# Riemann Mapping by Steepest Descent

By Mark Levi

In this issue’s column we outline a quick constructive proof of the Riemann mapping theorem.

Here’s a statement of the theorem: Any simply connected open set \(D\) in the plane that is not the entire plane can be mapped conformally and one-to-one onto an open unit disk, taking any point \(p \in D\) to the disk’s center, with the derivative at \(p\) being real and positive.

Our map is produced via a physically motivated argument: We think of a heat-conducting plate \(D\), insulated everywhere except for the boundary, as shown in Figure 1.

**Figure 1.** Heat-conducting plate, insulated everywhere but the boundary.

We place a heat sink at \(z = 0\), drawing \(2\pi\) calories per second, and maintain \(u = 0\) on \(\partial D\). Figure 2 shows the graph of the resulting stationary temperature distribution. Formally, we define

\[\begin{equation}\tag{1}

u(z) = \mathrm{ln} | z | + u_0 (z),

\end{equation}\]

where \(u_0 (z)\) is a *harmonic function* with boundary conditions chosen to cancel the logarithm on \(\partial D\).

Level curves of (1) are approximately circles near \(z = 0\); more precisely, the set \(D_t = \{u \leq −t\}\) for large \(t\) is approximately a small disk \(| z | \leq e^{−t − {u_{0}}(0)}\) (see Figure 2).

**Figure 2.** Constructing the conformal map from *D* to a disk. **Left:** The modified gradient flow preserves level curves. **Right:** Mapping by the modified gradient flow.

Remarkably, the flow \(\varphi^t\) of the modified gradient field

\[\begin{equation}\tag{2}\dot{z} = - \frac{1}{{|\nabla u|}^2} \nabla u

\end{equation}\]

shrinks \(D\) into \(D_t\) and does so conformally—i.e., \(\varphi^t\) is almost the desired map! Indeed, \(du/dt = −1\) along (2), and thus \(\varphi^t D = D_t\) . And because the right-hand side of (2) is analytic, \(\varphi^t\) is conformal. By dilating \(D_t\) we obtain the desired map \(f\) in the limit:

\[\begin{equation}\tag{3}

f(z) = \lim_{t \rightarrow \infty} e^t \phi^t z.

\end{equation}\]

The missing details of the proof, which are routine, can be found in [1].

Here we use the existence of solutions of the Dirichlet problem, which limits some generality on \(D\). Details can be found in F. John,

*Partial Differential Equations*, 4th ed., Springer, New York, 1991.

Since it can be written as \(1\sqrt{\nabla u} = 1/(u_x – iu_y)\), where \(U = u_x\) and \(V = −u_y\) satisfy the Cauchy–Riemann equations because \(u\) is harmonic.

**Acknowledgments:** The work from which these columns are drawn is funded by NSF grant DMS-1412542.

**References**

[1] M. Levi, *Riemann mapping by steepest descent*, Amer. Math. Monthly, 114:3 (2007), 246-251.

Mark Levi (levi@math.psu.edu) is a professor of mathematics at The Pennsylvania State University.