# 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.