Riemann mapping theorem and Dirichlet's principle

In his 1851 PhD thesis, Riemann introduced the following theorem concerning simply connected regions in $\mathbb C$:

Riemann mapping theorem: For any nonempty simply connected open subset $\Omega\subsetneq\mathbb C$, there exists a analytic $f:\Omega\to\mathbb C$ such that

1. $f(\Omega)$ is the open unit disk centered at the origin.
2. $f$ is one-to-one on $\Omega$.

This indicates that to study the properties of simply connected regions in $\mathbb C$, it is sufficient to investigate $\mathbb C$ itself and the unit disk. In modern textbooks on complex analysis, this result is usually proved via the theory of normal families.

In this article, we present how Riemann originally investigated this theorem. This would allow us to see the connection of three seemingly separate areas of mathematics: Complex Analysis, Geometry, and Calculus of Variations.

For simplicity, we assume $\partial\Omega$ to be smooth. The general case can be discussed by making smooth approximation to $\partial\Omega$.

Bounding the image

The maximum modulus principle indicates that the maximum of $\vert f(z)\vert$ can only be achieved when $z$ lies on the boundary $\partial \Omega$. As a result, we can require

\[\vert f(z)\vert=1\quad\forall z\in\partial \Omega.\]

so that the image of $f(\Omega)$ will surely be contained within the unit disk. Now, we can look at the one-to-one property.

One-to-one mappings in $\mathbb C$

Before constructing the desired $f$, we need to look at some necessary conditions satisfied by $f$. In this section, we prove in particular that its derivative is nonvanishing throughout $\Omega$.

Suppose $f'(z_0)=0$ for some $z_0\in \Omega$. Then $f(z)-f(z_0)$ has a zero of order $\ge2$. Because the zeros of analytic functions are isolated, we can find some $m>0$ and $r>0$ such that $f'(z)\ne0$ on $0<\vert z-z_0\vert\le r$ and $\vert f(z)-f(z_0)\vert\ge m$ on $\vert z-z_0\vert=r$, so for any complex $\zeta$ satisfying $\vert\zeta\vert<m$, we have $\vert\zeta\vert<\vert f(z)-f(z_0)\vert\ge m$ on $\vert z-z_0\vert=r$. Thus, it follows from Rouché's theorem that, the function $f(z)-f(z_0)+\zeta$ has $\ge2$ zeros in $0<\vert z-z_0\vert<r$. Since in this punctured region $f'(z)\ne0$, we conclude that there are distinct points $z_1,z_2\in \Omega$ such that $f(z_1)=f(z_2)=f(z_0)-\zeta$, which contradicts the one-to-one property.

This allows us to deduce the following sufficient condition for Riemann's theorem:

Lemma 1: Riemann's theorem will follow from the existence of a analytic $f:\Omega\to\mathbb C$ satisfying the following:

1. There exists a unique $z_0\in \Omega$ such that $f(z_0)=0.$
2. $\vert f(z)\vert=1$ on $\partial \Omega$ and $f'(z)\ne0$ in $\Omega$.

Proof. For any $w$ inside the open unit disk, we have $\vert w\vert<\vert f(z)\vert$ on $\partial \Omega$, so by Rouché's theorem, the function $f(z)-w$ and $f(z)$ should have the same number of zeros in $\Omega$. Due to the first condition, this indicates that there is a unique $z_1\in \Omega$ satisfying $f(z_1)=w$. Because $w$ is chosen arbitrarily, we deduce that $f$ is one-to-one and $f(\Omega)$ is the open unit disk.

Transition to harmonic functions

By our investigations in the previous section, the condition for Lemma 1 will be satisfied if $f(z_0)=0$ and there exists some analytic $h$ on $\Omega$ satisfying

\[f(z)=(z-z_0)e^{h(z)},\quad z\in \Omega\tag1\]

and

\[\Re[h(z)]=-\log\vert z-z_0\vert,\quad z\in\partial \Omega.\tag2\]

Cauchy-Riemann equations indicate that the real parts of analytic functions are harmonic. Conversely, for a given harmonic function, we can always construct its corresponding analytic function by recovering the harmonic conjugate via the Cauchy-Riemann equations. Therefore, we have

Lemma 2: Riemann's theorem will follow from the existence of a harmonic function $u$ on $\Omega$ satisfying $u=-\log\vert z-z_0\vert$ on $\partial \Omega$.

Dirichlet's problem

In general, if $\Omega$ is subset of $\mathbb R^n$ and $g$ is smooth on $\partial\Omega$, the problem of solving

\[\Delta u|_\Omega=0,\quad u|_{\partial\Omega}=g\tag3\]

is known as the Dirichlet problem, so we can complete the proof of Riemann's mapping theorem by solving this partial differential equation.

Explicit methods for solving are available only when $\Omega$ is in particlar shape (e.g. Poisson kernel for disks), but to prove Riemann's theorem, we only need to establish the existence of solutions. Interestingly, this was first achieved by minimizing a certain energy integral.

Energy integral and Dirichlet's principle

For a smooth function $\psi:[a,b]\to\mathbb R$, a method to determine how frequently $\psi$ varies in its domain is to consider the quantity

\[\int_a^b\vert\psi'(x)\vert^2\mathrm dx.\]

In general, for a smooth function $\psi$ defined on $\Omega\subset\mathbb R^n$, we consider its Dirichlet energy defined by

\[D[\psi]=\int_\Omega\vert\nabla\psi\vert^2\mathrm dV.\tag4\]

If no extra conditions are imposed on $D[\psi]$, it is clear that its minimum is attained whenever $\psi$ is constant, so Dirichlet introduced the boundary conditions for $\psi$. This produces the following optimization problem:

\[\min D[\psi]\quad s.t.\quad\psi|_{\partial\Omega}=g.\tag5\]

In the remaining part of this section, we will show that this optimization problem is equivalent to the partial differential equation (3).

Dirichlet's principle: A function solves the optimization problem (5) if and only if it solves the Dirichlet problem (3).

For the sake of convenience, we let $A$ denote the set of all smooth functions $\psi:\Omega\to\mathbb R$ satisfying $\psi=g$ on $\partial\Omega$ and $B$ denote the set of all smooth functions $v:\Omega\to\mathbb R$ that vanish on $\partial\Omega$. Thus, every difference of two functions in $A$ is an element of $B$.

By definition, $u\in A$ solves (5) if and only if for any $v\in B$, there is $D[u+v]\ge D[u]$. This indicates that if $v\in B$ is fixed and $t$ is any real number, the function $S(t)=D[u+tv]$ has a minimum at $t=0$. This means that $S'(0)=0$. By definition, there is

\[S(t)-S(0)=\int_\Omega[2t(\nabla u)\cdot(\nabla v)+t^2\vert\nabla v\vert^2]\mathrm dV,\]

so $u$ being a minimizer of (5) implies that

\[\int_\Omega(\nabla u)\cdot(\nabla v)\mathrm dV=0\quad\forall v\in B.\tag6\]

Because (6) suggests $D[u+v]=D[u]+D[v]$ for all $v\in B$, we conclude that the optimization problem (5) is equivalent to finding a $u\in A$ to fulfill (6).

Application of the divergence theorem

If $\rho$ is a scalar field and $\mathbf F$ is a vector field, then it is known that

\[\nabla\cdot(\rho\mathbf F)=\mathbf F\cdot\nabla\rho+\rho\nabla\cdot\mathbf F,\]

so plugging $\rho=v$ and $\mathbf F=\nabla u$ gives

\[\begin{aligned} (\nabla u)\cdot(\nabla u) &=\nabla\cdot(v\nabla u)-v\nabla\cdot(\nabla u) \\ &=\nabla\cdot(v\nabla u)-v\Delta u. \end{aligned}\]

Hence, by divergence theorem, there is

\[\begin{aligned} \int_\Omega(\nabla u)\cdot(\nabla v)\mathrm dV &=\int_\Omega[\nabla\cdot(v\nabla u)-v\Delta u]\mathrm dV \\ &=\int_{\partial\Omega}v\nabla u\mathrm dS-\int_\Omega v\Delta u\mathrm dV. \end{aligned}\]

Because $v\in B$, we find that the integral over $\partial\Omega$ vanishes, so (6) is equivalent to the statement that

\[\int_\Omega v\Delta u\mathrm dV=0\quad\forall v\in B.\tag7\]

Proof of Dirichlet's principle

At present, it is clear that solutions to the Dirichlet problem always satisfy (7), so it remains to prove that every $u\in A$ satisfying (7) must fulfill (3). Suppose $\Delta u\ne0$ at some $\boldsymbol{x_0}\in\Omega$. Then by continuity, we can find some $r>0$, $m>0$, and $\eta\in\lbrace1,-1\rbrace$ such that $\eta\Delta u(\boldsymbol x)\ge m$ whenever $\vert\boldsymbol x-\boldsymbol{x_0}\vert\le r$ ($r$ small enough so that the entire ball is contained in $\Omega$).

Let

\[\varphi(y)= \begin{cases} e^{-1/y} & y>0, \\ 0 & y\le 0. \end{cases}\]

Then it is well known that $\varphi(y)$ is a non-negative smooth function on $\mathbb R$, so

\[v(\boldsymbol x)=\varphi(r-\vert\boldsymbol x-\boldsymbol{x_0}\vert)\in B.\]

This indicates that

\[\left\vert\int_\Omega v\Delta u\mathrm dV\right\vert=\int_{\vert\boldsymbol x-\boldsymbol{x_0}\vert\le r}v\vert\Delta u\vert\mathrm dV\ge m\int_{\vert\boldsymbol x-\boldsymbol{x_0}\vert\le r}v\mathrm dV>0,\]

which is a contradiction. This indicates that we must have $\Delta u=0$ in $\Omega$, thereby completing the proof.

Riemann's proof of the mapping theorem

By Lemma 1, Lemma 2, and Dirichlet's principle, we realize that Riemann's theorem can be deduced as long as we can find a minimizer for the following problem

\[\min\int_\Omega\vert\nabla\psi\vert^2\mathrm dV,\quad \psi|_{\partial\Omega}=-\log\vert z-z_0\vert.\tag8\]

Riemann claimed that the existence of minimizer will immediately follow from the fact that the above integral is bounded below (i.e. infimum exists), thereby proving the theorem.

Flaws and remedies

Riemman's original proof is incomplete because his justification on the existence of minimizer is flawed. In fact, Weierstrass presented the following optimization problem

\[\min J[\psi]=\int_{-1}^1 x^2\vert\psi'(x)\vert^2\mathrm dx,\quad\psi(-1)=-1,\psi(1)=1\tag9\]

for continously differentiable $\psi$. Clearly $J[\psi]\ge0$. In addition, when $\psi$ is taken to be the sequence

\[\psi_N(x)={\arctan(Nx)\over\arctan N},\quad N=1,2,3,\dots\]

$\psi_N$ fulfills the boundary conditions and $J[\psi_N]\to0$ as $N\to+\infty$, which means $\inf J[\psi]=0$. However, this infimum cannot be attained by any continuously differentiable function for otherwise $\psi$ needs to be constant, violating the boundary conditions. This indicates that no minimizer exists to fulfill (9).

Although Weierstrass's example did not disprove the existence of minimizer to (8), it demonstrates that it is not reasonable for Riemann to conclude the existence of minimizer merely from the existence of infimum. Because many important results of Riemann were proved assuming such minimizer exists for the Dirichlet energy, Weierstrass's criticism caused mathematicians at the time to search for rigorous justification of Riemann's arguments in order to salvage Riemann's works.

In 1900, by developing a rigorous theory of functional analysis Hilbert showed that the minimizer to (5) exists provided that $\partial\Omega$ is smooth enough. Because a simply connected open set in $\mathbb C$ can have very complicated boundary behavior, this does prove the full form of the mapping theorem. Instead of tackling with the integral $D[\psi]$, the first rigorous proof of the mapping theorem by W. F. Osgood¹ in 1900 is based on a direct attack to the Dirichlet problem itself.

Conclusion

In this article, we began our discussion from Riemann's mapping theorem and eventually transformed the proof of this theorem into solving an optimization problem over space of functions. Finally, we present Riemann's proof of the mapping theorem and discussed its flaws and remedies.

1. Osgood, W. F. (1900). On the Existence of the Green’s Function for the Most General Simply Connected Plane Region. Transactions of the American Mathematical Society, 1(3), 310. ↩