Misuse of bi-

The prefix bi- is apt to cause confusion; how often does a bimonthly event occur? Its usage in mathematics is not free of inconsistency, either. Compare:

• Biholomorphic: a holomorphic map $f$ such that $f^{-1}$ is also holomorphic
• Biharmonic: a function $u$ such that $\Delta \Delta u=0$

Both of these usages are well established. Let’s switch them around just for the fun of it.

(1) Recall that $f\colon \Omega\to\mathbb C$ (where $\Omega\subset \mathbb C$) is holomorphic if $\displaystyle\frac{\partial }{\partial \bar z}f=0$. Following the second usage pattern, we would call $f$ “biholomorphic” if $\displaystyle \frac{\partial}{\partial \bar z}\frac{\partial}{\partial \bar z} f=0$. What can we say about such functions? Let $\displaystyle g=\frac{\partial }{\partial \bar z}f$. Clearly, $g$ is holomorphic. Now introduce $h(z)=f(z)-\bar zg(z)$. This function satisfies $\displaystyle \frac{\partial }{\partial \bar z}h(z) = g(z)-g(z)=0$, i.e., it is also holomorphic. Conclusion: the solutions of the equation $\displaystyle \left(\frac{\partial}{\partial \bar z}\right)^2 f=0$ are precisely the functions of the form $f(z)=h(z)+\bar z g(z)$ where $h$ and $g$ are holomorphic.

This representation formula tells us a lot about “biholomorphic” functions. They are very smooth: the real and imaginary parts are real-analytic. They are locally invertible outside of the set $\{z\colon |h'(z)+\bar zg'(z)|=|g(z)|\}$, which is usually 1-dimensional (compare to the discreteness of the branch set of holomorphic functions). They are not open in general: consider $f(z)=z\bar z = |z|^2$ for example. The modulus $|f|$ does not satisfy the maximum principle: consider $f(z)=1-|z|^2$. But Liouville’s theorem does hold: if $f$ is bounded in $\mathbb C$, it is constant.

(2) A map $f\colon \Omega\to\mathbb C$ (where $\Omega\subset \mathbb C$) is harmonic if $\Delta f=0$, or, put another way, if $f=u+iv$ where $u$ and $v$ are real harmonic functions. Following the first usage pattern, we would call $f$ “biharmonic” if $f^{-1}$ exists and is also harmonic. Any invertible holomorphic map clearly satisfies this definition. Anything else? Well, there are affine maps $f(z)=a z+b\bar z+c$ which are invertible as long as $|a|\ne |b|$ (here $a,b,c\in\mathbb C$ are constant.) The inverse of an affine map is also affine, hence harmonic. Anything else? Nothing comes to mind.

But in 1945 Gustave Choquet gave this example:

$\displaystyle x+iy\mapsto x+iv,\qquad \tan v \tan y = \tanh x$

That’s not a typo: both the regular tangent and its hyperbolic cousin appear in the formula. It is not at all obvious (but true) that $v$ is a harmonic function of $x$ and $y$. It is rather obvious that the map is an involution, hence “biharmonic” according to our definition.

Now that we have one “biharmonic” map that’s neither holomorphic nor affine, are there others? No (apart from trivial variations of $f$). Choquet attributed this uniqueness result to Jacques Deny who did not publish it. The first published classification of “biharmonic” maps appeared in a 1987 paper by Edgar Reich in a rather different form:

If $f$ and $f^{-1}$ are harmonic, then $f$ is either holomorphic, affine, or of the form $\displaystyle f(z)=\alpha(\beta z+2i\arg (\gamma-e^{\beta z}))+\delta$ where $\alpha,\beta,\gamma,\delta$ are complex constants such that $\alpha\beta\gamma\ne 0$ and $|e^{-\beta z}|<\gamma$ in the domain of $f$.

The proof can be found in Peter Duren’s book from which I quoted the statement of this result.