Commit 1627ea30 by Jan-Bernhard Kordaß

 ... @@ -160,8 +160,12 @@ Thus, this number only depends on the homotopy class of $f$. ... @@ -160,8 +160,12 @@ Thus, this number only depends on the homotopy class of $f$. \chapter{Harmonic Maps [Andy Sanders]} \chapter{Harmonic Maps [Andy Sanders]} Also consider the notes at \href{www.mathi.uni-heidelberg.de/~asanders/harmonicmaps.htm}. \section{Basics of harmonic maps} \section{Basics of harmonic maps} In the following let every manifold be oriented (for integration safety reasons). \subsection{Background differential geometry} \subsection{Background differential geometry} Let $E \to M$ be an $\R$-vector bundle over $M$ (second countable, hausdorff manifold) of rank $r$. Let $E \to M$ be an $\R$-vector bundle over $M$ (second countable, hausdorff manifold) of rank $r$. ... @@ -311,8 +315,8 @@ With respect to these, we have ... @@ -311,8 +315,8 @@ With respect to these, we have \overset{(*)}{=} \int_M \left<\nu, \delta^{\nabla^{f^{*}\T N}} \dop f\right> \dop V_g\\ \overset{(*)}{=} \int_M \left<\nu, \delta^{\nabla^{f^{*}\T N}} \dop f\right> \dop V_g\\ & = -\int_M \left< \nu, \tr_g(\nabla \dop f) \right> \dop V_g & = -\int_M \left< \nu, \tr_g(\nabla \dop f) \right> \dop V_g = - \int_M \left< \nu, \tau(f)\right> \dop V_g, = - \int_M \left< \nu, \tau(f)\right> \dop V_g, \end{align*} \end{align*} where $(*)$ follows by a calculation in local co-ordinates. where $(*)$ follows by a calculation in local co-ordinates. \end{proof} \end{proof} \begin{cor*}[Fundamental theorem of the calculus of variations] \begin{cor*}[Fundamental theorem of the calculus of variations] ... @@ -335,6 +339,187 @@ Then $\dop f = \partial_if^{alpha} \dop x^i \otimes \frac{\partial}{\partial y^{ ... @@ -335,6 +339,187 @@ Then$\dop f = \partial_if^{alpha} \dop x^i \otimes \frac{\partial}{\partial y^{ Thus $\tau(f) = (\Delta_gf^{\gamma} + \Gamma_{\alpha\beta}^{\gamma}(f) \partial_if^{\alpha}\partial_jf^{\beta}g^{ij})$. Thus $\tau(f) = (\Delta_gf^{\gamma} + \Gamma_{\alpha\beta}^{\gamma}(f) \partial_if^{\alpha}\partial_jf^{\beta}g^{ij})$. \section{Example and the Bochner formula (a glimpse of rigidity)} Recall that above we considered $C^2$-maps $f \colon (M,g) \to (N,h)$ with tension field \begin{align*} \tau(f) := \tr_g(\nabla \dop f) = 0 \in \Omega^0(M,f^*\T N). \end{align*} In local co-ordinates $\{x^i\}$ on $M$ and $\{y^{\alpha}\}$ on $N$ this means \footnote{Use roman indices for the $M$ and greek ones for $N$.} \begin{align*} \tau(f)^{\gamma} \frac{\partial}{\partial y^{\gamma}} = (\Delta_g f^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(f) \partial_if^{\alpha}\partial_jf^{\beta}g^{ij})\partial_{\gamma} = 0, \end{align*} where $\tilde \Gamma$ are the Christoffel symbols on $(N,h)$. \begin{expl*} \begin{enumerate}[label=\Roman*.] \item Let $(M,g) = (\R, \dop t^2)$ and let $\eta \colon \R \to (N,h)$. From above we know that the Laplace-Beltrami operator here reads \begin{align*} \Delta_gf = g^{ij} (\partial_i\partial_jf - \Gamma_{ij}^k\partial_kf) = g^{ij}(\partial_i\partial_jf) = \partial_t^2f \end{align*} for \begin{align*} \Gamma_{ij}^k = \frac{g^{km}}{2}(\partial_ig_{im} + \partial_jg_{im} - \partial_m(g_{ij}) \end{align*} ($=0$ in $\{g_{ij}\}$ is constant)\todo{repair} and $\{g_{ij}\} = g_{11} = f(\partial_t,\partial_t) = \dop t^2(\partial_t,\partial_t) = 1$. Hence \begin{align*} \tau(\eta)^{\gamma}\partial_{\gamma} = (\ddot \eta^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(\eta) \dot\eta^{\alpha}\dot\eta^{\beta})\partial_{\gamma} = 0, \end{align*} which is if and only if $\eta$ is a geodesic, i.e. the covariant derivate along $M$ of the curves speed vanishes: $\frac{\Dop}{\dop t} \dot \eta = 0$ and thus $E(\eta)|_a^b = \frac{1}{2}\int_a^b\|\dot\eta\|^2\dop t$. \item Now let $f \colon (M,g) \to \R$. Here $\tau(f) = \Delta_gf = 0$. \begin{prop*} If $M$ is closed, then the energy harmonic functions are constant. \end{prop*} \begin{proof} By Green's theorem (integration by parts) \begin{align*} \int_M \underbrace{g(\nabla f, \nabla f)}_{=\|\nabla f\|^2} \dop V_g = - \int \Delta_g f \cdot f \dop V_g = 0 \end{align*} for $\dop V_g = \sqrt{\det (\{g_{ij}\})} \dop x^1 \wedge \cdots \wedge \dop x^n$. Thus $\|\nabla f\|^2 = 0$ and $f$ must be constant. \end{proof} In our example, this shows $\Delta_g f = \lambda f$. \item Let $f \colon (M,g) \to (N,h)$ be an isometric immersion, i.e. $\dop f$ is injective and $g = f^{*}h = h(\dop f, \dop f)$. Then we have \begin{align*} e(f) & = \frac{1}{2} \|\dop f\|^2 = \frac{1}{2}h_{\alpha\beta} \partial_if^{\alpha}\partial_jf^{\beta}g^{ij} = \frac{1}{2}\partial_if^{\alpha}\partial_jf^{\beta}h(\partial_{\alpha},\partial_{\beta}) g^{ij}\\ & = \frac{1}{2}h(\partial_if^{\alpha}\partial_{\alpha},\partial_jf^{\beta}\partial_{\beta})g^{ij} = \frac{1}{2}h(\dop f(\partial_i),\dop f(\partial_j)) g^{ij} = \frac{1}{2}g_{ij}g^{ij} = \frac{m}{2} \end{align*} and hence $E(f) = \frac{m}{2}\Vol(f)$, where $\Vol(f) = \int_M\dop V_{f^{*}h} = \int_M\dop V_g$. This shows that $f$ is critical for $E$ if and only if $f$ is critical for $\Vol \colon \Imm(M,N) \to \R_+$. The latter is clearly if and only if $f$ is a \textbf{minimal submanifold}. Examples of minimal submanifolds in $\R^3$ include the 2-plane, or the helicoid. \end{enumerate} \end{expl*} \subsection{Composition laws for harmonic maps} Consider the composition \begin{align*} (M,g) \xrightarrow{f} (N,h) \xrightarrow{u} (Z,b). \end{align*} In general, if $f,u$ are harmonic, this needs not be harmonic again, which can be considered a bug or a feature''. \begin{align*} B_{u \circ f}(X,Y) = B_u(\dop f(X), \dop f(Y)) + \dop u (B_f(X,Y)) \end{align*} for $X,Y \in \T_pM$ and thus $B_{u \circ f} = \nabla^{\T^{*}M \otimes (u \circ f)^{*}\T N}(\dop(u \circ f))$. Hence $\tau(u \circ f) = \dop (\tau(f)) + \tr_g(f^{*}B_u)$. If $f$ is harmonic, then $\tau(u \circ f) = \tr_g(f^{*}B_u)$. \begin{prop*} If $f \colon M \to N$, is harmonic and $u \colon N \to Z$ is totally geodesic, i.e. $B_u = 0$. Then $u \circ f$ is harmonic. \end{prop*} What if $u \colon N \to \R$ is a function and $f$ is harmonic? Then \begin{align*} \tau(u \circ f) = \tr_g(f^{*}B_u) = \tr_g(f^{*}(\Hess(u)) = \sum_{i = 1}^n f^{*}(\Hess(u)) (E_i,E_i). \end{align*} Recall that a function $u \colon (N,h) \to \R$ is convex, if $\Hess(u)$ is positive definite. If $f$ is harmonic and $u$ is convex, then $\tau(u \circ f) = \nabla_g u \circ f \geq 0$ (these are called \CmMark{subharmonic functions}). \begin{thm*} A map is harmonic if and only if it pulls back germs of convex functions to germs of subharmonic functions. \end{thm*} There are various useful applications of the synthetic view'' on harmonic functions (e.g. Gromov-Shane). \begin{thm*} Suppose $(M,g)$ is closed, connected and $(N,h)$ is $1$-connected with non-positive curvature. Then every harmonic map $f \colon (M,g) \to (N,h)$ is constant. \end{thm*} \begin{proof} The distance function $N \to \R_{\geq 0}, x \mapsto \dop_N(p,x)^2$ for every $p \in N$ is actually smooth and strictly convex, e.g. $\dop_{\R^n}(0,x)^2 = x_1^2 + \cdots + x_n^2$. In case $f$ is harmonic, we have \begin{align*} \Delta_gu \circ f = \tau(u \circ f) = \tr_g(f^{*}B_u) \geq 0 \end{align*} and \begin{align*} -\int \| \dop(u \circ f)\|^2 \dop V_g = \int_M \Delta_gu \circ f \dop V_g \geq 0. \end{align*} Thus $\|\dop (u \circ f)\| = 0$ and hence $u \circ f$ is constant. \end{proof} \subsection{Bochner formulas} Let $(E,\nabla,a)$ be a riemannina vector bundle, i.e. $a$ is a metric on $E$, $\nabla$ is a connection on $E$ preserving $a$ ($\nabla a = 0$) and there is a vector bundle projection map $E \to (M,g)$. Let $\omega \in \Omega^p(M,E)$ and let $\nabla$ be a connection on $\Omega^p(M,E)$. \begin{align*} \hat\nabla \colon \Omega^p(M,E) \to \Omega^p(M,\T^{*}M \otimes \T^{*}M \otimes E), \quad \omega \mapsto ( (X,Y) \mapsto \nabla_X\nabla_Y\omega - \nabla_{\nabla_XY}\omega ) \end{align*} The \CmMark{trace Laplacian} is the operator \begin{align*} \nabla^2 \colon \Omega^p(M,E) \to \Omega^p(M,E), \quad \omega \mapsto \tr_g(\hat\nabla \omega). \end{align*} Recall that the \CmMark{Hodge Laplacian} was the operator \begin{align*} \dop^{\nabla} \colon \Omega^p(M,E) \to \Omega^{p+1}(M,E), \quad \alpha \otimes u \mapsto \dop \alpha \otimes u + (-1)^p \alpha \wedge \nabla u. \end{align*} With respect to the $L^2$-pairing $\beta \otimes v \mapsto \int_Mg(\alpha,\beta) a(u,v)\dop V_g$ it has a formal adjoint \begin{align*} \delta^{\nabla}\colon \Omega^{p+1}(M,E) \to \Omega^p(M,E). \end{align*} The Hodge Laplacian is the degree preserving operator given by $\dop^{\nabla} \circ \delta^{\nabla} + \delta^{\nabla} \circ \dop^{\nabla} =: \Delta_a$. The \CmMark[Bochner-Lichnerowicz formula]{(generalized) Bochner-Lichnerowicz formula} is given by \begin{align*} \nabla_a \omega = - \nabla^2\omega + S_{\omega}. \end{align*} for $S_{\omega} \in \Omega^p(M,E)$ with \begin{align*} S_{\omega}(X_1, \cdots, X_p) = \sum_{k = 1}^p\sum_{i = 1}^m(-1)^k(R^{\tilde \nabla}(e_i, X_k)\omega) (e_i,X_1, \ldots, \hat X_k, \ldots, X_n) \end{align*} for $X_i \in \T_pM$, $m = \dim M$ and $\{e_i\}$ an orthonormal frame around $p$.\footnote{Hat ($\hat X_k$), as always, means to omit the k-th term.} \begin{cor*} Let $f \colon (M,g) \to (N,h)$ be harmonic. Then \begin{align*} \Delta_ge(f) = \|B_f\|^2 - \sum_{ij}\underbrace{h(R^h(f_{*}e_i,f_{*}e_j) f_{*}e_j, f_{*}e_i))}_{= \lambda \sec(e_i,e_j)} + \sum_ih(f_{*}(\Ric^g(e_i)),f_ke_i) \end{align*} for an orthonormal frame $\{e_i\}$. \end{cor*} The key observation for an application of this is that, if $\Ric^g$ is a positive operator, then the latter sum is positive. \begin{thm*}[Eells-Sampson] Let $(M,g)$ be a closed with non-negative Ricci curvature and let $(N,h)$ have non-positive sectional curvature. \begin{enumerate}[label=(\roman*)] \item Then any harmonic map $f \colon (M,g) \to (N,h)$ is totally geodesic, i.e. $\nabla \dop f = B_f = 0$. \item If $\Ric^g$ is positive at any point, then $f$ is constant. \item If the sectional curvature of $(N,h)$ is strictly negative, then $f$ is constant or $f(M)$ is closed geodesic. \end{enumerate} \end{thm*} \begin{proof} The first statement easily follows from the corollary and $\int_M \left< \nabla u, \nabla v\right> \dop V_g = - \int_M \Delta u \cdot v \dop V_g$. The second is also not that hard and the last requires some work. \end{proof} %%% Local Variables: %%% Local Variables: %%% mode: latex %%% mode: latex %%% TeX-master: "skript-rtg-lectures-ws1617" %%% TeX-master: "skript-rtg-lectures-ws1617" ... ...