P,\mu))$ will have a unique convex 3D realization. The convex body may be degenerate , meaning that it lies in a plane and the sphere is viewed as the double of a convex planar region. Conversely, the surface of any compact, convex body in $ \R^3$ (not contained in a line) may be endowed with a complex structure and uniformized so that it is isometric to the Riemann sphere with a conformal metric of non-negative curvature; see, e.g. [Reshetnyak1993]. Thus, the existence of $ \rho(P,\mu)$ may be viewed as a problem of "folding" the shape $ P$ into $ \R^3$ and taking its convex hull, in such a way that the curvature of the resulting convex body is given by $ 4\pi \mu$.

The complement of $ P$ in $ (\Chat, \rho(P,\mu))$ will be called the cap of $ (P, \mu)$ and denoted by $ \hat{P}_\mu$. By construction, the metric on the cap is flat, so there is a locally isometric development map \begin{eqnarray*} D: \hat{P}_\mu \to (\Cak, |dz|). \end{eqnarray*} We say the cap is planar if the development $ D$ is injective.

Our first observation is that there always exists a probability measure $ \mu$ supported on $ \del P$ so that the metric $ \rho(P,\mu)$ exists (see Sect. 2.1 for a simple but degenerate construction). We also observe that not all caps are planar, and we give examples in Sect. 2.

The harmonic cap. We are especially interested in the case where $ P$ is a connected filled Julia set $ K(f)$ of a polynomial $ f: \Cak\to\Cak$ and the prescribed measure $ \mu$ is the measure of maximal entropy supported on the boundary of $ K(f)$; see details in Sect. 2.4. This metrized sphere was defined in ([DeMarco2003], Section 12) for an arbitrary rational map $ f: \P^1 \to \P^1$ of degree $ >1$. Questions about the features of its 3-dimensional realization were first posed by McMullen and Thurston.

To this end, we examine arbitrary planar shapes $ P\subset \Cak$, and we let $ \mu$ be the harmonic measure for the domain $ \Chat{\setminus} P$ relative to $ \infty$. By definition, $ \mu$ is the push-forward of the Lebesgue measure on the unit circle $ S^1$ (normalized to have total mass 1) under a conformal isomorphism $ \Phi : \Cak{\setminus} \overline{\D} \to \Cak{\setminus} P$; the measure $ \mu$ is well defined even if $ \Phi$ is not everywhere defined on $ S^1$. In this setting, the metric $ \rho(P,\mu)$ is simply an extension of the Euclidean metric $ |dz|$ on $ P$; it can be expressed in terms of the Green function \begin{eqnarray*} G_P(z) = \log|\Phi^{-1}(z)| \end{eqnarray*} for $ z \in \Cak{\setminus} P$. Setting $ G_P(z) = 0$ for $ z\in P$, we have \begin{eqnarray*} \rho(P,\mu) = e^{-2G_P(z)}|dz|. \end{eqnarray*} Observe that the metric $ \rho(P,\mu)$ is continuous on all of $ \Chat$: $ G_P$ is continuous on $ \Cak$ (by solvability of the Dirichlet problem on simply-connected domains), and it grows as $ \gamma + \log|z| + o(1)$ as $ z\to\infty$ for some $ \gamma\in\R$. The cap $ \hat{P}_\mu$ is called the harmonic cap of $ P$.

Theorem 1.1 allows one to appeal to the theory of univalent functions for conditions on $ P$ that guarantee planarity of the harmonic cap. If the harmonic cap is planar, then the construction can be iterated, to find the harmonic cap of the development of a harmonic cap. It would be interesting to understand the properties of this dynamical system on a class of planar shapes. (The closed unit disk is a fixed point of this operation; see Example 4.1.)

Constructing a cap. Given the data of a conformal metric $ (\Chat, \rho)$ with non-negative curvature distribution, it is a notoriously difficult problem to construct the 3D realization, even for polyhedral metrics (as we discuss below). But it turns out that a development of a cap $ \hat{P}_\mu$ in $ \Cak$ can be easily produced on the computer.

For planar shapes that are Jordan domains with rectifiable boundaries, a cap $ \hat{P}_\mu$ will have boundary of the same length as $ \del P$. A perimeter gluing of $ P$ and $ \hat{P}_\mu$ is the boundary identification (by arclength) between $ \del P$ and $ \del \hat{P}_\mu$ that produces $ (\Chat, \rho(P,\mu))$.

Given an arbitrary planar shape $ P$, we can approximate it by a shape $ P'$ with piecewise-differentiable Jordan curve boundary and approximate any given measure $ \mu$ on $ \del P$ with a probability measure supported on the boundary of $ P'$. In this way, Theorem 1.2 supplies a straightforward strategy to illustrate the caps. In practice, we use polygonal approximations to the planar shape $ P$ with discrete curvature supported on the vertices. See Figs. 10 and 2. A theorem of Reshetnyak states that weak convergence of the curvature distributions as measures on $ \Chat$ implies convergence of the metrics ([Reshetnyak1993], Theorem 7.3.1; [Rešetnjak1960]).

For polygonal planar shapes with arbitrary probability measures $ \mu$ supported on their vertices, our cap-drawing algorithm (which follows the proof of Theorem 1.2) can be used to draw the parametrization $ \hat{s}$, independent of the existence of the metric extension $ \rho(P,\mu)$. For many examples, the curve $ \hat{s}$ fails to form a closed loop or has a shape that cannot be the boundary parametrization of any Euclidean development of a cap (e.g., it may have positive winding number around a point in the plane, while the boundary of a cap development, traversed in the clockwise direction, will wind non-positively around all points). For example, if $ P$ is a triangle, there is a unique measure $ \mu$ supported on the vertices of $ P$ that gives rise to a cap: any associated cap is necessarily a triangle whose sidelengths are the same as those of $ P$, implying the cap is a reflected copy of $ P$, the convex shape is degenerate, and $ \mu(v) = (\pi - \theta)/(2\pi)$ where $ v$ is a vertex of $ P$ with internal angle $ \theta$. In general, the questions of when the metric $ \rho(P,\mu)$ exists and when the cap $ \hat{P}_\mu$ is planar are quite delicate, even in the polygonal setting.

Problem 1.3 is related to the geometry of the space of polygons with fixed side lengths and no boundary crossings, which, to our knowledge, has never been described. See [Connelly et al.2003] where it is proved that the space is connected and contractible. The 3-dimensional realization. Recall, by Alexandrov's theorems ([Alexandrov2005], [Aleksandrov1948], [Alexandroff1942, Pogorelov1973]), for nonnegative $ \mu$ there is a unique way to fold the Euclidean development of $ P$ and $ \hat{P}_\mu$ to form the boundary surface of a convex shape in $ \R^3$. We may view the output of the cap-drawing algorithm, as in Figs. 10 and 2, as paper cut-outs to be creased and glued to form the desired shape. Unfortunately, the exact shape of the 3-dimensional realization is not at all clear from the development alone. Even the set of folding lines inside $ P$ and $ \hat{P}_\mu$ is a mystery in general. Quoting from Alexandrov in translation ([Alexandrov2005], p.100), "To determine the structure of a polyhedron from a development, i.e., to indicate its genuine edges in the development, is a problem whose general solution seems hopeless." But in the case of harmonic measure on a planar shape, especially when the shape is the filled Julia set of a polynomial, there may be specialized ways to attack the problem.

Not long ago, Bobenko and Izmestiev devised an illuminating and constructive proof of Alexandrov's realization theorem for polyhedral metrics ([Bobenko and Izmestiev2008]), implementing their algorithm and making it publicly available. Unfortunately, the algorithm was not practical for the polyhedra that closely approximate the metrics for polynomial Julia sets ([Bartholdi2015]). Laurent Bartholdi modified their strategy to handle some dynamical examples, such as the filled Julia set of $ f(z) = z^2-1$ shown in Fig. 3.

Formally, the convex 3D realization of $ (\Chat, \rho(P, \mu))$ determines a Euclidean lamination on the interiors of $ P$ and $ \hat{P}_\mu$, consisting of the geodesic line segments that must be folded to form the 3D shape. We call this the bending lamination of the pair $ (P, \mu)$. If one also retains the data of the dihedral angles (the amount of the fold along each leaf of the lamination), we obtain a measured lamination, uniquely determined by the pair $ (P, \mu)$. We leave the following as an open problem:

Our research was supported by the National Science Foundation and the Simons Foundation.

In this section, we observe that for every planar shape $ P$, there is a probability measure $ \mu$ on its boundary so that the metric $ \rho(P,\mu)$ on $ \Chat$ exists, by simple constructions in $ \R^2$. We provide examples to illustrate the failure of planarity of a cap. We conclude the section with examples of harmonic caps coming from polynomial dynamical systems $ f: \Cak\to \Cak$. Formal definitions and the proofs of our theorems will be given in Sects. 3 and 4.

Start with a convex polygonal shape in the plane with an external angle of about $ \pi/16$ at one vertex. Remove two very thin spiral channels from the polygon that begin on adjacent edges of the polygon and spiral around one another, as in the left image of Fig. 4. If the spirals are sufficiently intertwined, then the spiral flaps on the developed naive cap will overlap. The right side of Fig. 4 shows the spirals reflected across the edges of the polygon.

For the harmonic cap, it is possible to construct an example similar to that of Sect. 2.2. Indeed, very skinny channels removed from any planar shape will have negligible harmonic measure, and so we can arrange for overlapping spirals in the cap.

More precisely, begin with a square planar shape and choose a tiny $ \eps>0$. The harmonic cap for the square is shown in Fig. 10. Now remove two very skinny spiral channels from the square, emanating from a single edge, as in the left image of Fig. 5; the openings of each channel should have width smaller than $ \eps$. The openings of the two spiral channels can be placed at a specified distance apart from one another, so that the harmonic measure of the interval between them is approximately equal to $ 1/32$ of the total mass. (The number $ 1/32$ is chosen because it is $ 1/4\pi$ times the curvature of $ \pi/8$ for the polygon vertex shown in Fig. 4). We can choose $ \eps>0$ as small we wish so that the harmonic measure along the spiral boundaries is almost 0. Indeed, as the width of the spiral channels shrinks to 0, the domains $ \Chat{\setminus} P$ are converging in the Carath[U+00B4]eodory sense to the complement of the square; see, e.g., ([Duren1983], [U+00A7]3.1).

Recall that the boundary of the cap development is parameterized by the formula of Theorem 1.2. The parametrization of the spirals on the cap, which will lie outside the clover-like harmonic cap for the square, will be essentially equal to a reflection of their original parametrizations (because $ \kappa$ will be essentially constant along their boundaries, having chosen the harmonic measure of the spirals to be near 0). On the other hand, the non-trivial portion of harmonic measure on the boundary of the square between the spiral-channel openings will curve the boundary of the cap so the spirals overlap. The change in tangent direction of the clover cap between the two attaching points of the spirals will be $ \pi/8$, by construction. See Fig. 5.

Now assume that $ f: \Cak\to\Cak$ is a complex polynomial of degree $ d\geq 2$. Its filled Julia set is \begin{eqnarray*} K(f) = \left\{z \in \Cak: \sup_n |f^n(z)| < \infty\right\}. \end{eqnarray*} Assume that $ K(f)$ is connected, so it is a planar shape. A planar development of its cap is given by the formula of Theorem 1.1. We can parameterize the boundary of the cap's development for smooth or polygonal approximations to $ K(f)$ using Theorem 1.2.

The Green function for $ K(f)$ can be computed dynamically, as \begin{eqnarray*} G_f(z) = \lim_{n\to\infty} \frac{1}{d^n} \log^+|f^n(z)|. \end{eqnarray*} The harmonic measure $ \mu_f = \frac{1}{2\pi} \Delta G_f$ is the unique measure of maximal entropy for $ f$, and its support is equal to the Julia set $ J(f) = \del K(f)$ ([Brolin1965, Ljubich1983, Freire et al.1983]). The metric on $ \Chat$ is defined by \begin{eqnarray*} \rho_f = e^{-2G_f(z)}|dz| \end{eqnarray*} for $ z\in \Cak$, with curvature distribution $ \omega_f = - \Delta \log \rho_f(z) = 4\pi \mu_f$.

In this section, we formalize the notions of curvature and metric from the point of view of Euclidean geometry, and we prove Theorem 1.2. In Proposition 3.1, we present an asymptotic formula for curvature when the boundary of the planar shape is a smooth Jordan curve, in terms of the circumference of small circles.

A convex polyhedron in $ \R^3$ is the intersection of finitely many closed halfspaces. It is said to be degenerate if it lies in a plane. When the polyhedron is non-degenerate and bounded, its boundary surface is topologically a sphere, and the Euclidean metric from $ \R^3$ induces an intrinsic path metric on the sphere. If the polyhedron is degenerate and bounded, but not contained in a line, we will still view its boundary as a topological sphere, doubling the planar polygon and gluing along the polygonal boundary.

Abstractly, a convex polyhedral metric on a 2-dimensional sphere is an intrinsic metric with non-negative curvature concentrated at finitely many points. In other words, in a small neighborhood of all but finitely many points, the surface is isometric to a region in $ \R^2$. In a neighborhood of each of the finitely many cone points , the surface is isometric to the point of a cone. The curvature of a cone point is equal to the angle deficit at the point; that is, if the circumference of any small circle of radius $ r$ centered at the cone point is equal to $ C(r)$, then the curvature is equal to $ (2\pi r - C(r))/r$. By the Gauss-Bonnet formula, the sum of the curvatures over all cone points on the sphere is equal to $ 4\pi$. [Alexandrov2005] examines the geometry of convex polyhedra in detail. He presents his proof from [Alexandroff1942] that any abstract polyhedral metric on a sphere is isometric to the boundary of a (possibly degenerate) convex polyhedron. Furthermore, the polyhedron in $ \R^3$ is unique, up to Euclidean isometries. Given a polyhedral metric on the sphere, and a simply-connected subset $ U$ of the sphere minus its cone points, a Euclidean development of $ U$ is a local isometry $ U \to \R^2$. Suppose we are given the image $ \mathcal{I}\subset \R^2$ of a Euclidean development of a full-area, simply-connected subset $ U$ of the sphere. Then, as a consequence of Alexandrov's theorem, the convex polyhedron in $ \R^3$ is uniquely determined by $ \mathcal{I}$ and the gluing along its boundary (that reconstructs the topological sphere). In particular, the planar development and the gluing information will uniquely determine the edges of the polyhedron and their dihedral angles in $ \R^3$---information that is not locally apparent.

Curvature is carefully treated by Alexandrov. It is defined by an additive set function $ \omega$ as follows. The curvature of a point is, as for a polyhedron, $ 2\pi$ minus the cone angle of the point. That is,

\begin{eqnarray}\label{pointcurvature} \omega(\{x\}) = \lim_{r\to 0^+} \frac{2\pi r - C(x,r)}{r} \end{eqnarray} | (3.1) |

Reshetnyak, who was a student of Alexandrov, reformulated Alexandrov's theory of metrics and curvature on a surface in complex-analytic language, expressing curvature as a finite Borel measure ([Reshetnyak1993]). We exploit this useful point of view in Sect. 4.

Suppose that a planar shape $ P$ is the closure of a Jordan domain with a piecewise-differentiable boundary. Fix a nonnegative Borel measure $ \mu$ on the boundary of $ P$. Let $ L$ be the length of $ \del P$. Let $ s$ be a piecewise-differentiable parametrization by arclength of the boundary of $ P$, in the counterclockwise direction, and write \begin{eqnarray*} s'(t) = e^{i \alpha(t)} \end{eqnarray*} for a piecewise-continuous function $ \alpha: [0,L] \to \R$. For $ t \in [0, L]$, we define a curvature function $ \kappa: [0,L] \to [0, 4\pi]$ by $ \kappa(0) = 0$ and

\begin{eqnarray} \label{curvaturefunction} \kappa(t) = 4\pi \mu(s(0,t]) \end{eqnarray} | (3.2) |

If $ \hat{P}_\mu$ exists, then it has a polygonal boundary with the same edge lengths as $ P$. We label its vertices in the clockwise direction by $ \hat{v}_0, \hat{v}_1, \ldots, \hat{v}_N = \hat{v}_0$. We may assume for simplicity that $ \hat{v}_0 = v_0$ and $ \hat{v}_1 = v_1$. The curvature condition implies that the internal angle $ \hat{\theta}_j$ at vertex $ \hat{v}_j$ must satisfy \begin{eqnarray*} 4\pi \mu(v_j) = 2\pi - \theta_j - \hat{\theta}_j. \end{eqnarray*} Therefore, the clockwise parametrization $ \hat{s}$ of $ \hat{P}_\mu$ will satisfy $ \hat{s}'(t) = e^{i \hat{\alpha}(t)}$ with \begin{eqnarray*} \hat{\alpha}(t) &=& - \sum_{j=1}^{k-1} (\pi - \hat{\theta}_j) \quad \mbox{for } \sum_{j=1}^{k-1} \ell_j \leq t < \sum_{j=1}^k \ell_j \\ &=& \alpha(t) - \sum_{j=1}^{k-1} 4\pi \mu(v_j) \quad \mbox{for } \sum_{j=1}^{k-1} \ell_j \leq t < \sum_{j=1}^k \ell_j \\ &=& \alpha(t) - \kappa(t) \end{eqnarray*} In other words, the parametrization of the boundary of $ \hat{P}_\mu$ is given in a clockwise orientation by \begin{eqnarray*} \hat{s}(t) = \int_0^t e^{i (\alpha(x) - \kappa(x))} \, dx. \end{eqnarray*}

If $ P$ is an arbitrary planar shape with piecewise-differentiable boundary, and if $ \mu$ is any probability measure supported on the boundary of $ P$, then the pair $ (P, \mu)$ can be approximated by a sequence of polygons $ (P_n, \mu_n)$ so that the vertices of $ P_n$ lie in $ \del P$ for all $ n$, and $ \mu_n$ is a discrete probability measure supported on the vertices of $ P_n$. We may construct the polygons $ P_n$ so that the arclength parametrizations $ s_n$ of $ \del P_n$ converge uniformly to $ s$ and that the angle functions $ \rho_n \to \rho$ uniformly. Furthermore, by choosing the vertices of $ P_n$ carefully, we may assume that for every $ \eps>0$, all atoms of mass at least $ \eps$ for $ \mu$ are vertices of $ P_n$ and atoms of $ \mu_n$ for all $ n\geq n(\eps) > 0$. In this way, we can also arrange that the curvature functions $ \kappa_n$ converge uniformly to the curvature function $ \kappa$. These choices for $ (P_n, \mu_n)$ imply that the integrals \begin{eqnarray*} \int_0^t e^{i (\rho_n(x) - \kappa_n(x))} \, dx \longrightarrow \int_0^t e^{i (\rho(x) - \kappa(x))} \, dx \end{eqnarray*} as $ n\to \infty$ for all $ t \in [0, |\del P|]$. In other words, if the cap $ \hat{P}_\mu$ exists , then the desired boundary parametrization will be uniformly approximated by the curves $ \hat{s}_n$ defined by \begin{eqnarray*} \hat{s}_n(t) = \int_0^t e^{i (\rho_n(x) - \kappa_n(x))} \, dx. \end{eqnarray*} Note that the curves $ \hat{s}_n$ are not necessarily closed loops, as the approximating polygonal caps $ \hat{P}_{\mu_n}$ may not exist. $ \square$ ⬜

If the boundary of the planar domain $ P$ and the measure $ \mu$ are smooth enough, then the curvature of Sect. 3.2 satisfies the following relation, as a consequence of Theorem 1.2.

It is interesting to compare the statement of Proposition 3.1 to the formula (3.1) for the Alexandrov curvature of a point, \begin{eqnarray*} \omega(\{x\}) = \lim_{r\to 0^+} \frac{2\pi r - C(x,r)}{r}, \end{eqnarray*} and to the Bertrand--Puiseux formula for the Gaussian curvature $ \kappa$ when the metric on a surface is smooth, \begin{eqnarray*} \kappa(x) = \; \lim_{r\to 0^+} \; 3 \, \frac{2\pi r - C(x,r)}{\pi r^3} \end{eqnarray*} ([Spivak1979], page 147). In our setting, the curvature of the surface is supported on a 1-dimensional curve, so the circumference discrepancy is proportional to $ r^2$.

We begin with a simple geometric lemma.

The curvature function of Eq. (3.2) is computed as \begin{eqnarray*} \kappa(t) = \mu(s(0,t]) = \int_0^t \delta(x) \,dx. \end{eqnarray*} For each $ t\in [0,L]$ and each small $ r>0$, the circumference $ C(s(t),r)$ is the sum of the lengths of two circular arcs: the arc in $ P$ to the "left" of $ s(t)$ (relative to the counterclockwise orientation on $ \partial P$), whose length we will denote by $ C_r(t)$, and the arc in $ \hat{P}_\mu$ to the "right" of $ \hat{s}(t)$ (relative to the clockwise orientation on $ \partial \hat{P}_\mu$), whose length we will denote by $ \hat{C}_r(t)$. Classical plane geometry tells us that the radius of the osculating circle to the plane curve $ s$ at $ s(t)$ is $ 1/| s^{\prime \prime}(t) | = 1/|\alpha^{\prime}(t)|$, using the notation of Theorem 1.2. Likewise, from Theorem 1.2, the radius of the osculating circle to the plane curve $ \hat{s}$ at $ \hat{s}(t)$ equals $ 1/ | \hat{s}^{\prime \prime}(t)| = 1/ | \alpha^{\prime}(t) - \kappa^{\prime}(t)|$.

For $ \alpha^{\prime}(t) > 0$, the osculating circle is to the left of $ s(t)$, so \begin{eqnarray*} \lim_{r \rightarrow 0} \frac{\pi r - C_r(t)}{r^2} = |\alpha^{\prime}(t)| = \alpha^{\prime}(t) \end{eqnarray*} by Lemma 3.2. For $ \alpha^{\prime}(t) < 0$, the osculating circle is to the right of $ s(t)$, so \begin{eqnarray*} \lim_{ r\rightarrow 0} \frac{\pi r - C_r(t)}{r^2} = \lim_{r \rightarrow 0} \frac{ \pi r - \left(2\pi r - A\left(\frac{1}{|\alpha^{\prime}(t)|},r\right)\right)}{r^2} = - | \alpha^{\prime}(t)| = \alpha^{\prime}(t) \end{eqnarray*} by Lemma 3.2. Thus $ \lim_{r \rightarrow 0} \frac{ \pi r - C_r(t)}{r^2} = \alpha^{\prime}(t)$, regardless of the sign of $ \alpha'(t)$. Similarly, \begin{eqnarray*} \lim_{r \rightarrow 0} \frac{ \pi r - \hat{C}_r(t)}{r^2} = -(\alpha^{\prime}(t) - \kappa'(t)) = \delta(t) - \alpha^{\prime}(t) \end{eqnarray*} regardless of the sign of $ \alpha^{\prime}(t) - \kappa'(t)$. Hence, \begin{eqnarray*} \lim_{r \rightarrow 0} \frac{2\pi r - C(s(t),r)}{r^2}& =& \lim_{r \rightarrow 0} \frac{ \pi r - C_r(t)}{r^2} + \lim_{r \rightarrow 0} \frac{ \pi r - \hat{C}_r(t)}{r^2}\\ & =& \alpha^{\prime}(t) + \delta(t) - \alpha^{\prime}(t) =\delta(t). \end{eqnarray*} $ \square$ ⬜

In this section, we present curvature in the setting of conformal metrics, allowing us to use tools from complex analysis to address our geometric questions. This perspective was first formalized by [Reshetnyak1993]. We present the proof of Theorem 1.1 and derive an alternative proof of the parametrization of the harmonic cap from Theorem 1.2. Finally, we revisit the general problem of existence of the metric $ \rho(P,\mu)$ in Proposition 4.2.

A smooth conformal metric on a domain in $ \Cak$ can be expressed as \begin{eqnarray*} \rho(z) |dz| \end{eqnarray*} for a smooth and positive function $ \rho$. The metric has non-negative curvature if $ U(z) = -\log \rho(z)$ is a subharmonic function. Working with a more general class of metrics, we will only require that $ U$ be subharmonic, not necessarily differentiable or everywhere finite. We will also require that all pairs of points have finite distance from one another. These requirements can be formulated in terms of the curvature of the metric, as we explain below.

Formally, a conformal metric $ \rho$ on $ \Chat$ is a (singular) Hermitian metric on the tangent bundle $ T\Chat \simeq \mathcal{O}_{\P^1}(2)$, and the curvature form of the metric is the positive measure given in local coordinates by \begin{eqnarray*} \omega_\rho = - \Delta \log \rho \end{eqnarray*} (with $ \Delta = 2i \del \delbar$ taken in the sense of distributions), so that \begin{eqnarray*} \int_{\Chat} \omega_\rho = 4\pi. \end{eqnarray*} In more classical terms, for a smooth metric $ \rho$, the Gaussian curvature is computed locally as \begin{eqnarray*} \kappa_\rho = \frac{-\Delta \log \rho}{\rho^2}. \end{eqnarray*} See, for example, ([Ahlfors1973], [U+00A7]1.5) or ([Hubbard2006], [U+00A7]2.2).

That $ U = -\log\rho$ is subharmonic guarantees that the curvature form $ \omega_\rho \geq 0$ as a distribution. Finite diameter is guaranteed by the assumption that $ \omega_\rho(\{z_0\}) < 2\pi$ for all $ z_0 \in \Chat$ ([Reshetnyak1993], p. 100). Recall from Sect. 3.4 that concentrated curvature, at points $ z_0 \in\Chat$ where $ 0 < \omega_{\rho}(\{z_0\}) < 2\pi$, corresponds to cone points in the local geometry. Also in this setting, a computation shows that the circumference $ C(z_0, r)$ of a small circle around $ z_0$ of radius $ r>0$ will satisfy ([Reshetnyak1993], Lemma 8.1.1) \begin{eqnarray*} \lim_{r \to 0^+} \frac{2\pi r - C(z_0,r)}{r} = \omega_{\rho}(\{z_0\}). \end{eqnarray*}

Conversely, every probability measure $ \mu$ on $ \Chat$ with $ \mu(\{z\}) < 1/2$ for all $ z$ gives rise to a conformal metric of finite diameter with curvature distribution $ 4\pi \mu$, unique up to scale. Indeed, there is a one-to-one correspondence between probability measures $ \mu$ on $ \Chat$ and their potentials, up to an additive constant, which can be viewed as logarithmically-homogeneous, plurisubharmonic functions $ G_\mu$ on the tautological line bundle $ \Cak^2{\setminus} \{(0,0)\} \to \P^1$; see, e.g., ([Forn[U+00E6]ss and Sibony1993], Theorem 5.9) and ([DeMarco2003], Section 12). The function $ G_\mu$ will satisfy $ (2\pi)^{-1} \Delta G_\mu(z,1) = \mu$ in local coordinates $ z$ on $ \Chat$, and the conformal metric is expressed as \begin{eqnarray*} \rho_\mu = e^{-2 G_\mu(z,1)} |dz|. \end{eqnarray*} The identification between measures and their potentials is continuous, taking the $ L^1_{loc}$ topology on potentials and the weak topology on measures. Moreover, convergence of curvatures implies convergence of the metrics ([Reshetnyak1993], Theorem 7.3.1).

Let $ P$ be a compact, connected set in $ \Cak$ containing at least 2 points, so that $ P$ is a planar shape as defined in Sect. 1. Let $ G_P: \Cak\to \R$ be the Green function for $ P$; it is the unique continuous function on $ \Cak$ satisfying (1) $ G_P \equiv 0$ on $ P$, (2) $ G_P(z) = \log|z| + O(1)$ for $ z$ near $ \infty$, and (3) $ G_P$ is harmonic on $ \Cak{\setminus} P$. Then define a metric on $ \Cak$ by \begin{eqnarray*} \rho_P = e^{-2G_P(z)} |dz|. \end{eqnarray*} By elementary potential theory, the function $ G_P$ satisfies $ G_P(z) = \log(z) + \gamma + o(1)$ for $ z$ near $ \infty$ for some real number $ \gamma$, so the metric extends uniquely by continuity across $ z=\infty$. Note that this metric is flat (with 0 curvature) away from the boundary $ \del P$. Its curvature form $ \omega_P = 2 \Delta G_P$ is equal to ($ 4\pi$ times) the harmonic measure on $ \del P$ (more precisely, the harmonic measure for the domain $ \Chat{\setminus} P$, relative to the point $ \infty$).

Let $ P$ be any planar shape. Let $ \Phi$ be the
Riemann map from the complement of the unit disk to the complement of
$ P$, sending infinity to infinity. Consider the holomorphic 1-form
\begin{eqnarray*} \eta = \frac{1}{(\Phi^{-1}(z))^2} \; dz \end{eqnarray*} on the complement of $ P$. Since the Green function
satisfies \begin{eqnarray*} G_P(z) = \log|\Phi^{-1}(z)| \end{eqnarray*} on $ \Chat{\setminus} P$, we see that $ |\eta|$ is precisely
the conformal metric $ \rho_P$ defined above, when restricted to the
complement of $ P$. Recall that Theorem 1.1 asserts that
a Euclidean development of the harmonic cap of $ P$ is given by the
locally univalent function $ g: \D\to\Cak$ defined by \begin{eqnarray*} g(z) = \int_0^z \Phi'(1/x) \, dx. \end{eqnarray*} It also asserts
that there exist examples where the locally univalent $ g$ fails to be
univalent.

It remains to observe that there exist planar shapes $ P$ for which the development $ g$ fails to be injective. We constructed such an example in Sect. 2.3, where $ P$ is a square minus two thin spiral channels. $ \square$ ⬜

Here we present an alternative proof of the cap parametrization in Theorem 1.2, in the special setting of harmonic measure.

As in Theorem 1.2, assume that $ P$ has a piecewise-differentiable boundary which is a Jordan curve parameterized by arclength by $ s: [0,L] \to \Cak$. Recall that $ s^{\prime}(t) = e^{i \alpha(t)}$ for some piecewise continuous function $ \alpha:[0,L] \rightarrow \R$. Let $ \Phi$ be a Riemann map from the complement of the unit disk to the complement of $ P$, sending infinity to infinity. Then $ \Phi$ extends to a homeomorphism from the unit circle to the boundary of $ P$. Define the conformal angle $ \theta: [0,L] \rightarrow \R$ by \begin{eqnarray*} \theta(t) := \mathrm{arg}(\Phi^{-1}(s(t))). \end{eqnarray*} Without loss of generality, we may assume $ \theta(0)=0$ so that $ \theta$ defines a homeomorphism from $ [0,L]$ to $ [0,2\pi]$. It follows that the curvature function of (3.2) for the harmonic measure $ \mu$ on $ \del P$ is equal to \begin{eqnarray*} \kappa(t) = 4\pi \mu(s(0,t]) = 2\theta(t). \end{eqnarray*} Therefore, from Theorem 1.2, we know that the parametrization of the boundary of the harmonic cap is given by

\begin{eqnarray} \label{shattheta} \hat{s}(t) = \int_0^t e^{i(\alpha(x) - 2\theta(x))} \, dx. \end{eqnarray} | (4.1) |

Theorem 1.1 grants an alternate proof of (4.1). Indeed, with the $ g: \D \to \Cak$ of Theorem 1.1, a parametrization of the boundary of the harmonic cap is given by \begin{eqnarray*} \hat{s}(t) = -g(1/\Phi^{-1}(s(t))) = -g(e^{-i\theta(t)}). \end{eqnarray*} Moreover, the derivative of $ g$ is $ g'(z) = \Phi'(1/z)$, and therefore, \begin{eqnarray*} \hat{s}'(t) &=& -g'(1/\Phi^{-1}(s(t)))\frac{-(\Phi^{-1})'(s(t)) \; s'(t)}{\Phi^{-1}(s(t))^2} \\ &=& \frac{-\Phi'(\Phi^{-1}(s(t)))}{-\Phi'(\Phi^{-1}(s(t)))} \; \frac{s'(t)}{\Phi^{-1}(s(t))^2} \\ &=& e^{i(\alpha(t) - 2\theta(t))}. \end{eqnarray*}

We conclude by returning to our original problem about the existence of a metric $ \rho(P,\mu)$, for the case where $ P$ is a planar shape with Jordan curve boundary and the probability measure $ \mu$ is arbitrary.

Suppose that $ J$ is a Jordan curve in $ \Chat$, cutting the sphere into Jordan domains $ A$ and $ B$. We may assume that $ 0 \in A$ and $ \infty \in B$. Suppose that $ \nu$ is a probability measure supported on $ J$, and let \begin{eqnarray*} U(z) = \int_\Cak \log|z-w| \, d\nu(w) \end{eqnarray*} be a potential function for $ \nu$ with logarithmic singularity at $ \infty$. The conformal metric \begin{eqnarray*} e^{-2U(z)}|dz| \end{eqnarray*} on $ \Cak$ extends to $ \Chat$ and has curvature distribution equal to $ 4\pi\nu$. Since $ A$ is simply connected, there exists a non-vanishing analytic function $ \phi: A \to \Cak$ so that \begin{eqnarray*} U(z) = \log|\phi(z)|. \end{eqnarray*} The function $ \phi$ is determined uniquely, up to postcomposition by a rotation. Set \begin{eqnarray*} f_\nu(z) = \int_0^z \frac{dz}{\phi(z)^2} \end{eqnarray*} for $ z \in A$. Then $ f_\nu: A \to \Cak$ is a locally-univalent Euclidean development of $ A$ into the plane. It extends continuously to the boundary curve $ J$. This proves the following proposition.

When $ \mu$ is the harmonic measure on $ \del P$, observe that we may take $ J = \del P$ and $ \nu = \mu$ in the statement of Proposition 4.2. Indeed, the potential function for harmonic measure satisfies $ U \equiv 0$ on $ P$ so that $ f_\nu = \mathrm{Id}$.

[Ahlfors1973] Ahlfors, L.V.: Conformal Invariants: Topics in Geometric
Function Theory, McGraw-Hill Series in Higher Mathematics. McGraw-Hill
Book Co., New York (1973)

[Aleksandrov1948] Aleksandrov, A.D.: Vnutrennyaya Geometriya Vypuklyh Poverhnoste[U+02D8][U+0131]. OGIZ, Moscow (1948)

[Alexandroff1942] Alexandroff, A.: Existence of a convex polyhedron and of a convex surface with a given metric. Rec. Math. [Mat. Sbornik] N.S. 11 (53), 15--65 (1942)

[Alexandrov2005] Alexandrov, A.D.: Convex polyhedra. Springer Monographs in Mathematics. Springer, Berlin (2005) (translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky, with comments and bibliography by V. A. Zalgaller and appendices by L. A. Shor and Yu. A. Volkov)

[Bartholdi2015] Bartholdi, L.: Personal communication (2015)

[Bobenko and Izmestiev2008] Bobenko, A.I., Izmestiev, I.: Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes. Ann. Inst. Fourier (Grenoble) 58 (2), 447--505 (2008)

[Brolin1965] Brolin, H.: Invariant sets under iteration of rational functions. Ark. Mat. 6 (103--144), 1965 (1965)

[Connelly et al.2003] Connelly, R., Demaine, E.D., Rote, G.: Straightening polygonal arcs and convexifying polygonal cycles. Discrete Comput. Geom. 30 (2), 205--239 (2003) [U.S.-Hungarian Workshops on Discrete Geometry and Convexity (Budapest, 1999/Auburn, AL, 2000)]

[Demaine and O'Rourke2005] Demaine, E.D., O'Rourke, J.: A survey of folding and unfolding in computational geometry. In: Combinatorial and Computational Geometry, Math. Sci. Res. Inst. Publ., vol. 52, pp. 167--211. Cambridge Univ. Press, Cambridge (2005)

[Demaine and O'Rourke2007] Demaine, E.D., O'Rourke, J.: Geometric Folding Algorithms. Cambridge University Press, Cambridge (2007) (Linkages, origami, polyhedra)

[DeMarco2003] DeMarco, L.: Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity. Math. Ann. 326 (1), 43--73 (2003)

[Duren1983] Duren, P.L.: Univalent functions. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259. Springer, New York (1983)

[Forn[U+00E6]ss and Sibony1993] Forn[U+00E6]ss, J.E., Sibony, N.: Complex dynamics in higher dimensions. In: Complex Potential Theory (Montreal, PQ, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 439, pp. 131--186. Kluwer Acad. Publ., Dordrecht (1994) (Notes partially written by Estela A. Gavosto)

[Freire et al.1983] Freire, A., Lopes, A., Ma~né, R.: An invariant measure for rational maps. Bol. Soc. Brasil. Mat. 14 (1), 45--62 (1983)

[Hubbard and Papadopol1994] Hubbard, J., Papadopol, P.: Superattractive fixed points in $ { c}^n$. Indiana Univ. Math. J. 43 , 321--365 (1994)

[Hubbard2006] Hubbard, J.H.: Teichm[U+00A8]uller theory and applications to geometry, topology, and dynamics, vol. 1. Matrix Editions, Ithaca (2006). Teichm[U+00A8]uller theory, with contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, with forewords by William Thurston and Clifford Earle

[Ljubich1983] Ju, M.: Ljubich. Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dyn. Syst. 3 (3), 351--385 (1983)

[Pogorelov1973] Pogorelov, A.V.: Extrinsic Geometry of Convex Surfaces. American Mathematical Society, Providence (1973) (translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, vol. 35)

[Pommerenke1992] Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299. Springer, Berlin (1992)

[Rešetnjak1960] Rešetnjak, Ju.G.: Isothermal coordinates on manifolds of bounded curvature. I, II. Sibirsk. Mat. [U+02C7]Z. 1 , 88--116, 248--276 (1960)

[Reshetnyak1993] Reshetnyak, Yu.G.: Two-dimensional manifolds of bounded curvature. In: Geometry, IV, Encyclopedia Math. Sci., vol. 70 , pp. 3--163, 245--250. Springer, Berlin (1993)

[Series2006] Series, C.: Thurston's bending measure conjecture for once punctured torus groups. In: Spaces of Kleinian Groups, London Math. Soc. Lecture Note Ser., vol. 329, pp. 75--89. Cambridge Univ. Press, Cambridge (2006)

[Spivak1979] Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. II, 2nd edn. Publish or Perish Inc, Wilmington (1979)

[Thurston1998] Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. In: The Epstein Birthday Schrift, Geom. Topol. Monogr., vol. 1, pp. 511--549. Geom. Topol. Publ., Coventry (1998)

[The Sage Developers2016] The Sage Developers. Sage Mathematics Software (Version 6.10.beta7) (2016).

[Van Andel and Bradshaw2011] Van Andel, E., Bradshaw, R.: Riemann mapping [sage mathematics software package] (2011)

[Aleksandrov1948] Aleksandrov, A.D.: Vnutrennyaya Geometriya Vypuklyh Poverhnoste[U+02D8][U+0131]. OGIZ, Moscow (1948)

[Alexandroff1942] Alexandroff, A.: Existence of a convex polyhedron and of a convex surface with a given metric. Rec. Math. [Mat. Sbornik] N.S. 11 (53), 15--65 (1942)

[Alexandrov2005] Alexandrov, A.D.: Convex polyhedra. Springer Monographs in Mathematics. Springer, Berlin (2005) (translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky, with comments and bibliography by V. A. Zalgaller and appendices by L. A. Shor and Yu. A. Volkov)

[Bartholdi2015] Bartholdi, L.: Personal communication (2015)

[Bobenko and Izmestiev2008] Bobenko, A.I., Izmestiev, I.: Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes. Ann. Inst. Fourier (Grenoble) 58 (2), 447--505 (2008)

[Brolin1965] Brolin, H.: Invariant sets under iteration of rational functions. Ark. Mat. 6 (103--144), 1965 (1965)

[Connelly et al.2003] Connelly, R., Demaine, E.D., Rote, G.: Straightening polygonal arcs and convexifying polygonal cycles. Discrete Comput. Geom. 30 (2), 205--239 (2003) [U.S.-Hungarian Workshops on Discrete Geometry and Convexity (Budapest, 1999/Auburn, AL, 2000)]

[Demaine and O'Rourke2005] Demaine, E.D., O'Rourke, J.: A survey of folding and unfolding in computational geometry. In: Combinatorial and Computational Geometry, Math. Sci. Res. Inst. Publ., vol. 52, pp. 167--211. Cambridge Univ. Press, Cambridge (2005)

[Demaine and O'Rourke2007] Demaine, E.D., O'Rourke, J.: Geometric Folding Algorithms. Cambridge University Press, Cambridge (2007) (Linkages, origami, polyhedra)

[DeMarco2003] DeMarco, L.: Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity. Math. Ann. 326 (1), 43--73 (2003)

[Duren1983] Duren, P.L.: Univalent functions. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259. Springer, New York (1983)

[Forn[U+00E6]ss and Sibony1993] Forn[U+00E6]ss, J.E., Sibony, N.: Complex dynamics in higher dimensions. In: Complex Potential Theory (Montreal, PQ, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 439, pp. 131--186. Kluwer Acad. Publ., Dordrecht (1994) (Notes partially written by Estela A. Gavosto)

[Freire et al.1983] Freire, A., Lopes, A., Ma~né, R.: An invariant measure for rational maps. Bol. Soc. Brasil. Mat. 14 (1), 45--62 (1983)

[Hubbard and Papadopol1994] Hubbard, J., Papadopol, P.: Superattractive fixed points in $ { c}^n$. Indiana Univ. Math. J. 43 , 321--365 (1994)

[Hubbard2006] Hubbard, J.H.: Teichm[U+00A8]uller theory and applications to geometry, topology, and dynamics, vol. 1. Matrix Editions, Ithaca (2006). Teichm[U+00A8]uller theory, with contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, with forewords by William Thurston and Clifford Earle

[Ljubich1983] Ju, M.: Ljubich. Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dyn. Syst. 3 (3), 351--385 (1983)

[Pogorelov1973] Pogorelov, A.V.: Extrinsic Geometry of Convex Surfaces. American Mathematical Society, Providence (1973) (translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, vol. 35)

[Pommerenke1992] Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299. Springer, Berlin (1992)

[Rešetnjak1960] Rešetnjak, Ju.G.: Isothermal coordinates on manifolds of bounded curvature. I, II. Sibirsk. Mat. [U+02C7]Z. 1 , 88--116, 248--276 (1960)

[Reshetnyak1993] Reshetnyak, Yu.G.: Two-dimensional manifolds of bounded curvature. In: Geometry, IV, Encyclopedia Math. Sci., vol. 70 , pp. 3--163, 245--250. Springer, Berlin (1993)

[Series2006] Series, C.: Thurston's bending measure conjecture for once punctured torus groups. In: Spaces of Kleinian Groups, London Math. Soc. Lecture Note Ser., vol. 329, pp. 75--89. Cambridge Univ. Press, Cambridge (2006)

[Spivak1979] Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. II, 2nd edn. Publish or Perish Inc, Wilmington (1979)

[Thurston1998] Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. In: The Epstein Birthday Schrift, Geom. Topol. Monogr., vol. 1, pp. 511--549. Geom. Topol. Publ., Coventry (1998)

[The Sage Developers2016] The Sage Developers. Sage Mathematics Software (Version 6.10.beta7) (2016).

[Van Andel and Bradshaw2011] Van Andel, E., Bradshaw, R.: Riemann mapping [sage mathematics software package] (2011)

- 99
- × Ahlfors, L.V.: Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York (1973)
- × Aleksandrov, A.D.: Vnutrennyaya Geometriya Vypuklyh Poverhnoste[U+02D8][U+0131]. OGIZ, Moscow (1948)
- × Alexandroff, A.: Existence of a convex polyhedron and of a convex surface with a given metric. Rec. Math. [Mat. Sbornik] N.S. 11 (53), 15--65 (1942)
- × Alexandrov, A.D.: Convex polyhedra. Springer Monographs in Mathematics. Springer, Berlin (2005) (translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky, with comments and bibliography by V. A. Zalgaller and appendices by L. A. Shor and Yu. A. Volkov)
- × Bartholdi, L.: Personal communication (2015)
- × Bobenko, A.I., Izmestiev, I.: Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes. Ann. Inst. Fourier (Grenoble) 58 (2), 447--505 (2008)
- × Brolin, H.: Invariant sets under iteration of rational functions. Ark. Mat. 6 (103--144), 1965 (1965)
- × Connelly, R., Demaine, E.D., Rote, G.: Straightening polygonal arcs and convexifying polygonal cycles. Discrete Comput. Geom. 30 (2), 205--239 (2003) [U.S.-Hungarian Workshops on Discrete Geometry and Convexity (Budapest, 1999/Auburn, AL, 2000)]
- × Demaine, E.D., O'Rourke, J.: A survey of folding and unfolding in computational geometry. In: Combinatorial and Computational Geometry, Math. Sci. Res. Inst. Publ., vol. 52, pp. 167--211. Cambridge Univ. Press, Cambridge (2005)
- × Demaine, E.D., O'Rourke, J.: Geometric Folding Algorithms. Cambridge University Press, Cambridge (2007) (Linkages, origami, polyhedra)
- × DeMarco, L.: Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity. Math. Ann. 326 (1), 43--73 (2003)
- × Duren, P.L.: Univalent functions. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259. Springer, New York (1983)
- × Forn[U+00E6]ss, J.E., Sibony, N.: Complex dynamics in higher dimensions. In: Complex Potential Theory (Montreal, PQ, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 439, pp. 131--186. Kluwer Acad. Publ., Dordrecht (1994) (Notes partially written by Estela A. Gavosto)
- × Freire, A., Lopes, A., Ma~né, R.: An invariant measure for rational maps. Bol. Soc. Brasil. Mat. 14 (1), 45--62 (1983)
- × Hubbard, J., Papadopol, P.: Superattractive fixed points in $ { c}^n$. Indiana Univ. Math. J. 43 , 321--365 (1994)
- × Hubbard, J.H.: Teichm[U+00A8]uller theory and applications to geometry, topology, and dynamics, vol. 1. Matrix Editions, Ithaca (2006). Teichm[U+00A8]uller theory, with contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, with forewords by William Thurston and Clifford Earle
- × Ju, M.: Ljubich. Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dyn. Syst. 3 (3), 351--385 (1983)
- × Pogorelov, A.V.: Extrinsic Geometry of Convex Surfaces. American Mathematical Society, Providence (1973) (translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, vol. 35)
- × Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299. Springer, Berlin (1992)
- × Rešetnjak, Ju.G.: Isothermal coordinates on manifolds of bounded curvature. I, II. Sibirsk. Mat. [U+02C7]Z. 1 , 88--116, 248--276 (1960)
- × Reshetnyak, Yu.G.: Two-dimensional manifolds of bounded curvature. In: Geometry, IV, Encyclopedia Math. Sci., vol. 70 , pp. 3--163, 245--250. Springer, Berlin (1993)
- × Series, C.: Thurston's bending measure conjecture for once punctured torus groups. In: Spaces of Kleinian Groups, London Math. Soc. Lecture Note Ser., vol. 329, pp. 75--89. Cambridge Univ. Press, Cambridge (2006)
- × Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. II, 2nd edn. Publish or Perish Inc, Wilmington (1979)
- × Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. In: The Epstein Birthday Schrift, Geom. Topol. Monogr., vol. 1, pp. 511--549. Geom. Topol. Publ., Coventry (1998)
- × The Sage Developers. Sage Mathematics Software (Version 6.10.beta7) (2016).
- × Van Andel, E., Bradshaw, R.: Riemann mapping [sage mathematics software package] (2011)

Theorem 1.1 Theorem
1.2 Figure 1 Figure 2 Problem
1.3 Figure 3 Problem
1.4

2. Caps, Spirals, and Julia Sets
2.1. The Naive Cap

2.2. The Naive Cap Is Not Always Planar

3. Metrics and Curvature
Figure 4

2.3. Non-planar Example for Harmonic Measure
Figure 5

2.4. Polynomial Julia Sets
Example 2.1 Example
2.2 Figure 6 Example
2.3 Figure 7

3.1. Polyhedra and Cone Angles

3.2. More General Metrics of Non-negative Curvature

3.3. Parametrization of the Cap

3.4. Circumference and Curvature

4. Harmonic Measure and Holomorphic 1-Forms
Proposition 3.1 Lemma
3.2

4.1. Complex-Analytic Point of View

4.2. Harmonic Measure as Curvature

4.4. Harmonic Cap Boundary Parametrization

4.5. Metric Existence for General Measures

Example 4.1

4.3. The Harmonic Cap
Proposition 4.2