SLE revision 2: Loewner’s Differential Equation

Last time I set up the geometric notions of probability that will be needed to proceed with the course material. Now we consider the deterministic differential equation due to Loewner (1923) which he used to make progress on the Bieberbach Conjecture, but which will also underpin the construction of SLE. This proof is adapted for this specific case from the slightly more general argument in Duren’s Univalent Functions (Section 3.3). Because in that setting the result concerns an infinite domain, readers should beware that though I am using identical notation, in about half the cases, my function are the inverse and my sets the complement of what they are in Duren.

To explain the construction, as with so many things, a picture speaks a thousand words. Unfortunately I have neither the software nor, right now, the time to produce the necessary diagrams, so the following will have to suffice.

Consider a deterministic simple curve in the unit disc, $(\gamma(t): t\in[0,\infty))$. Removing initial segments of the curve gives the nested simply-connected regions:

$U_t:=\mathbb{U}\backslash \gamma[0,t]$.

Then define as in the previous post, the unique conformal map

$f_t: U_s\rightarrow \mathbb{U}$ such that $f_t(0)=0, f_t'(0)\in\mathbb{R}^+$,

and furthermore set $\xi_t$ to be the image of $\gamma(t)$ under this map. (Note that though the conformal map is not defined on the boundary, it must extend continuously).

$f_t'(0)$ is increasing.

Very informally, this derivative records how much twisting is required at the origin to turn the slit domain into the open disk. Extending the path will demand further twisting. More rigorously, set:

$g_t=f_t^{-1}:\mathbb{U}\rightarrow U_t$.

Then the $(g_t)$ are injective functions from the unit disc to itself which preserve the origin, so Schwarz’s lemma applies. They are clearly not rotations, so

$|g_t'(0)|<1$.

By the inverse function theorem, $|f_t'(0)|>1$ (*). Now, given $t>s$, can decompose:

$f_t=f_s\circ \tilde{f}$,

and $\tilde{f}$ has this useful Schwarz property (*) also. By applying the chain rule, noting that $f_s(0)=0$, we deduce that

$|f_t'(0)|>|f_s'(0)|$.

This means we are free to demand that the curve has time parameter such that

$|f_t'(0)|=e^t$.

A reminder of the statement of Schwarz’s Lemma: Continue reading