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Riemann Mapping by Steepest Descent

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

$\tag{1} u(z) = \mathrm{ln} | z | + u_0 (z),$

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

$\tag{2}\dot{z} = - \frac{1}{{|\nabla u|}^2} \nabla u$

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:

$\tag{3} f(z) = \lim_{t \rightarrow \infty} e^t \phi^t z.$

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 ([email protected]) is a professor of mathematics at The Pennsylvania State University.