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# Evolution of multiple closed knotted curves in space

Miroslav Kolář Department of Mathematics, Faculty of Nuclear Sciences and
Physical Engineering Czech Technical University in Prague, Trojanova 13,
Prague, 12000, Czech Republic Daniel Ševčovič Department of Applied
Mathematics and Statistics, Faculty of Mathematics Physics and Informatics,
Comenius University, Mlynská dolina, 842 48, Bratislava, Slovakia.
Corresponding author<EMAIL_ADDRESS>
###### Abstract

We investigate a system of geometric evolution equations describing a
curvature and torsion driven motion of a family of 3D curves in the normal and
binormal directions. We explore the direct Lagrangian approach for treating
the geometric flow of such interacting curves. Using the abstract theory of
nonlinear analytic semi-flows, we are able to prove local existence,
uniqueness, and continuation of classical Hölder smooth solutions to the
governing system of non-linear parabolic equations modelling $n$ evolving
curves with mutual nonlocal interactions. We present several computational
studies of the flow that combine the normal or binormal velocity and
considering nonlocal interaction.

###### keywords:

Curvature driven flow, binormal flow, nonlocal flow, Biot-Savart law,
interacting curves, analytic semi-flows, Hölder smooth solutions, flowing
finite volume method.

2010 MSC. Primary: 35K57, 35K65, 65N40, 65M08; Secondary: 53C80.

##  1 Introduction

In this work, we focus on the evolution of space curves involving
interactions. These one-dimensional structures, which form space curves, are
frequently encountered in various scientific and engineering challenges.
Connections to dislocation dynamics are discussed in [10, 11] along with
additional references. Historical research into the dynamics of vortex
structures and rings, which align with one-dimensional curves, was initiated
by Helmholtz [16]. The significance of vortex structures in aerospace
technology is highlighted in several foundational studies (refer to Thomson
[29], Da Rios [24], Betchov [6], Arms and Hama [3], and Bewley [7]). Vortex
structures are known to maintain stability over time. This stability is
evident in studies of tornadoes and descriptions of volcanic activities (see
Fukumoto _et al._ [13, 14], Hoz and Vega [18], Vega [30]). Specific
interactions between linear vortex structures, exhibiting dynamic behaviors
such as ’frog leaps’, are noted (refer to Mariani and Kontis [21]). For an
overview of vortex dynamics and further discussions on the evolution of closed
curves, please see our latest publication [9] by Beneš, Kolář, and Ševčovič.

The structure of the paper is as follows. Section 2 revisits the Lagrangian
framework for evolving curve families. It introduces a set of evolutionary
equations governing the dynamics of interacting curve systems, together with
recent findings on the existence and uniqueness of classical Hölder continuous
solutions. The proof technique employs the abstract theory of analytic semi-
flows in Banach spaces, due to Angenent [2, 1]. Section 3 concentrates on the
numerical discretization approach, utilizing the flowing finite-volume method
for spatial derivative discretization and the method of lines to address the
resulting ODE systems. Section 6 showcases examples of the dynamics of
interacting curves, with interactions shaped by the Biot-Savart nonlocal law,
and discusses the development of 3D evolving knotted curves.

##  2 Lagrangian description of evolving curves

We investigate the family of curves $\\{\Gamma_{t},t\geq 0\\}$ evolving in
space $\mathbb{R}^{3}$. We employ the Lagrangian description of curves, in
which the curve is described by a position vector
${\mathbf{X}}={\mathbf{X}}(t,u)$ for $t\geq 0$ and $u\in I$, where
$I=\mathbb{R}/\mathbb{Z}\simeq S^{1}$ is the unit circle. The curve
$\Gamma_{t}$ is then parameterized by $\Gamma_{t}=\\{{\mathbf{X}}(t,u),u\in
I\\}$. The unit tangent vector ${\mathbf{T}}$ to $\Gamma_{t}$ is defined as
${\mathbf{T}}=\partial_{s}{\mathbf{X}}$, where $s$ is the unit arc-length
parametrization defined by $ds=gdu$ where $g=|\partial_{u}{\mathbf{X}}|$ is
the relative local length of the curve. Here, $|\cdot|$ denotes the Euclidean
norm. The curvature $\kappa$ of a curve $\Gamma_{t}$ is defined as
$\kappa=|\partial_{s}{\mathbf{X}}\times\partial^{2}_{s}{\mathbf{X}}|=|\partial_{s}^{2}{\mathbf{X}}|$.
If $\kappa>0$, we can define the Frenet frame along the curve $\Gamma_{t}$
with unit normal ${\mathbf{N}}={\kappa}^{-1}\partial_{s}^{2}{\mathbf{X}}$ and
binormal vectors ${\mathbf{B}}={\mathbf{T}}\times{\mathbf{N}}$, respectively.
We study a coupled system of evolutionary equations that describes the
evolution of closed 3D curves evolving in normal and binormal directions, and
the scalar quantity $\varrho$ calculated over the evolving family of curves.
More specifically, we focus on the analysis of the motion of a family of $n$
curves evolving in 3D and satisfying the system of geometric equations:

$\partial_{t}{\mathbf{X}}^{i}=a^{i}\partial^{2}_{s^{i}}{\mathbf{X}}^{i}+b^{i}(\partial_{s^{i}}{\mathbf{X}}^{i}\times\partial^{2}_{s^{i}}{\mathbf{X}}^{i})+{\mathbf{F}}^{i}+\alpha^{i}{\mathbf{T}}^{i},\qquad{\mathbf{X}}^{i}(\cdot,0)={\mathbf{X}^{i}_{0}}(\cdot),\quad
i=1,\ldots,n,$ (1)

which is subject to initial conditions at the origin $t=0$ representing
parametrization of the family of initial curves $\Gamma^{i}_{0},i=1,\ldots,n$.
Here $a^{i}=a^{i}({\mathbf{X}}^{i},{\mathbf{T}}^{i})\geq 0$, and
$b^{i}=b^{i}({\mathbf{X}}^{i},{\mathbf{T}}^{i})$ are bounded and smooth
functions of their arguments, ${\mathbf{T}}^{i}$ is the unit tangent vector to
the curve and $s^{i}$ is the unit arc-length parametrization of the curve
$\Gamma^{i}$. The source forcing term ${\mathbf{F}}^{i}$ is assumed to be a
smooth and bounded function. Here
${\mathbf{F}}^{i}={\mathbf{F}}^{i}({\mathbf{X}}^{i},{\mathbf{T}}^{i},\gamma^{i1},\ldots,\gamma^{in})$
is the forcing term and $\gamma^{ij}=\gamma^{ij}({\mathbf{X}}^{i},\Gamma^{j})$
may depend on the position vector ${\mathbf{X}}^{i}\in\mathbb{R}^{3}$ and the
entire curve $\Gamma^{j}$.

As an example of nonlocal source terms ${\mathbf{F}}^{i},i=1,\ldots,n,$ one
can consider a flow of $n=2$ interacting curves evolving in 3D according to
the geometric equations:

$\begin{split}\partial_{t}{\mathbf{X}}^{1}&=a^{1}\partial^{2}_{s^{1}}{\mathbf{X}}^{1}+b^{1}(\partial_{s^{1}}{\mathbf{X}}^{1}\times\partial^{2}_{s^{2}}{\mathbf{X}}^{1}+\gamma^{12}({\mathbf{X}}^{1},\Gamma^{2}),\\\
\partial_{t}{\mathbf{X}}^{2}&=a^{2}\partial^{2}_{s^{2}}{\mathbf{X}}^{2}+b^{2}(\partial_{s^{2}}{\mathbf{X}}^{2}\times\partial^{2}_{s^{2}}{\mathbf{X}}^{2})+\gamma^{21}({\mathbf{X}}^{2},\Gamma^{1}),\end{split}$

where the nonlocal source term

$\gamma^{ij}({\mathbf{X}},\Gamma^{j})=\oint_{\Gamma^{j}}\frac{({\mathbf{X}}-{\mathbf{X}}^{j}(s^{j}))\times{\mathbf{T}}^{j}(s^{j})}{|{\mathbf{X}}-{\mathbf{X}}^{j}(s^{j})|^{3}}ds^{j}$

represents the Biot-Savart force measuring the integrated influence of points
${\mathbf{X}}^{j}$ belonging to the second curve
$\Gamma^{j}=\\{{\mathbf{X}}^{j}(u),u\in[0,1]\\}$ at a given point
${\mathbf{X}}\in\mathbb{R}^{3}$.

The tangential velocity $\alpha^{i}$ that appears in geometric evolution (1)
has no impact on the shape of the evolving family of closed curves
$\Gamma^{i}_{t},t\geq 0$. This means that the curves $\Gamma^{i}_{t},t\geq 0,$
evolving according to (1) do not depend on a particular choice of the total
tangential velocity
$v_{T}^{i}={\mathbf{F}}^{i}\cdot{\mathbf{T}}^{i}+\alpha^{i}$. On the other
hand, the tangential velocity has a significant impact on the analysis of
evolution of curves from both the analytical and numerical points of view (see
e.g., Hou et al. [17], Kimura [20], Mikula and Ševčovič [26, 22, 23], Yazaki
and Ševčovič [27]. Barrett _et al._ [4, 5], Elliott and Fritz [12],
investigated the gradient and elastic flows for closed and open curves in
$\mathbb{R}^{d},d\geq 2$. They constructed a numerical approximation scheme
using a suitable tangential redistribution. Kessler _et al._ [19] and Strain
[28] illustrated the role of a suitably chosen tangential velocity in
numerical simulation of two-dimensional snowflake growth and unstable
solidification models. Garcke _et al._ [15] applied the uniform tangential
redistribution in the theoretical proof of the non-linear stability of
stationary solutions for curvature driven flow with triple junction in the
plane.

It is known that the relative local length $g^{i}/L^{i},$ is constant with
respect to time $t$, i.e.,
$\frac{g^{i}(u,t)}{L(\Gamma^{i}_{t})}=\frac{g^{i}(u,0)}{L(\Gamma^{i}_{0})},\quad
u\in I,t\geq 0$ provided that the total tangential velocity $v^{i}_{T}$
satisfies
$\partial_{s^{i}}v^{i}_{T}=\kappa^{i}v^{i}_{N}-\frac{1}{L^{i}}\int_{\Gamma^{i}}\kappa
v^{i}_{N}ds^{i}$ (see, e.g., [17], [20], [26], [22], [23]). Another suitable
choice of the total tangential velocity $v^{i}_{T}$ is the so-called
asymptotically uniform tangential velocity proposed and analyzed by Mikula and
Ševčovič in [22, 23]. It satisfies
$\lim_{t\to\infty}\frac{g^{i}(u,t)}{L(\Gamma^{i}_{t})}=1$ uniformly with
respect to $u\in[0,1]$. This means that the redistribution becomes
asymptotically uniform.

In [9], Beneš, Kolář and Ševčovič generalized methodology and techniques of
proofs of the local existence, uniqueness and continuation of solutions from
our previous paper [8] to the case of combined motion of closed space curves
evolving in normal and binormal direction and taking into account mutual
nonlocal interactions. We proved the result on existence and uniqueness of
classical solutions for a system on $n$ evolving curves in $\mathbb{R}^{3}$
with mutual nonlocal interactions including, in particular, the vortex
dynamics evolved in the normal and binormal directions and external force of
the Biot-Savart type, or evolution of interacting dislocation loops.

In the rest of this section we state the result on the existence and
uniqueness of classical Hölder smooth solutions to the system of governing
equations (1). The method of the proof is based on the abstract theory of
analytic semi-flows and the theory of maximal regularity in Banach spaces due
to Angenent [2, 1]. First, we introduce the function space setting. By
$h^{k+\varepsilon}(S^{1})$ we denote the so-called little Hölder space, i.e.
the Banach space which is the closure of $C^{\infty}$ smooth functions in the
norm Banach space of $C^{k}$ smooth functions defined on the periodic domain
$S^{1}$, and such that the $k$-th derivative is $\varepsilon$-Hölder smooth.
Here $0<\varepsilon<1$ and $k$ is a non-negative integer. The norm is given as
a sum of the $C^{k}$ norm and the Hölder semi-norm of the $k$-th derivative.
Next, we introduce the scale of Banach spaces of Hölder continuous functions
defined in the periodic domain $S^{1}$:

$E_{k}=(h^{2k+\varepsilon}(S^{1}))^{3},\quad{\mathcal{E}}_{k}=\underbrace{E_{k}\times\ldots\times
E_{k}}_{n-times},\quad k=0,\ 1/2,\ 1$

We assume that the functions $a^{i}>0,b^{i}$, and ${\mathbf{F}}^{i}$ are
sufficiently smooth and globally Lipschitz continuous functions (see [9,
assumtions (H)] for details). Moreover, we $\alpha^{i},i=1,\ldots,n$, is the
tangential velocity that preserves the relative local length. Assume that the
parametrization ${\mathbf{X}}_{0}\equiv({\mathbf{X}}^{i}_{0})_{i=1}^{n},$ of
initial curves $\Gamma^{i}_{0}$ belongs to the Hölder space
${\mathcal{E}}_{1}$, and it is a uniform parametrization, that is,
$|\partial_{u}{\mathbf{X}}^{i}_{0}(u)|=L(\Gamma^{i}_{0})>0$ for all $u\in I$
and $i=1,\ldots,n$. With regard to [9, Theorem 4.1], there exists $T>0$ and
the unique family of curves $\\{\Gamma^{i}_{t},t\in[0,T]\\},i=1,\ldots,n$,
evolving in 3D according to the system of nonlinear nonlocal geometric
equations (1). Their parametrization satisfies
${\mathbf{X}}=({\mathbf{X}}^{i})_{i=1}^{n}\in C([0,T],{\mathcal{E}}_{1})\cap
C^{1}([0,T],{\mathcal{E}}_{0})$, and ${\mathbf{X}}(\cdot,0)={\mathbf{X}}_{0}$.
If the maximal time of existence $T_{max}<\infty$ is finite then $\lim_{t\to
T_{max}}\max_{i,\Gamma^{i}_{t}}|\kappa^{i}(\cdot,t)|=\infty$.

##  3 Flowing finite volumes numerical discretization scheme

Figure 1: Discretization of a segment of a 3D curve by the method of flowing
finite volumes.

In this section, we present a numerical discretization scheme for solving the
system of equations (1) enhanced by the tangential velocity $\alpha^{i}$. Our
discretization scheme is based on the method of lines with spatial
discretization obtained by means of the finite-volume method. For simplicity,
we consider one evolving curve $\Gamma$ (omitting the curve index $i$) and
rewrite the abstract form of (1) in terms of the principal parts of its
velocity.

$\partial_{t}{\mathbf{X}}=a\partial_{s}^{2}{\mathbf{X}}+b(\partial_{s}{\mathbf{X}}\times\partial_{s}^{2}{\mathbf{X}})+{\mathbf{F}}+\alpha{\mathbf{T}}.$
(2)

We consider $M$ discrete nodes ${\mathbf{X}}_{k}={\mathbf{X}}(u_{k})$,
$k=0,1,2,\ldots,M$ along the curve $\Gamma$. The dual nodes are defined as
${\mathbf{X}}_{k\pm\frac{1}{2}}={\mathbf{X}}(u_{k}\pm h/2)$ (see Figure 1)
where $h=1/M$ and $({\mathbf{X}}_{k}+{\mathbf{X}}_{k+1})/2$ denote averages in
the segments that connect the discrete nodes near them and differs from
${\mathbf{X}}_{k\pm\frac{1}{2}}\in\Gamma$. The $k$-th segment
$\mathcal{S}_{k}$ of $\Gamma$ between the nodes ${\mathbf{X}}_{k-1}$ and
${\mathbf{X}}_{k}$ represents the finite volume. The integration of equation
(2) in the segment of $\Gamma$ between the nodes
${\mathbf{X}}_{k+\frac{1}{2}}$ and ${\mathbf{X}}_{k-\frac{1}{2}}$ yields

$\begin{split}\int_{u_{k-\frac{1}{2}}}^{u_{k+\frac{1}{2}}}\partial_{t}{\mathbf{X}}|\partial_{u}{\mathbf{X}}|du=&\int_{u_{k-\frac{1}{2}}}^{u_{k+\frac{1}{2}}}a\frac{\partial}{\partial_{u}}\left(\frac{\partial_{u}{\mathbf{X}}}{|\partial_{u}{\mathbf{X}}|}\right)du+\int_{u_{k-\frac{1}{2}}}^{u_{k+\frac{1}{2}}}b(\partial_{s}{\mathbf{X}}\times\partial_{s}^{2}{\mathbf{X}})|\partial_{u}{\mathbf{X}}|du\\\
&+\int_{u_{k-\frac{1}{2}}}^{u_{k+\frac{1}{2}}}{\mathbf{F}}|\partial_{u}{\mathbf{X}}|du+\int_{u_{k-\frac{1}{2}}}^{u_{k+\frac{1}{2}}}\alpha\partial_{u}{\mathbf{X}}du.\end{split}$
(3)

Let us denote $d_{k}=|{\mathbf{X}}_{k}-{\mathbf{X}}_{k-1}|$ for
$k=1,2,\ldots,M,M+1$, where ${\mathbf{X}}_{0}={\mathbf{X}}_{M}$ and
${\mathbf{X}}_{1}={\mathbf{X}}_{M+1}$ for closed curve $\Gamma$ and we
approximate the integral expressions in (3) by means of the finite volume
method along $\Gamma$ as follows:

$\begin{split}&\int_{u_{k-\frac{1}{2}}}^{u_{k+\frac{1}{2}}}\partial_{t}{\mathbf{X}}|\partial_{u}{\mathbf{X}}|du\approx\frac{d{\mathbf{X}}_{k}}{dt}\frac{d_{k+1}+d_{k}}{2},\\\
&\int_{u_{k-\frac{1}{2}}}^{u_{k+\frac{1}{2}}}a\partial_{u}\left(\frac{\partial_{u}{\mathbf{X}}}{|\partial_{u}{\mathbf{X}}|}\right)du\approx
a_{k}\left(\frac{{\mathbf{X}}_{k+1}-{\mathbf{X}}_{k}}{d_{k+1}}-\frac{{\mathbf{X}}_{k}-{\mathbf{X}}_{k-1}}{d_{k}}\right),\\\
&\int_{u_{k-\frac{1}{2}}}^{u_{k+\frac{1}{2}}}b(\partial_{s}{\mathbf{X}}\times\partial_{s}^{2}{\mathbf{X}})|\partial_{u}{\mathbf{X}}|du\approx
b_{k}\frac{d_{k+1}+d_{k}}{2}\kappa_{k}({\mathbf{T}}_{k}\times{\mathbf{N}}_{k}),\\\
&\int_{u_{k-\frac{1}{2}}}^{u_{k+\frac{1}{2}}}{\mathbf{F}}|\partial_{u}{\mathbf{X}}|du\approx{\mathbf{F}}_{k}\frac{d_{k+1}+d_{k}}{2},\qquad\int_{u_{k-\frac{1}{2}}}^{u_{k+\frac{1}{2}}}\alpha\partial_{u}{\mathbf{X}}du\approx\alpha_{k}\frac{{\mathbf{X}}_{k+1}-{\mathbf{X}}_{k-1}}{2}.\end{split}$

The approximation of the nonnegative curvature $\kappa$, tangent vector
${\mathbf{T}}$ and the normal vector ${\mathbf{N}}$,
$\kappa{\mathbf{N}}=\partial_{s}{\mathbf{T}}$ reads as follows:

$\begin{split}\kappa_{k}&\approx\left|\frac{2}{d_{k}+d_{k+1}}\left(\frac{{\mathbf{X}}_{k+1}-{\mathbf{X}}_{k}}{d_{k+1}}-\frac{{\mathbf{X}}_{k}-{\mathbf{X}}_{k-1}}{d_{k}}\right)\right|,\\\
{\mathbf{T}}_{k}&\approx\frac{{\mathbf{X}}_{k+1}-{\mathbf{X}}_{k-1}}{d_{k+1}+d_{k}},\quad{\mathbf{N}}_{k}\approx\kappa_{k}^{-1}\frac{2}{d_{k}+d_{k+1}}\left(\frac{{\mathbf{X}}_{k+1}-{\mathbf{X}}_{k}}{d_{k+1}}-\frac{{\mathbf{X}}_{k}-{\mathbf{X}}_{k-1}}{d_{k}}\right).\end{split}$

To discretize the governing equations, we assume that
$\partial_{t}{\mathbf{X}},\partial_{u}{\mathbf{X}},{\mathbf{F}}$ and $\alpha$
are constant over the finite volume between the nodes
${\mathbf{X}}_{k+\frac{1}{2}}$ and ${\mathbf{X}}_{k-\frac{1}{2}}$, taking
values
$\partial_{t}{\mathbf{X}}_{k},\partial_{u}{\mathbf{X}}_{k},{\mathbf{F}}_{k}$
and $\alpha_{k}$, respectively. In approximation ${\mathbf{F}}_{k}$ of the
non-local vector-valued function ${\mathbf{F}}$, we assume that the curve
$\Gamma$ entering the definition of ${\mathbf{F}}$ is approximated by the
polygonal curve with vertices
$({\mathbf{X}}_{0},{\mathbf{X}}_{1},\ldots,{\mathbf{X}}_{M})$. To find the
approximation $\alpha_{k}$ of the tangential velocity, we apply a simple
trapezoidal integration formula. The values $\alpha_{0}=\alpha_{M}$ are chosen
so that $\sum_{j=1}^{M}\alpha_{j}d_{j}=0$. In summary, the semi-discrete
scheme for solving (2) can be written as follows.

$\displaystyle\frac{d{\mathbf{X}}_{k}}{dt}\frac{d_{k+1}+d_{k}}{2}$
$\displaystyle=$ $\displaystyle
a_{k}\left(\frac{{\mathbf{X}}_{k+1}-{\mathbf{X}}_{k}}{d_{k+1}}-\frac{{\mathbf{X}}_{k}-{\mathbf{X}}_{k-1}}{d_{k}}\right)+b_{k}\frac{d_{k+1}+d_{k}}{2}\kappa_{k}({\mathbf{T}}_{k}\times{\mathbf{N}}_{k})$
$\displaystyle+{\mathbf{F}}_{k}\frac{d_{k+1}+d_{k}}{2}+\alpha_{k}\frac{{\mathbf{X}}_{k+1}-{\mathbf{X}}_{k-1}}{2},$
$\displaystyle{\mathbf{X}}_{k}(0)$ $\displaystyle=$
$\displaystyle{\mathbf{X}}_{ini}(u_{k}),\quad\text{for}\ k=1,\ldots,M.$ (5)

Resulting system (3–5) of ODEs is solved numerically by means of the 4th-order
explicit Runge-Kutta-Merson scheme with automatic time stepping control and
the tolerance parameter $10^{-3}$ (see [25]). We chose the initial time step
as $4h^{2}$, where $h=1/M$ is the spatial mesh size.

##  4 Examples of evolution of linked Fourier curves under Biot-Savart
external force

As an example of a non-local source term ${\mathbf{F}}$ we consider the
external force corresponding to the Biot-Savart law. It represents the
integrated influence of all points belonging to the curve
$\Gamma=\\{{\mathbf{X}}(s),s\in[0,L(\Gamma)]\\}$ at a given point
${\mathbf{X}}\in\mathbb{R}^{3},{\mathbf{X}}\not\in\Gamma$. It is given as a
line integral:

${\mathbf{F}}({\mathbf{X}},\Gamma)=\int_{\Gamma}\frac{({\mathbf{X}}-{\mathbf{X}}(s))\times\partial_{s}{\mathbf{X}}(s)}{|{\mathbf{X}}-{\mathbf{X}}(s)|^{3}}ds.$
(6)

Let $\Gamma^{1}$ and $\Gamma^{2}$ be two non-intersecting closed curves in 3D.
The Biot-Savart force is connected with the Gauss linking number and the
integral $link(\Gamma^{1},\Gamma^{2})$ of $\Gamma^{1}$ and $\Gamma^{2}$ can be
defined as follows:

$\displaystyle
link(\Gamma^{1},\Gamma^{2})=\frac{1}{4\pi}\oint_{\Gamma^{1}}\oint_{\Gamma^{2}}\frac{det\left(\partial_{s_{1}}{\mathbf{X}}^{1}(s_{1}),\partial_{s_{2}}{\mathbf{X}}^{2}(s_{2}),{\mathbf{X}^{1}}(s_{1})-{\mathbf{X}^{2}}(s_{2})\right)}{|{\mathbf{X}^{1}}(s_{1})-{\mathbf{X}^{2}}(s_{2})|^{3}}ds_{1}ds_{2}$
$\displaystyle\hskip-14.22636pt=-\frac{1}{4\pi}\oint_{\Gamma^{1}}{\mathbf{F}}({\mathbf{X}}^{1}(s_{1}),\Gamma^{2})\cdot\partial_{s_{1}}{\mathbf{X}}^{1}(s_{1})ds_{1}=-\frac{1}{4\pi}\oint_{\Gamma^{2}}{\mathbf{F}}({\mathbf{X}}^{2}(s_{2}),\Gamma^{1})\cdot\partial_{s_{2}}{\mathbf{X}}^{2}(s_{2})ds_{2},$

where the closed curves $\Gamma^{1}$ and $\Gamma^{2}$ are parameterized by
${\mathbf{X}}^{1}(s_{1})$ and ${\mathbf{X}}^{2}(s_{2})$, respectively. The
linking number $link(\Gamma^{1},\Gamma^{2})$ belongs to $\mathbb{Z}$.

A Fourier curve is a closed curve in 3D that can be parameterized by a finite
Fourier series in the parameter $u\in[0,1]$. In Fig. 2 a) we show two linked
circles $\Gamma^{1}$ and $\Gamma^{2}$ with the linking number
$link(\Gamma^{1},\Gamma^{2})=-1$ for parameterization (7), and
$link(\Gamma^{1},\Gamma^{2})=1$ when parameterized by (8):

$\displaystyle\hskip-28.45274pt{\mathbf{X}}^{1}(u)=(\cos(2\pi u),\sin(2\pi
u),0),\ \ {\mathbf{X}}^{2}(u)=(1+\cos(2\pi u),0,\sin(2\pi u)),$ (7)
$\displaystyle\hskip-28.45274pt{\mathbf{X}}^{1}(u)=(-\sin(2\pi u),\cos(2\pi
u),0),{\mathbf{X}}^{2}(u)=(1+\cos(2\pi u),0,-\sin(2\pi u)).$ (8)

In Fig. 2 b) we present two linked circles, and Biot-Savart force vector field
induced by $\Gamma^{2}$ acting on points belonging to $\Gamma^{1}$.

a) b)

Figure 2: Two linked circles a) and the Biot-Savart force vector field induced
by $\Gamma^{2}$ acting on points of $\Gamma^{1}$, b).

a) b)

Figure 3: The Listing’s 8-knot curve linked with a circle a) and an ellipse b)

The explicit parametrization of the Listing’s 8-knot curve ${\mathbf{X}}^{1}$
is given by

${\mathbf{X}}^{1}(u)=(3\cos(4\pi u),2\sin(6\pi u+1/2),(\cos(10\pi
u+1/2)+\sin(6\pi u+1/2))/2).$ (9)

The Listing’s 8-knot curve parameterized by (9) is shown in Fig. 3 a) with a
linked-in circle (linking number 0) that is parameterized by:

${\mathbf{X}}^{2}(u)=(\cos(2\pi u),\ 0\ ,\sin(2\pi u)).$

The Listing’s 8-knot curve ${\mathbf{X}}^{1}$ with a linked-in ellipse
(linking number -2) that is parameterized by:

${\mathbf{X}}^{2}(u)=(\cos(2\pi u)-1.5,\ 0.5\ ,0.8\sin(2\pi u))$

is shown in Fig. 3 b).

In what follows, we present results of numerical approximation of solutions to
the coupled system of governing PDEs:

$\begin{split}\partial_{t}{\mathbf{X}}^{1}&=\partial^{2}_{s^{1}}{\mathbf{X}}^{1}+\delta{\mathbf{F}}({\mathbf{X}}^{1},\Gamma^{2}),\\\
\partial_{t}{\mathbf{X}}^{2}&=\partial^{2}_{s^{2}}{\mathbf{X}}^{2}+\delta{\mathbf{F}}({\mathbf{X}}^{2},\Gamma^{1}),\end{split}$

which is subject to initial conditions ${\mathbf{X}}^{1}(\cdot,0)$ and
${\mathbf{X}}^{2}(\cdot,0)$ at the origin $t=0$. As a forcing term, we
consider the Biot-Savart force ${\mathbf{F}}({\mathbf{X}}^{i},\Gamma^{j})$
scaled by the factor $\delta=0.1$.

In Fig. 4 we present the time evolution of two linked circles $\Gamma^{1}$ and
$\Gamma^{2}$ parameterized by (7) with the linking number
$link(\Gamma^{1},\Gamma^{2})=-1$. In Fig. 5 we present the time evolution of
two linked circles $\Gamma^{1}$ and $\Gamma^{2}$ parameterized by (8) with the
linking number $link(\Gamma^{1},\Gamma^{2})=1$. In Fig. 6 we present the time
evolution of the initial Listing’s 8-curve $\Gamma^{1}$ linked with a circle
$\Gamma^{2}$ parameterized by (4) shown in Fig. 3 a) with the linking number
$link(\Gamma^{1},\Gamma^{2})=0$. In Fig. 7 we present the time evolution of
the initial Listing’s 8-curve $\Gamma^{1}$ linked with an ellipse circle
$\Gamma^{2}$ shown in Fig. 3 b) with the linking number
$link(\Gamma^{1},\Gamma^{2})=-2$.

$t=0$ $t=0.031$ $t=0.062$

$t=0.093$ $t=0.124$ $t=0.146$

Figure 4: Evolution of the initial linked circles parameterized by (7).

$t=0$ $t=0.031$ $t=0.062$

$t=0.093$ $t=0.124$ $t=0.146$

Figure 5: Evolution of the initial linked circles parameterized by (8).

$t=0$ $t=0.077$

$t=0.166$ $t=0.255$

$t=0.344$ $t=0.433$

Figure 6: Evolution of the initial Listing’s 8-curve linked with a circle
(blue) (3) a) and the external force given by the regularized Biot-Savart
force given by (6).

$t=0$ $t=0.049$

$t=0.166$ $t=0.149$

$t=0.198$ $t=0.249$

Figure 7: Evolution of the initial Listing’s 8-curve linked with an ellipse
(blue) (3) b) and the external force given by the regularized Biot-Savart
force given by (6).

##  5 Conlusions

In this article, we investigated a set of geometric evolution equations that
describe the curvature-driven motion of a family of 3D curves along the normal
and binormal directions. An evolving family of curves can interact in either
local or non-local ways. In particular, we analyzed evolving pairs of closed
linked curves that form knots in 3D. We utilized the direct Lagrangian method
to solve the geometric flow of these interacting curves. We applied the
abstract theory of nonlinear analytic semi-flows to prove the local existence,
uniqueness, and continuation of classical Hölder smooth solutions for the
system of nonlinear parabolic equations in question. Using the finite-volume
method, we proposed an effective numerical method for solving the governing
system of parabolic partial differential equations. Finally, we provided
multiple computational studies on the flow of linked curves.

## Acknowledgments

The first author received support from the project 21-09093S of the Czech
Science Foundation. The third author received support from the Slovak Research
and Development Agency project APVV-20-0311.

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# On the convolution of convex $2$-gons

Martin Chuaqui  Martin Chuaqui, Facultad de Matemáticas, Pontificia
Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
<EMAIL_ADDRESS>,  Rodrigo Hernández  Rodrigo Hernández, Facultad de
Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña
del Mar, Chile<EMAIL_ADDRESS>,  Adrián Llinares  Adrián Llinares,
Department of Mathematics and Mathematical Statistics, Umeå University,
SE-90187 Umeå, Sweden<EMAIL_ADDRESS>and  Alejandro Mas  Alejandro
Mas, Departamento de Análisis Matemático, Universidad de Valencia, 46100
Burjassot, Spain<EMAIL_ADDRESS>
###### Abstract.

We study the convolution of functions of the form

$f_{\alpha}(z):=\dfrac{\left(\frac{1+z}{1-z}\right)^{\alpha}-1}{2\alpha},$

which map the open unit disk of the complex plane onto polygons of 2 edges
when $\alpha\in(0,1)$. We extend results by Cima by studying limits of
convolutions of finitely many $f_{\alpha}$ and by considering the convolution
of arbitrary unbounded convex mappings. The analysis for the latter is based
on the notion of angle at infinity, which provides an estimate for the growth
at infinity and determines whether the convolution is bounded or not. A
generalization to an arbitrary number of factors shows that the convolution of
$n$ randomly chosen unbounded convex mappings has a probability of $1/n!$ of
remaining unbounded. We also extend Cima’s analysis on the coefficients of the
functions $f_{\alpha}$ by providing precise asymptotic behavior for all
$\alpha$.

###### Key words and phrases:

Convolution, convex mappings, 2-gons

###### 2020 Mathematics Subject Classification:

30C45

##  1\. Introduction

As usual, let $\mathbb{D}$ and $\mathbb{T}$ be the open unit disk and the unit
circle in the complex plane, respectively. If $f$ and $g$ are two holomorphic
functions in $\mathbb{D}$, we define their _convolution_ as

$(f*g)(z):=\dfrac{1}{2\pi
i}\int_{|\zeta|=\frac{1+|z|}{2}}f(\zeta)g\left(z\zeta^{-1}\right)\,\dfrac{d\zeta}{\zeta}.$

Moreover, it can be checked that

$(f*g)(z)=\sum_{n=0}^{\infty}a_{n}b_{n}z^{n},$

where $\\{a_{n}\\}_{n\geq 0}$ and $\\{b_{n}\\}_{n\geq 0}$ denote the sequences
of Taylor coefficients of $f$ and $g$, so the convolution is also known as
_Hadamard product_ in the bibliography.

The convolution of functions satisfying certain geometric properties has been
extensively studied (see for instance [8] or Chapter 14 of [4] for some
general references). For this paper, it is especially relevant the following
result, which was originally conjectured by Pólya and Schoenberg [5, Appendix
II] and was proved by Ruscheweyh and Sheil-Small [7].

######  Theorem A (Ruscheweyh, Sheil-Small).

Let $f$ and $g$ belong to the class $\mathcal{C}$ of univalent functions in
$\mathbb{D}$ whose ranges are convex sets. Then, $f*g\in\mathcal{C}$ as well.

In this paper we will focus on the mappings

$f_{\alpha}(z):=\dfrac{\left(\frac{1+z}{1-z}\right)^{\alpha}-1}{2\alpha},\quad
0<\alpha<1.$

The mappings $f_{\alpha}$ are univalent and $f_{\alpha}(\mathbb{D})$
corresponds to the convex $2$-gon with vertex at $-\frac{1}{2\alpha}$ and
edges with directions $e^{\frac{\alpha\pi}{2}i}$ and
$e^{-\frac{\alpha\pi}{2}i}$.

If we allow $\alpha$ to assume the value 1, it is clear that

$f_{1}(z)=\dfrac{z}{1-z}=\sum_{k=1}^{\infty}z^{k},$

so $f_{1}$ turns out to be the identity element for the convolution in the
subspace of analytic functions vanishing at the origin. On the other hand,
taking limits when $\alpha\rightarrow 0$ we get that

$f_{0}(z):=\lim_{\alpha\rightarrow
0^{+}}f_{\alpha}(z)=\dfrac{1}{2}\log\left(\frac{1+z}{1-z}\right),$

which is again one-to-one and
$f_{0}(\mathbb{D})=\\{z\in\mathbb{C}:|\operatorname{Im}z|<\frac{\pi}{4}\\}$.

Much of the motivation of the present paper stems from Cima’s paper [2]
studying the convolution $f_{\alpha}*f_{\beta}$. We will collect several of
the results therein and bring attention to his comment on the technical
difficulties faced in the analysis. The paper is organized as follows. In
Section 2, we extend the results in [2] by giving precise asymptotic behavior
for the coefficients of the function $f_{\alpha}$. We determine when
$f_{\alpha}*f_{\beta}$ is bounded or not for all choices of the parameters,
and study all possible limits of the convolution of finitely many mappings
$f_{\alpha}$. In Section 3, we recall the notion of angle at infinity and use
it to establish bounds for the growth of general unbounded convex mappings.
From this, we find the probability that the convolution of finitely many
randomly chosen unbounded convex mappings is bounded.

### Acknowledgements

We would like to thank I. Efraimidis for his careful reading and helpful
comments.

The work of the third author was funded by Grant 275113 of the Research
Council of Norway through the Alain Bensoussan Fellowship Programme from
ERCIM, and his current research is funded by the postdoctoral scholarship
JCK22-0052 from The Kempe Foundations. The fourth author is partially
supported by Grant PID2022-136619NB-I00 from Ministerio de Ciencia e
Innovación. The third and fourth authors are partially supported by Grant
PID2019-106870GB-I00 from Ministerio de Ciencia e Innovación.

Part of this research was conducted during the stay of A. Llinares and A. Mas
at the Pontificia Universidad Católica de Chile and the Universidad Adolfo
Ibáñez. These visits as well as the work of the first two authors were
partially funded by Fondecyt Grants 1190380 and 1190756.

##  2\. Growth of convolutions of 2-gons

###  2.1. Asymptotic estimates

For any $\alpha\in(0,1)$, let $\\{g_{n}(\alpha)\\}_{n\geq 0}$ be the sequence
of Taylor coefficients of $f_{\alpha}$. From the power series of
$F_{\alpha}(z):=\left(\frac{1+z}{1-z}\right)^{\alpha}$, it can be seen that

$g_{n}(\alpha)=\dfrac{1}{2\alpha}\sum_{k=1}^{n}2^{k}\binom{\alpha}{k}\binom{n-1}{n-k},\quad
n\geq 2,$

where $\binom{a}{b}=\frac{\Gamma(a+1)}{\Gamma(b+1)\Gamma(a-b+1)}$ and $\Gamma$
denotes the Euler Gamma function. However, we have found more useful to follow
Cima’s approach [2] of treating $f_{\alpha}$ as the solution of the
differential equation

$f_{\alpha}^{\prime}(z)=\dfrac{2\alpha f_{\alpha}(z)+1}{1-z^{2}},\quad\forall
z\in\mathbb{D},$

and $f_{\alpha}(0)=0$. Since $g_{0}(\alpha)=0$ and $g_{1}(\alpha)=1$, the
recursion relation

(1) $g_{n+2}(\alpha)=\dfrac{2\alpha
g_{n+1}(\alpha)+ng_{n}(\alpha)}{n+2},\quad\forall n\geq 0,$

can be derived and from it we deduce the following lemma.

######  Lemma B (Cima [2]).

The sequence $\\{g_{n}(\alpha)\\}_{n\geq 0}$ has the following properties:

(i) For every $n\geq 1$, $g_{n}(\alpha)$ is a polynomial of degree $n-1$ with
non-negative coefficients. In particular, $0<g_{n}(\alpha)<1$ for all
$\alpha\in(0,1)$ and $n\geq 2$.

(ii) The subsequences $\\{g_{2m}(\alpha)\\}_{m\geq 1}$ and
$\\{g_{2m-1}(\alpha)\\}_{m\geq 1}$ are decreasing. If
$\alpha\in\left(\frac{1}{2},1\right)$, then the chain of inequalities

$g_{n}(\alpha)>g_{n+1}(\alpha)>\dfrac{n}{n+2-2\alpha}g_{n}(\alpha)$

holds for every $n\geq 2$.

(iii) For every $\alpha\in(0,1)$, there exists a constant $C=C(\alpha)$ such
that

(2) $g_{n}(\alpha)\leq\dfrac{C}{n^{1-\alpha}},\quad\forall n\geq 0.$

Observe that, in particular, this last inequality yields that
$f_{\alpha}*f_{\beta}$ is bounded if $\alpha+\beta<1$, as was explicitly
mentioned in [2, Theorem 2]. Our aim is to complement this theorem by
considering the case $\alpha+\beta\geq 1$, showing in fact that a lower bound
of the same type as in (2) is also true.

######  Theorem 2.1.

The sequence $\\{n^{1-\alpha}g_{n}(\alpha)\\}_{n\geq 1}$ is uniformly
convergent in intervals of the form $(\alpha_{0},1)$, $0<\alpha_{0}<1$.

###### Proof.

To simplify the notation, we let $G_{n}(\alpha):=n^{1-\alpha}g_{n}(\alpha)$,
$n\geq 1$. It follows from (1) that

$G_{n+2}(\alpha)-G_{n}(\alpha)=\dfrac{2\alpha
g_{n+1}(\alpha)+n\big{(}1-(1+2/n)^{\alpha}\big{)}g_{n}(\alpha)}{(n+2)^{\alpha}}.$

Since $\alpha\in(0,1)$, we have that

$-\dfrac{2\alpha}{n}+\dfrac{2\alpha(1-\alpha)}{n^{2}}\geq
1-\left(1+\dfrac{2}{n}\right)^{\alpha}\geq-\dfrac{2\alpha}{n},\quad\forall
n\geq 3$

and then

$\dfrac{2\alpha\big{(}g_{n+1}(\alpha)-g_{n}(\alpha)\big{)}}{(n+2)^{\alpha}}+\dfrac{2\alpha(1-\alpha)g_{n}(\alpha)}{n(n+2)^{\alpha}}\geq
G_{n+2}(\alpha)-G_{n}(\alpha)\geq\dfrac{2\alpha\big{(}g_{n+1}(\alpha)-g_{n}(\alpha)\big{)}}{(n+2)^{\alpha}}.$

Assume first that $\alpha\geq\frac{1}{2}$. From Lemma B, we see that

(3)
$|g_{n+1}(\alpha)-g_{n}(\alpha)|=g_{n}(\alpha)-g_{n+1}(\alpha)<\dfrac{2(1-\alpha)}{n+2(1-\alpha)}g_{n}(\alpha).$

Since $g_{k}(\alpha)$ is an increasing function of $\alpha$, then
$0<g_{k}(\alpha)<1$. Consequently, if $m>j$ we have that

$\displaystyle|G_{2m}(\alpha)-G_{2j}(\alpha)|$ $\displaystyle\leq$
$\displaystyle\sum_{k=j}^{m-1}|G_{2k+2}(\alpha)-G_{2k}(\alpha)|$
$\displaystyle\leq$
$\displaystyle\sum_{k=j}^{m}\left(\dfrac{2\alpha|g_{2k+1}(\alpha)-g_{2k}(\alpha)|}{(2k+2)^{\alpha}}+\dfrac{2\alpha(1-\alpha)g_{2k}(\alpha)}{2k(2k+2)^{\alpha}}\right)$
$\displaystyle\leq$
$\displaystyle\alpha(1-\alpha)2^{1-\alpha}\sum_{k=j}^{m}\left(\dfrac{1}{\big{(}k+1-\alpha\big{)}(k+1)^{\alpha}}+\dfrac{1}{k(k+1)^{\alpha}}\right)$
$\displaystyle\leq$
$\displaystyle\sum_{k=j}^{m}\left(\dfrac{1}{(2k+1)\sqrt{k+1}}+\dfrac{1}{2k\sqrt{k+1}}\right),$

and hence $\\{G_{2m}(\alpha)\\}_{m\geq 1}$ is Cauchy in $m$ uniformly in
$\alpha$, and is therefore uniformly convergent. The same analysis applies to
$\\{G_{2m+1}(\alpha)\\}_{m\geq 0}$, and we will show that both limits
coincide. Indeed, Lemma B yields that

$G_{2m}(\alpha)\geq\left(\dfrac{2m}{2m+1}\right)^{1-\alpha}G_{2m+1}(\alpha)\geq\dfrac{m}{m+1-\alpha}G_{2m}(\alpha),\quad\forall
m\geq 1,$

and therefore $\\{G_{n}(\alpha)\\}_{n\geq 1}$ is uniformly convergent in
$\left(\frac{1}{2},1\right)$.

Now, assume that $0<\alpha<\frac{1}{2}$. Since
$2\alpha\big{(}g_{n+1}(\alpha)-g_{n}(\alpha)\big{)}$, $n\geq 2$, corresponds
to the $(n+1)$-th coefficient of $H_{\alpha}(z):=(1-z)F_{\alpha}(z)$, and
since

$H_{\alpha}(z)=(1+z)^{\alpha}(1-z)^{1-\alpha}=H_{1-\alpha}(-z),$

we see from (3) and the estimate $g_{n}(1-\alpha)=O(n^{-\alpha})$ that

$|g_{n+1}(\alpha)-g_{n}(\alpha)|=\dfrac{1-\alpha}{\alpha}|g_{n+1}(1-\alpha)-g_{n}(1-\alpha)|=O\left(\dfrac{1}{n^{1+\alpha}}\right).$

Hence again $\\{G_{2m}(\alpha)\\}_{m\geq 1}$ and
$\\{G_{2m+1}(\alpha)\\}_{m\geq 0}$ are uniformly convergent on the intervals
$\left(\alpha_{0},\frac{1}{2}\right]$, $0<\alpha_{0}<\frac{1}{2}$. Finally,
both limits must coincide because

$|G_{n+1}(\alpha)-G_{n}(\alpha)|\leq(n+1)^{1-\alpha}\left(\dfrac{2(1-\alpha)}{n+2\alpha}+1-\left(\dfrac{n}{n+1}\right)^{1-\alpha}\right)=O\left(\dfrac{1}{n^{\alpha}}\right).$

∎

Note that the proof provides additional qualitative information about the
limit $L(\alpha)=\lim_{n\rightarrow\infty}G_{n}(\alpha)$ as a function of
$\alpha$. This function must be continuous on $(0,1)$ and it is not difficult
to see that it must be strictly positive with $|1-L(\alpha)|=O(1-\alpha)$ when
$\alpha\rightarrow 1^{-}$. The following theorem states this more precisely.

######  Theorem 2.2.

For every $0<\alpha<1$, we have that

(4) $L(\alpha)=\dfrac{2^{\alpha-1}}{\Gamma(\alpha+1)}.$

###### Proof.

On the one hand, we trivially have

$\lim_{r\rightarrow
1^{-}}(1-r)^{\alpha}f_{\alpha}(r)=\dfrac{2^{\alpha-1}}{\alpha}.$

On the other hand, observe that
$c_{n}(\alpha):=\frac{\Gamma(n+\alpha)}{\Gamma(\alpha)n!}$ is the $n$-th
Taylor coefficient of $(1-z)^{-\alpha}$, $\alpha>0$. The Euler-Gauss formula
for the Gamma function (see for instance [3, p. 255]) implies that

$\lim_{n\rightarrow\infty}\dfrac{g_{n}(\alpha)}{c_{n}(\alpha)}=\Gamma(\alpha)L(\alpha).$

Let $\varepsilon>0$ and $n_{0}$ be such that

$\Gamma(\alpha)L(\alpha)-\varepsilon\leq\dfrac{g_{n}(\alpha)}{c_{n}(\alpha)}\leq\Gamma(\alpha)L(\alpha)+\varepsilon,\quad\forall
n\geq n_{0}.$

Then

$\lim_{r\rightarrow 1^{-}}(1-r)^{\alpha}f_{\alpha}(r)=\lim_{r\rightarrow
1^{-}}(1-r)^{\alpha}\sum_{n=n_{0}}^{\infty}g_{n}(\alpha)r^{n}\in[\Gamma(\alpha)L(\alpha)-\varepsilon,\Gamma(\alpha)L(\alpha)+\varepsilon],$

and by letting $\varepsilon\rightarrow 0$ we find that

$\Gamma(\alpha)L(\alpha)=\dfrac{2^{\alpha-1}}{\alpha},$

which implies (4). ∎

######  Remark 2.3.

Observe that, $L(1)=1$ because $g_{n}(1)=1$ for all $n$. However, because
$\lim_{\alpha\rightarrow 0^{+}}L(\alpha)=\frac{1}{2}$, $L$ is only defined on
$(0,1]$.

Following a similar analysis as in the proof of Theorem 2.2 along with
Carathéodory theorem [6, Theorem 2.1] (the unbounded version of which can
apparently be attributed to Torhorst [9] as well), we can finally state an
extension of Theorem 2 in [2].

######  Theorem 2.4.

For $\alpha,\beta\in(0,1]$ the following hold:

(i) If $\alpha+\beta<1$, then $f_{\alpha}*f_{\beta}$ is bounded.

(ii) If $\alpha+\beta=1$, then

$\lim_{z\rightarrow
1}\dfrac{(f_{\alpha}*f_{\beta})(z)}{-\log(1-|z|)}=\dfrac{1}{2\Gamma(\alpha+1)\Gamma(\beta+1)}.$

(ii) If $\alpha+\beta>1$, then

$\lim_{z\rightarrow
1}(1-|z|)^{\alpha+\beta-1}(f_{\alpha}*f_{\beta})(z)=2^{\alpha+\beta-2}\dfrac{\Gamma(\alpha+\beta-1)}{\Gamma(\alpha+1)\Gamma(\beta+1)}.$

###  2.2. Arbitrary number of convolutions

In this section we study the convolution of an arbitrary number of 2-gons,
including the limiting case $f_{0}$. For
$\alpha\in\mathbb{D}\setminus\\{0\\}$, we define the function $f_{\alpha}$ as
the rotation of $f_{|\alpha|}$ by $e^{i\arg\alpha}$, i.e.,

$f_{\alpha}(z):=e^{-i\arg\alpha}f_{|\alpha|}\big{(}e^{i\arg\alpha}z\big{)},\quad
z\in\mathbb{D}.$

We are interested in the set of all possible limits of finite convolutions of
2-gons given by

$\mathcal{L}:=\big{\\{}f\>:\>f=\lim_{n\rightarrow\infty}f_{\alpha_{1}}*\ldots*f_{\alpha_{n}},\\{\alpha_{n}\\}_{n\geq
1}\subset\mathbb{D}\setminus\\{0\\}\big{\\}}\subset\mathcal{C}.$

According to the lemma below, we may assume that all of the elements of the
sequence $\\{\alpha_{n}\\}_{n\geq 1}$, except perhaps $\alpha_{1}$, are
positive.

######  Lemma 2.5.

$\mathcal{L}=\big{\\{}f\>:\>f=\lim_{n\rightarrow\infty}f_{\alpha_{1}}*\ldots*f_{\alpha_{n}},\alpha_{1}\in\mathbb{D}\setminus\\{0\\},\\{\alpha_{n}\\}_{n\geq
2}\subset(0,1)\big{\\}}.$

###### Proof.

Let $\\{\alpha_{n}\\}_{n\geq 1}\subset\mathbb{D}\setminus\\{0\\}$ and consider
the function $f\in\mathcal{H}(\mathbb{D})$ such that

$f_{\alpha_{1}}*\ldots*f_{\alpha_{n}}\rightarrow f$

uniformly for all compact subset $K$ of $\mathbb{D}$. Since
$g_{2}(\alpha)=\alpha$, we have that the sequence of partial products
$\left\\{\prod_{j=1}^{n}\alpha_{j}\right\\}_{n\geq 1}$ is convergent. Let $p$
be its limit. At this point, we split the proof by cases. (i) If $p=0$, we are
going to show that $f(z)=z$. Without loss of generality, we might assume that
$|\alpha_{n}|\geq\frac{1}{2}$ for all $n\geq 1$. Because of Lemma B, we have
that

$\prod_{j=1}^{n}|\alpha_{j}|\geq\prod_{j=1}^{n}\left|g_{k}(|\alpha_{j}|)\right|,\quad
n\geq 1,k\geq 2,$

and thus it is clear that $f(z)=z$, which is obviously the limit of
“convolutioning” $f_{\frac{1}{2}}$, with itself infinitely many times.

(ii) On the other hand, if $p\neq 0$ we have that the infinite product
converges. By continuity, we have that $\prod_{n\geq 1}|\alpha_{n}|=|p|$, and
therefore the product $\lambda:=\prod_{n\geq 1}e^{i\arg\alpha_{n}}$ is also
convergent. Observe that $|\lambda|=1$, and consider the sequence
$\beta_{1}:=\lambda|\alpha_{1}|$ and $\beta_{n}:=|\alpha_{n}|$, $n\geq 2$. We
are going to show that

(5) $f=\lim_{n\rightarrow\infty}f_{\beta_{1}}*\ldots*f_{\beta_{n}}.$

Indeed, observe that the $k$-th Taylor coefficient of $f$ is given by

$\lim_{n\rightarrow\infty}\prod_{j=1}^{n}g_{k}(|\alpha_{j}|)e^{i(k-1)\arg\alpha_{j}}=\lambda^{k-1}\lim_{n\rightarrow\infty}\prod_{j=1}^{n}g_{k}(|\alpha_{j}|),$

which is trivially equal to the $k$-th coefficient of
$\lim_{n\rightarrow\infty}f_{\beta_{1}}*\ldots*f_{\beta_{n}}$ because of the
definition of the sequence $\\{\beta_{n}\\}_{n\geq 1}$. Therefore, it is clear
that the identity (5) holds. ∎

It may seem from this proof that $\\{\alpha_{n}\\}_{n\geq 1}$ and
$\\{\beta_{n}\\}_{n\geq 1}$ can give rise to the same limit as long as

$\lim_{n\rightarrow\infty}\prod_{j=1}^{n}\alpha_{j}=\lim_{n\rightarrow\infty}\prod_{j=1}^{n}\beta_{j},$

but this is not true. For instance, if $\\{\alpha_{n}\\}_{n\geq
1}\subset(0,1)$ is a sequence such that its infinite product is convergent,
and consider the sequence $\\{\beta_{n}\\}_{n\geq 1}$ defined as

$\beta_{2m-1}=\beta_{2m}=\sqrt{\alpha_{m}},\quad\forall m\geq 1.$

A straightforward computation shows that the function
$g_{3}(\alpha)=1-\frac{2}{3}(1-\alpha^{2})$ is a _supermultiplicative_
function. In other words,

$g_{3}(\alpha\beta)>g_{3}(\alpha)g_{3}(\beta)$

for all $\alpha,\beta\in(0,1)$. Then, we have that

$\prod_{j=1}^{n}g_{3}(\alpha_{j})>\prod_{j=1}^{2n}g_{3}(\beta_{j}),\quad\forall
n\geq 1,$

and hence the convergence of the product $\prod_{j\geq 1}\beta_{j}$ yields
that

$\prod_{j=1}^{\infty}g_{3}(\alpha_{j})>\prod_{j=1}^{\infty}g_{3}(\beta_{j})>0,$

which suffices to see that

$\lim_{n\rightarrow\infty}f_{\alpha_{1}}*\ldots*f_{\alpha_{n}}\neq\lim_{n\rightarrow\infty}f_{\beta_{1}}*\ldots*f_{\beta_{n}}.$

The following result highlights the effects of having a vanishing Taylor
coefficient in $\mathcal{L}$.

######  Proposition 2.6.

If $f\in\mathcal{L}$ and $f^{(k_{0})}(0)=0$ for some $k_{0}\geq 2$, then
$f(z)\equiv z$.

###### Proof.

Let $\alpha_{1}\in\mathbb{D}\setminus\\{0\\}$ and $\\{\alpha_{n}\\}_{n\geq
2}\subset(0,1)$ be such that
$f=\lim_{n\rightarrow\infty}f_{\alpha_{1}}*\ldots*f_{\alpha_{n}}$. Then, the
condition $f^{(k_{0})}(0)=0$ implies that

$\lim_{n\rightarrow\infty}\prod_{j=2}^{n}g_{k_{0}}(\alpha_{j})=0,$

since $g_{k}(|\alpha_{1}|)>0$. Without loss of generality, assume that
$f^{(k_{0}-1)}(0)\neq 0$.

Two cases are possible:

(i) If there exists a subsequence $\\{\alpha_{n_{j}}\\}_{j\geq 1}$ such that
$\alpha_{n_{j}}\leq\frac{1}{2}$ for all $j$, then the monotonocity of the
polynomials $\\{g_{k}(\alpha)\\}_{k\geq 2}$ with respect to $\alpha$ implies
that

$\lim_{n\rightarrow\infty}\prod_{j=1}^{n}g_{k}(\alpha_{j})=0,\quad\forall
k\geq 2,$

and hence $f$ must be the identity.

(ii) If there exists $N\geq 2$ such that $\alpha_{n}>\frac{1}{2}$ for all
$n\geq N$, we will show that

$\lim_{n\rightarrow\infty}\prod_{j=N}^{n}\alpha_{j}=0,$

and then Lemma B yields that $f(z)\equiv z$.

In order to get a contradiction, assume that $\prod_{n\geq 1}\alpha_{n}$ is
convergent. In other words, that $k_{0}>2$. Then, the product $\prod_{n\geq
1}\frac{k_{0}-1}{k_{0}+1-2\alpha_{n}}$ is also convergent. However, Lemma B
implies that

$\prod_{j=N}^{n}g_{k_{0}}(\alpha_{j})>\left(\prod_{j=N}^{n}\frac{k_{0}-1}{k_{0}+1-2\alpha_{n}}\right)\left(\prod_{j=N}^{n}g_{k_{0}-1}(\alpha_{j})\right),$

so the product on the left-hand side is bounded from below by the
multiplication of two convergent and positive products, which is not possible.
Thus, the partial products of the sequence $\\{\alpha_{n}\\}_{n\geq 1}$
converge to zero, which means that $f$ is the identity. ∎

The following analogue of Theorem 2.4 can also be drawn.

######  Theorem 2.7.

Let $\\{\alpha_{n}\\}_{n\geq 1}\subset(0,1]$ and $B:=\sum_{n\geq
1}(1-\alpha_{n})\in(0,\infty]$. Consider the function
$f:=\lim_{n\rightarrow\infty}f_{\alpha_{1}}*\ldots*f_{\alpha_{n}}$. Then, we
have the following:

(i) If $B>1$, then $f$ is bounded.

(ii) If $B=1$, then

(6) $\lim_{z\rightarrow
1}\dfrac{f(z)}{-\log(1-|z|)}=\dfrac{1}{2}\prod_{n=1}^{\infty}\dfrac{1}{\Gamma(\alpha_{n}+1)}.$

(iii) If $B<1$, then

$\lim_{z\rightarrow
1}(1-|z|)^{1-B}f(z)=\dfrac{\Gamma\left(1-B\right)}{2^{B}}\prod_{n=1}^{\infty}\dfrac{1}{\Gamma(\alpha_{n}+1)}.$

###### Proof.

(i) If $B>1$, then there exists $N$ big enough such that the partial sum
$\sum_{n=1}^{N}(1-\alpha_{n})$ is greater than $1$ as well. Thus, we have that

$\|f\|_{\infty}\leq\|f_{\alpha_{1}}*\ldots*f_{\alpha_{N}}\|_{\infty}<\infty.$

(ii) Since the sum is convergent observe $\alpha_{n}<\frac{1}{2}$ for, at
most, a finite number of indexes. Because $|1-L(\alpha)|=O(1-\alpha)$ in
$\left(\frac{1}{2},0\right)$, note that, in particular, we have that the
product $\prod_{n\geq 1}L(\alpha_{n})$ is convergent. As a consequence of the
uniform convergence of $\\{G_{k}(\alpha)\\}_{k\geq 1}$, we have that there
exists $k_{0}\geq 0$ such that

$\prod_{n=1}^{\infty}L(\alpha_{n})-\varepsilon\leq
k\prod_{n=1}^{\infty}g_{k}(\alpha_{n})\leq\prod_{n=1}^{\infty}L(\alpha_{n})+\varepsilon,\quad\forall
k\geq k_{0},$

and therefore (6) holds.

(iii) It is similar to the previous case. ∎

Although finite convolutions of elements of $\\{f_{\alpha}\\}_{0\leq\alpha\leq
1}$ do not belong to $\mathcal{L}$, the latter theorem can be also applied for
this case. Nevertheless, these finite convolutions may be approximated by
functions in $\mathcal{L}$, as it is implicitly shown in the following
proposition.

######  Proposition 2.8.

The functions $f_{0}$ and $f_{1}$ do not belong to $\mathcal{L}$, but they
belong to the closure of $\mathcal{L}$. In particular, $\mathcal{L}$ is not
closed.

###### Proof.

Since the coefficients of $f_{\alpha}$, $\alpha\in\mathbb{D}$, have moduli
strictly less than $1$, it is clear that $f_{1}\not\in\mathcal{L}$. On the
other hand, $f_{0}$ does not belong to $\mathcal{L}$ because of Proposition
2.6.

Now, consider the quantities

$\alpha_{n,j}:=\left(1-\dfrac{1}{j+1}\right)^{2^{-n}},\quad n,j\geq 1,$

and the sequence of functions
$F_{j}:=\lim_{n\rightarrow\infty}f_{\alpha_{1,j}}*\ldots*f_{\alpha_{n,j}}$. It
is clear that $\\{F_{j}\\}_{j\geq 1}\subset\mathcal{L}$.

Observe that $g_{k}(\alpha_{n,j})\leq g_{k}(\alpha_{n,j+1})<1$ for all
$n,j\geq 1$ and $k\geq 2$, so the sequence of $k$-th Taylor coefficients of
the functions $\\{F_{j}\\}_{j\geq 1}$ is convergent. Hence, the limit
$F=\lim_{j\rightarrow\infty}F_{j}$ exists. It remains to show that $F\equiv
f_{1}$, but this a consequence of the extremal properties of $f_{1}$. Indeed,
it is a well-known fact that

$|a_{n}|\leq 1,\quad\forall n\geq 2$

if $f(z)=z+\sum_{n\geq 2}a_{n}z^{n}\in\mathcal{C}$, and that equality is
attained if and only if $f(z)\equiv\overline{\lambda}f_{1}(\lambda z)$ for
some unimodular constant $\lambda$. It is clear that $F\in\mathcal{C}$ and, by
the construction of $\\{F_{j}\\}_{j\geq 1}$, its second Taylor coefficient is
equal to 1. Therefore $F$ must coincide with $f_{1}$.

Once we know that $f_{1}\in\overline{\mathcal{L}}$, it is easy to see that
$f_{0}\in\overline{\mathcal{L}}$ (for example, it can be checked that
$G_{j}:=f_{1/j}*F_{j}\in\mathcal{L}$ converges to $f_{0}$ when
$j\rightarrow\infty$) and consequently we have finished the proof. ∎

Unfortunately, a complete characterization of $\overline{\mathcal{L}}$ seems
to be out of reach. However, we can provide the following necessary condition.

######  Proposition 2.9.

Let $f\in\overline{\mathcal{L}}$. Then, one of the following statements is
true:

(i) All of the Taylor coefficients of $f$ are different than $0$.

(ii) The function $f$ is an odd function and all the odd Taylor coefficients
are different than 0.

(iii) $f(z)\equiv z$.

###### Proof.

Take $f\in\overline{\mathcal{L}}$ different than the identity. Consider a
sequence of functions $\\{F_{j}\\}_{j\geq 1}\subset\mathcal{L}$ such that
$F\leavevmode\nobreak\ =\leavevmode\nobreak\ \lim_{j\rightarrow\infty}F_{j}$.
Since $f(z)\not\equiv z$, we might assume that $F_{j}(z)\not\equiv z$ for all
$j$. Let $\\{\alpha_{n,j}\\}_{n\geq 1}$ (all of them positive except,
possibly, the first one) such that $F_{j}\leavevmode\nobreak\
=\leavevmode\nobreak\
\lim_{n\rightarrow\infty}f_{\alpha_{1,j}}*\ldots*f_{\alpha_{n,j}}$. Note that
we have that

(7)
$g_{k}(|\alpha_{1,j}|)\prod_{n=2}^{\infty}g_{k}(\alpha_{n,j})\geq|\alpha_{1,j}|^{k-1}\prod_{n=2}^{\infty}\alpha_{n,j}^{k-1},\quad\forall
k\geq 2,\forall j\geq 1,$

since the elementary estimate $g_{k}(\alpha)\geq\alpha^{k-1}$ holds for every
$0<\alpha<1$ and $k\leavevmode\nobreak\ \geq\leavevmode\nobreak\ 2$. Moreover,
Lemma B implies that

(8) $|\alpha_{1,j}|\prod_{n=2}^{\infty}\alpha_{n,j}\geq
g_{2m}(|\alpha_{1,j}|)\prod_{n=2}^{\infty}g_{2m}(\alpha_{n,j})$

and

$g_{3}(|\alpha_{1,j}|)\prod_{n=2}^{\infty}g_{3}(\alpha_{n,j})\geq
g_{2m+1}(|\alpha_{1,j}|)\prod_{n=2}^{\infty}g_{2m+1}(\alpha_{n,j})$

for all $j,m\geq 1$. Consequently, there is no odd number $k$ such that

$\lim_{j\rightarrow\infty}g_{k}(|\alpha_{1,j}|)\prod_{n=2}^{\infty}g_{k}(\alpha_{n,j})=0$

because $f(z)\not\equiv z$. If $k$ is an even number such that the $k$-th
Taylor coefficient of $f$ is zero, it follows from (7) and (8) that $f$ must
be an odd function. ∎

##  3\. Estimates for unbounded convex sets

In this section we deduce estimates for general unbounded convex sets. We
recall that the angle at infinity of a convex set is the infimum of the angles
of all sectors containing that set. It follows from convexity that an
unbounded convex set contains a $2$-gon of this amplitude if it is positive.

######  Theorem 3.1.

Let $0<\alpha<1$. Let $g$ be a unbounded convex mapping with $g(1)=\infty$. If
$g(\mathbb{D})$ has an angle at infinity of $\pi\alpha$, then there exists
$\delta>0$ such that

(9) $|1-z|^{\alpha}|g(z)|\geq\delta$

when $z\rightarrow 1$.

###### Proof.

Without loss of generality, we may assume that $\Omega=g(\mathbb{D})$ contains
a sector of the form $S_{\alpha}:=c+f_{\alpha}(\mathbb{D})$, that
$g(0)=c+f_{\alpha}(0)$ and that $0$ lies outside the closure
$\overline{g(\mathbb{D})}$. Thus, there exists an analytic function
$\varphi:\mathbb{D}\rightarrow\mathbb{D}$ with $\varphi(0)=0$ such that
$c+f_{\alpha}(z)=g\big{(}\varphi(z)\big{)}$ for all $z$. Since $g(1)=\infty$,
then we necessarily have that $\varphi(1)=1$ and the Julia-Wolff lemma [6,
Section 4.1] yields that the angular derivative

$\varphi^{\prime}(1):=\lim_{z\rightarrow
1,z\in\Delta}\dfrac{\varphi(z)-1}{z-1},$

where $\Delta$ is a Stolz angle at $1$, exists and
$|\varphi^{\prime}(1)|\in[1,\infty]$. If $z=\varphi(\zeta)$ for some
$\zeta\in\mathbb{D}$, we have that

$|1-z|^{\alpha}|g(z)|=|1-\varphi(\zeta)|^{\alpha}|c+f_{\alpha}(\zeta)|$

so

$\lim_{z\rightarrow
1,z\in\Delta\cap\varphi(\mathbb{D})}|1-z|^{\alpha}|g(z)|=\dfrac{2^{\alpha-1}|\varphi^{\prime}(1)|^{\alpha}}{\alpha}>0.$

It remains to analyze the behavior as $z\rightarrow 1$ with
$z\in\mathbb{D}\setminus D_{\alpha}$, where $D_{\alpha}:=\varphi(\mathbb{D})$.
Since $g$ is convex, then its Schwarzian norm $\|Sg\|$ is less than or equal
to 2. Because the angle at infinity is positive it follows also that
$g(\mathbb{D})$ is a quasidisk on the Riemann sphere, hence $\|Sg\|<2$ by
Theorem 4 in [1]. We conclude that the function

$G=\frac{1}{g}+1$

maps $\mathbb{D}$ onto a bounded quasidisk, in particular, $G(\mathbb{D})$ is
a John domain. If $B(z)$ denotes the closed Carleson box
$\left\\{\zeta\,:\,|z|\leq|\zeta|\leq
1,\left|\arg\left(\overline{\zeta}z\right)\right|\leq\pi(1-|z|)\right\\}$,
then there exists a uniform $C>0$ such that

$\operatorname{diam}G\big{(}B(z)\big{)}\leq Cd\big{(}G(z),\partial
G(\mathbb{D})\big{)}.$

Writing $z=re^{i\theta}$, this estimate implies that

$\dfrac{1}{|g(e^{i\theta})|}-\dfrac{1}{|g(z)|}\leq|G(z)-G(e^{i\theta})|\leq
C|G(z)-G(1)|=\dfrac{C}{|g(z)|}$

and therefore we have that $|g(e^{i\theta})|\geq\frac{|g(z)|}{1+C}$. Since
$|1-z|\sim|1-e^{i\theta}|$, we conclude that there is a $\delta>0$ such that

$0<\delta\leq|1-z|^{\alpha}|g(z)|\lesssim|1-e^{i\theta}|^{\alpha}|g(e^{i\theta})|,\quad
z\rightarrow 1,z\in D_{\alpha}.$

A similar argument replacing $e^{i\theta}$ by $\zeta\in\mathbb{D}\setminus
D_{\alpha}$ shows that

$\delta\leq|1-\zeta|^{\alpha}|g(\zeta)|,\quad\zeta\rightarrow 1^{-},$

and we are done. ∎

On the other hand, it follows from subordination that if the angle at infinity
of the unbounded convex region $\Omega=g(\mathbb{D})$ is $\pi\alpha$, then for
any $\beta>\alpha$ there exists $K=K(\beta)$ such that

(10) $|1-z|^{\beta}|g(z)|\leq K,\quad z\rightarrow 1.$

An interesting issue is whether there exists a sector of minimal amplitude
containing $\Omega$. This will happen only if the vertices of the sectors of a
minimizing sequence remain in a bounded region of the plane. For example, a
parabolic region has zero angle at infinity but no parallel strip contains it.
Simple modifications provide similar examples for positive angle at infinity
(for example when $\Omega$ is the intersection of angles of decreasing
amplitudes with aligned and divergent vertexes, like the one represented in
Figure 1), so in principle the choice $\beta=\alpha$ is not allowed.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$
Figure 1. Example of convex set with vertex at infinity.

We state now an analogue of Theorem 2.4. Although it can be derived from this
result and the fact that the convolution of convex mappings preserves
subordination (see Theorem 4.1 in [7]), we present an alternative and self-
contained proof.

######  Theorem 3.2.

Let $g_{1},g_{2}$ be unbounded convex mappings defined in $\mathbb{D}$ with
angles $\pi\alpha,\pi\beta$ at infinity, respectively.

(i) If $\alpha+\beta<1$ then $g_{1}*g_{2}$ is bounded.

(ii) If $\alpha+\beta\geq 1$ then $g_{1}*g_{2}$ is unbounded and the angle at
infinity is at least $\pi(\alpha+\beta-1)$.

The angle at infinity is equal to $\pi(\alpha+\beta-1)$ if both regions
$g_{1}(\mathbb{D}),g_{2}(\mathbb{D})$ admit sectors

of minimal amplitude containing them.

###### Proof.

We may assume that both $\alpha,\beta<1$. After a rotation, we may assume that
both regions contain the ray $(0,\infty)$, with $g_{1}(1)=g_{2}(1)=\infty$ and
$0\in g_{1}(\mathbb{T}),g_{2}(\mathbb{T})$ as shown in the Figure 2.

Figure 2. Example of $g_{j}(\mathbb{D})$ after the normalization.

Using the integral representation of the convolution, we have that

$(g_{1}*g_{2})(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}g_{1}\left(\frac{1+r}{2}e^{it}\right)g_{2}\left(\frac{2re^{i\theta}}{1+r}e^{-it}\right)\,dt,\quad
z=re^{i\theta}.$

If $z\rightarrow e^{it_{0}}\neq 1$ then the integral remains bounded because
the factors are each integrable and unbounded at different values of $t$. In
other words, $g_{1}*g_{2}$ can become unbounded only as $z\rightarrow 1$.
Because the convolution is continuous in the spherical metric, we may assume
that $z$ is real. For small $\delta>0$ fixed, the integral

$\int_{\delta<|t|\leq\pi}g_{1}\left(\frac{1+r}{2}e^{it}\right)g_{2}\left(\frac{2r}{1+r}e^{-it}\right)\,dt$

remains bounded as $r\rightarrow 1$, hence it suffices lo analyze the behavior
of

$I(r)=\int_{|t|\leq\delta}g_{1}\left(\frac{1+r}{2}e^{it}\right)g_{2}\left(\frac{2r}{1+r}e^{-it}\right)\,dt$

as $r\rightarrow 1$. For this we write $g_{1}=u_{1}+iv_{1}$,
$g_{2}=u_{2}+iv_{2}$, and express

$I(r)=\int_{|t|\leq\delta}\left[U(r,t)+iV(r,t)\right]\,dt,$

where

$U(r,t)=u_{1}\left(\frac{1+r}{2}e^{it}\right)u_{2}\left(\frac{2r}{1+r}e^{-it}\right)-v_{1}\left(\frac{1+r}{2}e^{it}\right)v_{2}\left(\frac{2r}{1+r}e^{-it}\right)$

and

$V(r,t)=u_{1}\left(\frac{1+r}{2}e^{it}\right)v_{2}\left(\frac{2r}{1+r}e^{-it}\right)+v_{1}\left(\frac{1+r}{2}e^{it}\right)u_{2}\left(\frac{2r}{1+r}e^{-it}\right).$

The normalizations of $g_{1},g_{2}$ imply that
$u_{1}(\frac{1+r}{2}e^{it}),u_{2}(\frac{2r}{1+r}e^{-it})>0$ while
$v_{1}(\frac{1+r}{2}e^{it})$ and $v_{2}(\frac{2r}{1+r}e^{-it})$ have the sign
of $t$. Furthermore, all four quantities $u_{1},u_{2},v_{1},v_{2}$ will be
comparable in size because the domains contain and are contained in certain
sectors.

For part (i) we use (10) for $g_{1}$ and $g_{2}$ with angles
$\alpha^{\prime}>\alpha$ and $\beta^{\prime}>\beta$ and
$\alpha^{\prime}+\beta^{\prime}<1$ to conclude that $I(r)$ will stay bounded
as $r\rightarrow 1$. For part (ii), we see from the above remarks on the size
and signs of the $u_{j},v_{k}$ that it suffices to analyze no only the real
part of $I(r)$, but just the integral

$J(r)=\int_{0}^{\delta}u_{1}\left(\frac{1+r}{2}e^{it}\right)u_{2}\left(\frac{2r}{1+r}e^{-it}\right)\,dt$

as $r\rightarrow 1$.

Next, a simple calculation shows that

$\frac{1}{|1-se^{it}|^{\gamma}}\sim\frac{1}{\left[(1-s)^{2}+t^{2}\right]^{\frac{\gamma}{2}}}$

as $s\rightarrow 1$, hence by (9) we have

$J(r)\gtrsim\int_{0}^{\delta}\frac{dt}{\left[(1-r)^{2}+t^{2}\right]^{\frac{\alpha+\beta}{2}}}=\frac{1}{(1-r)^{\alpha+\beta-1}}\int_{0}^{\delta/(1-r)}\frac{dx}{(1+x^{2})^{\frac{\alpha+\beta}{2}}}.$

This proves that $g_{1}*g_{2}$ becomes unbounded as $z\rightarrow 1$ and,
because of (10), that the angle at $(g_{1}*g_{2})(1)=\infty$ cannot be smaller
than $\pi(\alpha+\beta-1)$. Finally, if both domains admit minimizing sectors
containing them, we will also have $J(r)\lesssim(1-r)^{1-\alpha-\beta}$,
showing that the angle at infinity of the convolutions must be
$\pi(\alpha+\beta-1)$. ∎

######  Corollary 3.3.

Let $g_{1},\ldots,g_{n}$ be unbounded convex mappings with angles at infinity
$\pi\alpha_{1},\ldots,\pi\alpha_{n}$. Then the probability that
$g_{1}*\cdots*g_{n}$ is unbounded is equal to $1/n!$.

###### Proof.

Each stage of the successive convolutions must be unbounded, otherwise the
final convolution will be bounded. Therefore, $\alpha_{1}+\alpha_{2}\geq 1$,
then $\alpha_{1}+\alpha_{2}+\alpha_{3}\geq 2$, and so on. We conclude that
$\alpha_{1}+\cdots+\alpha_{n}\geq n-1$. A standard calculation shows that the
volume of this region in the $n$-cube $[0,1]^{n}$ is equal to $1/n!$, as
claimed.

∎

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