$f \colon(\Sigma, g)\to(M,h)$ is harmonic if and only if $f \colon(\Sigma, e^{2u}g)\to(M,h)$ is harmonic.

\end{cor*}

Thus, the notion of harmonicity only depends on the ``conformal class'' of $g$.

A \CmMark{complex structure} on $\Sigma$ is a maximal atlas of charts into $\C$ with holomorphic (conformal) transition functions.

It yields an almost complex structure given by $J \colon\T\Sigma\to\T\Sigma$ with $J^2=-\id$.

Let $\mathcal M(\Sigma)$ be the space of riemannian metrics on $\Sigma$.

Then $C^{\infty}(M)$ acts on $\mathcal(\Sigma)$ by $u \cdot g \mapsto e^{2u}g$.

Further denote by $\mathcal C(\Sigma)$ the set of complex structures on $\Sigma$ (viewed as $J \colon\T\Sigma\to\T\Sigma$ as above), which agree with the orientation.

In both cases we consider the $C^{\infty}$-topology on these spaces.

\begin{thm*}

There is a homeomorphism $C^{\infty}(\Sigma)\backslash\mathcal M(\Sigma)\leftrightarrow\mathcal C(\Sigma)$.

\end{thm*}

\begin{proof}[Proof (sketch)]

Consider the map $\mathcal C(\Sigma)\to\mathcal M(\Sigma)/C^{\infty}(\Sigma), \sigma\mapsto$ all metrics $g$ such that $g$ defines the same angle as $\sigma$.

Another result from geometric analysis yields so-called ``isothermal coordinates'' $\mathcal M(\Sigma)/C^{\infty}(\Sigma)\to\mathcal C(\Sigma)$.

For $g \in\mathcal M(\Sigma)/C^{\infty}(\Sigma)$ there exists coordinates $\{\dop x, \dop y\}$ on about any $p \in\Sigma$ such that $g = e^{2u}(\dop x^2+\dop y^2)$.

If $(x,y)$ and $(\tilde x, \tilde y)$ are competing isothermal coordinate systems, then the transition function between them is conformal.

Thus we obtain a map $g \mapsto\{$ isothermal coordinate atlas $\}$, where $e^{2u}g$ is mapped to the same atlas as $g$.

\end{proof}

Denote by $\mathcal M_{-1} :=\{ h \in\mathcal M \mid\sec(h)=-1\}\subset\mathcal M$ the subspace of hyperbolic metrics.

\begin{thm*}[Application of Eells-Sampson / Hartmann]

Given $h \in\mathcal M_{-1}$ and a diffeomorphism $f \colon\Sigma\to\Sigma$ there exists (exactly one\footnote{This uniqueness is due to Hartmann}) a harmonic map $u \colon(\Sigma, \sigma)\to(\Sigma, h)$ such that $u$ is homotopic to $f$.

Moreover, for $\Diff_0(\Sigma)=\{\eta\colon\Sigma\to\Sigma\text{ diffeomorphism isotopic to the }\id\}$ the map $\psi$ is $\Diff_0(\Sigma)$-equivariant, i.e. $\psi(\eta\cdot h =\eta^{*}h)= u_h \circ\eta$ for $\eta\in\Diff_0\Sigma$.

Hence we obtain a map $\mathcal M_{-1}/\Diff_0\Sigma\xrightarrow{\psi} C^{\infty}(\Sigma,\Sigma)/\Diff_0\Sigma$.

But we can use the theory of Higgs bundles from the right hand side to some useful vector space.

For now, we will obtain a way to get around this and directly pass to a nice vector space.

Let $\mathrm{QD}(\sigma)=\{\alpha\mid\alpha\text{ is a holomorphic section of }\T_{\text{hol}}^{*}\Sigma^{\otimes2}\}$.

In local coordinates $z$ we have $\alpha(z)\dop z^2$ for $\alpha$ holomorphic.

One consequence of 19th century complex analysis, known as the Riemann-Roch theorem, implies that $\dim\mathrm{QD}(\sigma)=3g -3$.

\begin{thm*}[Hopf]

If $f \colon(\Sigma, \sigma)\to(M,h)$ is harmonic, then for a metric $g$ in the conformal class of $\sigma$ we have

\begin{align*}

f^{*}h = \alpha\dop z^2 + e(f) g + \bar\alpha\dop\bar z^2,

\end{align*}

where $\alpha\dop z^2$ is harmonic and $e(f)=\frac{1}{2}\|\dop f\|^2_{g,h}$, i.e. $\alpha=\alpha\dop z^2\in\mathrm{QD}(\sigma)$.

$\alpha$ is called the \CmMark{Hopf differential}.

\end{thm*}

\begin{thm*}[Mike Wolf]

Consider $\Phi\colon\mathcal M_{-1}/\Diff_0\Sigma\to\mathrm{QD}(\sigma), h \mapsto$ Hopf differential of $u_h \colon(\Sigma, \sigma)\to(\Sigma, h)$ is a diffeomorphism.

\end{thm*}

The strategy to proof this starts by noticing that $\mathcal M_{-1}/\Diff_0\Sigma$ is a finite dimensional manifold.

We want to show that $\Phi$ is injective and proper; thus, $\Phi$ is a covering map and hence $\mathrm{QD}(\sigma)$ is simply-connected which implies that $\Phi$ is a diffeomorphism.

Step 1: Injective (Sampson).

$u_h \colon(\Sigma, g)\to(\Sigma, h)$ with $\partial u \colon\T_{\text{hol}}\Sigma\to\T\Sigma\otimes_{\R}\C$ and $\dop u_{\C}(\frac{\partial}{\partial z})$.

Bochner formula for $\|\partial f\|^2$.

Fact 1 (Schoen-Yau). $\|\partial u\|^2$ is nowhere vanishing.