### Added draft of the second lecture of the third RTG day.

parent f7e4dc1a
 ... ... @@ -1021,6 +1021,161 @@ The key observation for an application of this is that, if $\Ric^g$ is a positiv The second is also not that hard and the last requires some work. \end{proof} \section{The Eells-Sampson existence theorem} \textbf{Story:} Given two manifolds $M,N$, is there a best map in a given free homotopy class $\beta \in [M,N]$, where $[M,N]$ denotes the free homotopy classes of smooth maps. From now on, best'' means harmonic with respect to some riemannian metric. \begin{expl*} If $M = S^n$, then it is a theorem that every homotopy class $\gamma \in [S^1,N]$ (for $N$ closed) admits a harmonic representative $\gamma \colon S^1 \to (N,h)$, i.e. is a closed geodesic. \end{expl*} \begin{expl*} What about $\dim(M) \geq 2$. In this case it depends on the curvature of $(N,h)$. Consider the flat torus $\mathbb T^2$ and the round sphere $\mathbb S^2$. For a dregree 1 map $\mathbb T^2 \to \mathbb S^2$ there is no harmonic map in the homotopy class (see the book by Lin on geometry of harmonic maps). \end{expl*} \begin{thm*}[Eells-Sampson 1964] Let $(M,g), (N,h)$ be closed manifolds and $h$ with non-positive sectional curvature. Then given any $f \colon M \to N$ $C^2$-map there exists a harmonic map $u \colon (M,g) \to (N,h)$ such that $u$ is freely homotopic to $f$. \end{thm*} Try to take $\tau(u) = 0$ for some $u \sim f$. In this approach $E \colon C^2(M,N) \to \R, f \mapsto \frac{1}{2} \int_M \|\dop f\|^2 dV_g$ such that $E(f_n) \to \inf_{f \in C^2}E(f)$, we would have to weaken to topology considering Sobolov spaces $W^{1,2}(M,N)$ The other approach using gradient flow goes as follows. Try to solve initival value problem (IVP). Let $f \colon M \times (0,\infty) \to N$, such that $\frac{\partial f}{\partial t} = \tau(f_t)$ and $f(\blank, 0) = f$. Recall the first variation of every $f_t \colon M \to N, \frac{\dop}{\dop t}f_t|_{t = 0} = 0$. Then $\delta E(\nu) = \frac{\dop}{\dop t}E(f_t)|_{t=0} = -\int_M \left<\tau(f),\nu\right>dV_g = -Q(\tau(f),\nu)$, where $\left<\blank,\blank\right>$ is the inner product on $f^{*}\T N$ induced by $h$. If we manage to solve $\frac{\partial f}{\partial t} = \tau(f)$, then \begin{align*} \frac{\dop}{\dop t} E(f_t)|_{t = t_0} = \int_M \left<\tau(f),\tau(f_t) \right> \dop V_g \leq 0 \end{align*} and equal to zero if and only if $\tau(f_{t_0}) = 0$. $\frac{\partial f^{\gamma}}{\partial t} = \Delta_gf^{\gamma} + \Gamma_{\alpha\beta}^{\gamma}(f)\partial_if^{\alpha}\partial_if^{\beta}g^{il}$. \subsection{1st short time existence} \begin{thm*} Suppose $f \colon M \to N$ is a $C^2$-map. Then there exists a $T_{\text{max}} > 0$ such that (IVP) \begin{align*} \frac{\partial f_t}{\partial t} = \tau(f_t) \text{ and } f_0 \equiv f \end{align*} has a solution on $[0, T_{\text{max}}]$. If $T_{\text{max}} < \infty$, then \begin{align*} \limsup_{t \nearrow T, x \in M}(f_t) = + \infty. \end{align*} \end{thm*} Note that there is no assumption on the curvature. \subsection{Need another Bochner formula} Let $(N,h)$ has non-positive sectional curvature and let $M$ be an $m$-dimensional manifold. Then we can calculate \begin{align*} & \frac{\partial}{\partial t} e(f_t) - \Delta_ge(f_t)\\ & \quad = -\underbrace{\|B_{f_t}\|^2}_{=\nabla \dop f_t} - \sum_{i=1}^n h(\sum_{j = 1}^m \dop f_t(\Ric^g(e_{i},e_j)e_j),\dop f_t(e_i))\\ & \qquad + \underbrace{\sum_{i,j = 1}^m h(R^h(\dop f_t(e_i), \dop f_t(e_j))\dop f_t(e_j),\dop f_t(e_i))}_{ \leq m}, \end{align*} where $\Ric^g \colon \T M \otimes \T M \to \R$ is the Ricci tensor, $R^h$ is the full curvature tensor of $(N,h)$ and $\{e_1, \ldots, e_m\}$ is an orthonormal frame of $N$. The latter summand is $\operatorname{const} \sec^h(\operatorname{span}(\dop f_t(e_i),\dop f_t(e_j)))$. We continue \begin{align*} \leq -\sum_{i = 1}^m h(\sum_{j=1}^m \dop f_t(\Ric^g(e_i,e_j)e_j),\dop f_t(e_i)) \leq C \sum_{i,j = 1}^m h(\dop f_t(e_i, \dop f_t(e_j)) \leq C e(f_t). \end{align*} Thus we get the following theorem. \begin{thm*} If $(N,h)$ has non-positive sectional curvature, then \begin{align*} \frac{\partial}{\partial t}e(f_t) - \Delta_ge(f_t) \leq C e(f_t). \end{align*} \end{thm*} \subsection{Moser-Harnack inequality} Let $z_0 = (x_0,t_0) \in M \times (0,T)$ and let $0 < R < \min\{ \text{inf radius of } g , t_0 \}$. The parabolic cylinder is given as follows \begin{align*} P_R(z_0) = \{ z = (x,t) \in M \times (0,\infty) \mid \dop g(x,x_0) < R, t_0 - R^2 \leq t \leq t_0 \}. \end{align*} \begin{thm*}[Moser] Suppose $v \in C^2(P_R(z_0))$ is non-negative and satisfies \begin{align*} \frac{\partial v}{\partial t} - \Delta_gv \leq Cv \text{ for } C > 0. \end{align*} Then there exists a $C_1 > 0$ such that \begin{align*} v(z_0) \leq C_1R^{2-m}\int_{P_R(z_0)}v \dop V_g \dop t. \end{align*} \end{thm*} If we apply this, we obtain \begin{align*} e(f_t)(z_0) \leq CR^{2-n} \int_{}\int_{} e(f_t) \dop V_g \dop t \leq C R^{2-n} \int_{t_0 - R^2 }^{t_0} E(f_t) \dop t \leq C R^{2-n} E(f) R^2. \end{align*} If $T_{\text{max}} < \infty$, then recall $\limsup_{t \nearrow T, x \in M} e(f_t) = +\infty$, but we have proved $e(f_t)$ is uniformly bounded, hence $T_{\text{max}} = + \infty$. \subsection{Black box \# 37} Since $e(f_t)$ is bounded for all time, elliptic regularity'' implies that for all $m > 0$ we have $\|\nabla^m \dop f_t\| \leq C_m$. For $f \colon M \times [0,\infty) \to N$ by Arzela-Ascoli, we know that there exists a subsequence $t_k \to \infty$ such that $f(\blank, t_k) \to u$ for $t_k \to \infty$ (in the sense of $C^2$-convergence). We calculate \begin{align*} \int_0^{t_0} \int_M \big\|\frac{\partial f}{\partial t}\big\|^2 \dop V_g\dop t & = \int_0^{t_0} \int_M \|\tau(f_t)\|^2 \dop V_g \dop t = - \int_0^{t_0} \frac{\partial E}{\partial t}(f_t)\dop t\\ & = -E(f_t) + E(f) \leq E(f) < \infty. \end{align*} Hence $\limsup_{t_0 \nearrow +\infty} \int_{t_0-2}^{t_0}\int_M \|\frac{\partial f}{\partial t}\|^2\dop V_g\dop t = 0$. Now one computes a Bochner formula for $\|\frac{\partial f}{\partial t}\|^2_{C^0}$. This yields an equality of the following form. For each $0 < < 1$ we have and each $t > 0$ \begin{align*} \|\tau(f_t)\|^2_{C^{\alpha}(M \times [t-1,t])} = \big\|\frac{\partial f}{\partial t}\big\|^2_{C^{\alpha}(M \times [t-1,t])} < C(\alpha) \big\|\frac{\partial f}{\partial t}\big\|^2_{L^2(M \times [t-2,t])} \xrightarrow{t \to +\infty} 0. \end{align*} Hence there exists a subsequence $t_i$ such that \begin{align*} \| \tau(f_i)\|_{C^{\alpha}} \xrightarrow{t_i \to \infty} 0 \end{align*} 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" ... ...
 ... ... @@ -6,16 +6,20 @@ % \includegraphics[scale=0.15]{mathe-logo.jpg} \vspace*{3cm} {\Large \textsc{Notes}}\\[0.8cm] {\huge \textsc{RTG Lectures}}\\[1.2cm] \textls{ {\Large \textsc{Notes taken at the}}\\[0.8cm] {\huge \textsc{RTG Lectures of the RTG 2227}}\\[2.2cm] {\Large \textsc{on the subjects of}}\\[1.8cm] {\Large \textsc{Torsion Invariants}}\\ \textsc{Prof. Dr. R. Sauer}\\[1.2cm] \textsc{by Prof. Dr. R. Sauer}\\[1.2cm] {\Large\textsc{Harmonic maps}}\\ \textsc{Dr. A. Sanders} \textsc{by Dr. A. Sanders} \vfill \textsc{Winter 2016/17} \textsc{\Large Karlsruhe and Heidelberg}\\[0.4cm] \textsc{\Large Winter 2016/17} } \vspace{3cm} \end{center} \end{titlepage} ... ...
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