Commit 15e2fdd0 authored by Jan-Bernhard Kordaß's avatar Jan-Bernhard Kordaß

Added Andy's 4th lecture.

parent 4cd68889
......@@ -1176,7 +1176,149 @@ and hence $0 = \lim \tau(f_i) = \tau(u)$.
Since we have $C^2$-convergence, we can conclude that $u \sim f$.
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\section{Harmonic maps and Teichmüller theory}
Classically Teichmüller theory belongs to complex analysis and has been studied for a long time from an analytic perspective.
But if we have a holomorphic function, then we know its real and imaginary part to be harmonic and we might hope to obtain a more general connection.
Let $\Sigma$ be a closed, oriented, connected smooth surface of genus $\geq 2$.
Further let $(M,h)$ be a riemannian manifold and let $f \colon (\Sigma, g) \to (M,h)$ be a smooth map and $U \colon \Sigma \to \R$ a $C^{\infty}$-function.
\begin{lemma*}
$\frac{1}{2}\int\|\dop f\|^2_{g,h}\dop V_g = E(f,g,h) = E(f,e^{2u}g,h)$.
\end{lemma*}
\begin{proof}
\begin{align*}
\|\dop f\|^2_{e^{2u}g,h} = e^{-2u}g^{ij}\partial_if^{\alpha}\partial_1f^{\beta}h_{\alpha\beta}(f) = e^{-2u}\|\dop f\|^2_{g,h}
\end{align*}
where $\dop V_{e^{2u}g} = \sqrt{\det(e^{2u}g_{ij})}\dop x^1\wedge \dop x^2 = e^{2u}\sqrt{\det(g_{ij})}\dop x^1 \wedge \dop x^2 = e^{2u}\dop V_g$ and thus
\begin{align*}
\|\dop f\|^2_{e^{2u}g,h} \dop V_{e^{2u}g} = \|\dop f\|^2_{g,h} \dop V_g.
\end{align*}
\end{proof}
\begin{cor*}
$f \colon (\Sigma, g) \to (M,h)$ 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 $f$.
\end{thm*}
We obtain a map
\begin{align*}
\psi \colon \mathcal M_{-1} \to C^{\infty}(\Sigma, \Sigma),
\quad
h \mapsto u_h
\end{align*}
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 $\mathrm{QD}(\sigma) = \{ \alpha \mid \alpha \text{ is a holomorphic section of } \T_{\text{hol}}^{*}\Sigma^{\otimes 2}\}$.
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.
\begin{align*}
\Delta_g\log \|\partial u\|^2 = 2 \|\partial u\|^2 - \frac{2}{\|\partial u\|^2} \|\alpha\|^2_g - 2.
\end{align*}
Assume: $u_1, u_2$ with the same Hopf differential
\begin{align*}
& 0 \geq \Delta_g(\log \|\partial u_1\|^2 - \log \|\partial u_2\|^2)\\
& \qquad = 2 \left( \|\partial u_1\|^2 - \| \partial u_2\|^2\right) - 2 \|\alpha\|_g^2 \left(\frac{1}{\|\partial u_1\|^2} - \frac{1}{\|\partial u_2\|^2}\right) > 0.
\end{align*}
Assume: $\|\partial u_1\| > \|\partial u_2\|$.
Thus, at a maximum of $\log \|\partial u_1\|^2 - \log \|\partial u_2\|^2 > 0$.
Hence $\|\partial u_1\| \leq \|u_2\|$.
Now do the same with $\log \|\partial u_2\| - \log \|\partial u_1\| \implies \|\partial u_1\| = \|\partial u_2\|$.
Similar calculations lead to $\|\dop u_1\|^2 = \|\dop u_2\|^2$ and another Bochner formula for $\eta \in \Diff_0\Sigma$ shows that $u_1 = u_2 \circ \eta$.
Step 2: Properness.
Note that the funcationals
\begin{align*}
& A \colon \mathcal M_{-1}/\Diff_0\Sigma \to \R,
\quad
h \mapsto \int \|\alpha\|_g^2 \dop V_g\\
& E \colon \mathcal M_{-1}/\Diff_0\Sigma \to \R,
\quad
h \mapsto E(u_h)
\end{align*}
are connected as follows
\begin{align*}
E(h) - 2\pi(2g-2) \leq A(h) \leq E(h) + 2\pi(2g -2).
\end{align*}
Thus, $A(h_n) \to \infty$ if and only if $E(h_n) \to \infty$.
Hence $\Phi$ is proper if and only if $E$ is proper.
\begin{thm*}[Wolf]
$E$ is proper, i.e. it takes a lot of energy to map dissimilar surfaces to one another.
\end{thm*}
Then, by the previous discussion $E$, hence $A$ and hence $\Phi$ is proper.
Thus, $\Phi$ is injective, proper and smooth and therefore a cover, but since $\mathrm{QD}(\sigma)$ is simply connected, it is a diffeomorphism.
\begin{cor*}
$\mathcal M_{-1}/\Diff_0\Sigma$ is a contractible manifold; in fact it is diffeomorphism to $C^{3g-3}$.
\end{cor*}
$\mathcal C(\Sigma) \hookrightarrow \mathcal M/C^{\infty}(\Sigma)$.
By uniformization, we have a section $\mathcal M/C^{\infty}(\Sigma) \to \mathcal M_{-1} \subset \mathcal M$ and thus
\begin{align*}
\mathrm{QD}(\sigma) \simeq \mathcal T(\Sigma) \simeq \mathcal C(\Sigma)/\Diff_0 \simeq \mathcal M_{-1}/\Diff_0(\Sigma) \simeq \C^{3g-3},
\end{align*}
where $\mathcal T(\Sigma)$ is Teichmüller space.
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