Ram On

This artistic drawing made it into a paper in Bulletin LMS (Iwaniec-K-Onninen; drawing by the first named author). What does it illustrate?

Any two simply connected domains $D,D'\subset \mathbb R^2$ are diffeomorphic: there exists a bijection $f\colon D\to D'$ such that both $f$ and $f^{-1}$ are continuously differentiable (make them infinitely differentiable if you prefer). If neither $D$ nor $D'$ is the entire plane, the Riemann mapping theorem furnishes a conformal $f$ that fits the task; and since the plane is obviously diffeomorphic to a disk (though not conformally diffeomorphic), this proves the claim. That being said, the behavior of diffeomorphism near the boundary is not guaranteed to be nice: the first derivatives of $f$ or of $f^{-1}$ may be unbounded. This causes problems if we want to use $f$, e.g., to transfer some integral from one domain to another via the change of variables formula.

So we want only “controlled diffeomorphisms”: those for which both $f$ and $f^{-1}$ have bounded first-order derivatives. With this constraint in mind, which domains $\Omega$ are equivalent to the disk $\mathbb D$ via a controlled diffeomorphism? Clearly, such $\Omega$ must be bounded. Any other restrictions?

Let $C$ be an upper bound for the derivatives $\|Df\|, \|Df^{-1}\|$. Our diffeomorphism must quasi-preserve the length of curves: $C^{-1}L(\gamma)\le L(f\circ \gamma)\le CL(\gamma)$. In particular, this holds for crosscuts, simple curves that divide the domain into two parts. Let’s compare the length of a crosscut $\gamma$ to the shorter of two boundary arcs between its endpoints. In the disk, the shorter boundary arc has the length at most $(\pi/2)L(\gamma)$. Therefore, the domain $\Omega$ must have the same property, except with a different constant such as $C^2\pi/2$. This is what is called an inner chordarc domain.

The definition can be extended to multiply connected domains by considering crosscuts that join two points of the same boundary components. The first drawing shows that an inner chordarc domain may have a jagged boundary: corners are allowed, as are inward cusps and logarithmic spirals. Only outward cusps are ruled out.

The definition can be put in practical terms. The domain is a lake, you are standing on its shore, and want to reach another point on the shore. You could either swim across or walk along the shore. The lake is inner-chordarc if a sufficiently slow swimmer (with a low swim/walk speed ratio) will always choose to walk.

Here is another real-life example: a simply connected inner chordarc domain that is not a Jordan domain (the boundary is not a closed curve).

Finally, the theorem that justifies the definition. (J. Väisälä, 1987): A simply connected domain admits a controlled diffeomorphism onto a disk if and only if it is an inner chordarc domain.