For a closed Riemann surface $\Sigma$ of genus $g \geq 2$, the space of holomorphic quadratic differentials on $\Sigma$ can be identified with the cotangent space $T\_\Sigma^\* \mathcal{T}\_g$: in other words, "holomorphic quadratic differentials parametrize the infinitesimal deformations of complex structures on $\Sigma$."


This is a bit different when $\Sigma = \mathbf{C}$, because this is a (non-compact) surface with genus $g = 0$. The Teichmüller space of the complex plane $\mathbf{C}$ contains only two elements: the standard complex structure and that inherited from the Poincare model.


However, there are many holomorphic quadratic differentials on $\mathbf{C}$, at least if one allows for these to be unbounded. Given any holomorphic function $\phi: \mathbf{C} \to \mathbf{C}$, the differential $\phi \, \mathrm{d} z^2$ is holomorphic. For example, we may take this to be a polynomial, say $\phi = p = a\_0 + a\_1 z + \cdots + a\_k z^k$.



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> Is there a "Teichmüllerian" interpretation of these polynomial quadratic differentials on $\mathbf{C}$, in the sense that they would parametrize an infinitesimal deformation (of some object)? In slightly more specific terms: given a differential $(a\_0 + \cdots + a\_k z^k) \mathrm{d} z^2$, is there a natural one-parameter family of objects, say $(\Omega\_t)$, for which $"\frac{d}{dt} \Omega\_t |\_{t = 0} = p \, \mathrm{d} z^2"$?
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The naive thing would be to put $g\_t = \lvert \mathrm{d} z \rvert^2 + t p \,\mathrm{d} z^2$, but this is not a metric on $\mathbf{C}$, as it is not positive definite for any $t \neq 0$ (provided $\operatorname{deg} p \geq 1$). For this reason I thought that this wasn't the right way of approaching the question.



Teichmuller interpretation of unbounded holomorphic quadratic differentials
These can be seen as global or infinitesimal deformations of harmonic maps $\mathbb{C} \to \mathbb{H}$ with ideal polygonal image. In particular, Han-Tam-Treibergs-Wan [show](https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/1995/0003/0001/CAG-1995-0003-0001-a003.pdf) that a harmonic injective immersion $h: \mathbb{C} \to \mathbb{H}$ has polynomial Hopf differential $\phi$ with $\text{deg}(\phi) = m$ if and only if $\text{Im}(h)$ is an ideal polygon with $m+2$ vertices (such harmonic maps with prescribed degree-$m$ polynomial Hopf differential always exist). I should remark that Hopf differentials of Harmonic maps come up elsewhere in Teichmüller theory: for a first example, they be used to parametrize Teichmüller spaces of closed surfaces (see Mike Wolf's [thesis](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-29/issue-2/The-Teichm%C3%BCller-theory-of-harmonic-maps/10.4310/jdg/1214442885.full)).


To answer your question a bit more directly, let $\mathcal{P}\_m$ be the space of holomorphic quadratic differentials on $\mathbb{C}$ of the form $(z^m + a\_{m-2}z^{m-2} + ... + a\_1)dz^2$, and let $\mathcal{H}\_m$ be the collection of harmonic injective immersions $\mathbb{C} \to \mathbb{H}$ with $\text{Im}(h)$ an ideal $m+2$-gon, $h(0) = 0$, and $h'(0) = 1$ (these are the forms you can arrange for by using conformal/isometric transformations of base/target). Then the map $\mathcal{H}\_m \to \mathcal{P}\_m$ given by taking Hopf differentials is a homeomorphism. The mappings, not just the images, do matter here.


So if you're given $\phi = a\_{k}z^k + ... + a\_0$, you can consider the $1$-parameter family $h\_t: \mathbb{C} \to \mathbb{H}$ of harmonic maps with Hopf differential $(z^{k+2} + t\phi) dz^2$ with $h\_t(0) = 0, h\_t'(0) = 1$, and recover $\phi dz^2$ as the derivative of the Hopf differentials of $h\_t$. Of course, better than you ask for happens here: the family $t\phi dz^2$ *globally* describes a family of harmonic mappings with polygonal images.


