Loewner differential equation

Remarks

For each $t$, the map $g_t$ is a conformal map of $\mathbb{H} \setminus K_t$ onto $\mathbb{H}$ with the normalisation $g_t(z) = z + \frac{2t}{z} + O(\abs{z}^{-2})$ as $z \to \infty$. The hulls $K_t$ increase with $t$, and the number $2t$ is their half-plane capacity. Example: for $U = 0$, the equation gives $\partial_t (g_t(z)^2) = 4$, so $g_t(z) = \sqrt{z^2 + 4t}$ with the branch that maps $\mathbb{H}$ into $\mathbb{H}$. A point $iy$ with $y > 0$ has $g_t(iy)^2 = 4t - y^2$, which reaches $0 = U(t)$ at $t = y^2/4$; so $K_t$ is the vertical segment from $0$ to $2i\sqrt{t}$. Non-example: the translations $f_t(z) = z + t$ are conformal self-maps of $\mathbb{H}$, but they do not come from the equation, since $f_t(z) - z = t$ does not tend to $0$ at infinity. The radial Loewner equation, which Loewner introduced first, is the analogue for the unit disc, with a driving function with values on the unit circle and maps normalised at $0$.