Do biholomorphic functions maintain ratio of diameters of ellipses?

Question

I am currently reading through the book of Farb&Magalit, where they define the dilation of a diffeomorphism as the ratio of the diameters of an image of a circle (which is then an ellipse). The problem I have here, is that this is not necessarily independent of the choice of charts. Hence my question:

Do biholomorphic functions (from $\mathbb{C}$ to $\mathbb{C}$) maintain the ratio of the two diameters (length of the two axis) of an ellipse with center 0.

I see that holomorphic functions maintain circles. But obviously the (non-bijective) holomorphic function $z\mapsto z^2$ does not maintain the ratio of the diameters (thinking of some standard ellips, where one diameter is 1 and the other one is 42). Is there a nice way to see, this nevertheless holds for biholomorphic functions?

Answer 1

I suspect you mean infinitesimal ellipses. That is the usual definition of a quasiconformal map (see https://en.wikipedia.org/wiki/Quasiconformal_mapping).

More precisely, if you have a diffeomorphism $f: U \to V$ where $U,V$ are domains in the plane (more generally, Riemann surfaces) then the differential of $f$ at $x \in U$ is a $\mathbb R$-linear map that will send a circle in the tangent plane $T_x U$ to an ellipse in the tangent plane $T_{f(x)} V$. You can then look at the eccentricity of the ellipse and call that the dilatation of $f$ at $x$. Then you can define the dilatation of $f$ as the sup of the dilatations of $f$ over the $x \in U$.

Notice that the dilatation of $f$ at $x$ will be zero if and only if $Df(x)$ is $\mathbb C$-linear, so the dilatation of $f$ is zero if and only if $f$ is conformal (biholomorphic). In fact, the dilatation $K(f)$ (when it is finite) measures how far $f$ is from being holomorphic (the smaller it is, the closer $f$ is to be holomorphic, hence the term quasiconformal).

For technical reasons, we are interested not only in diffeomorphisms in this context but also in homeomorphisms with nice enough distributional derivatives, for which this definition becomes a bit more technical.

As Robert Isreal noted, the question does not make much sense if you are talking about actual ellipses in the plane instead of ellipses in the tangent planes.

Answer 2

A biholomorphic function from (all of) $\mathbb C$ to $\mathbb C$ is of the form $f(z) = a + b z$, and thus a composition of translation ($z \to z + a$), rotation ($z \to \omega z$ where $|\omega|=1$) and dilation ($z \to r z$ where $r > 0$), all of which map ellipses to ellipses, preserving the ratio of major to minor axes. They don't preserve the centre, however. A biholomorphic function on an open subset of $\mathbb C$ will not in general map ellipses to ellipses (or circles to circles) at all.