... ... @@ -1176,7 +1176,149 @@ and hence $0 = \lim \tau(f_i) = \tau(u)$. Since we have $C^2$-convergence, we can conclude that $u \sim f$. %%% Local Variables: %%% mode: latex %%% TeX-master: "skript-rtg-lectures-ws1617" %%% End: \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.