Commit c64bf571 authored by Jan-Bernhard Kordaß's avatar Jan-Bernhard Kordaß

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"
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......@@ -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[20]{
{\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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