Commit 1627ea30 authored by Jan-Bernhard Kordaß's avatar Jan-Bernhard Kordaß
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Added Andy's section lecture.

parent 04ffa56d
......@@ -160,8 +160,12 @@ Thus, this number only depends on the homotopy class of $f$.
\chapter{Harmonic Maps [Andy Sanders]}
Also consider the notes at \href{www.mathi.uni-heidelberg.de/~asanders/harmonicmaps.htm}.
\section{Basics of harmonic maps}
In the following let every manifold be oriented (for integration safety reasons).
\subsection{Background differential geometry}
Let $E \to M$ be an $\R$-vector bundle over $M$ (second countable, hausdorff manifold) of rank $r$.
......@@ -181,7 +185,7 @@ where $\Omega^0(E)$ denotes smooth sections in $E$, such that
Let $q$ be an inner product on $E$.
We say that $\nabla$ is a \CmMark{metric connection} for $q$, if for all $s,t \in \Omega^0(E)$ we have
\begin{align*}
\dop q(s,t) = q(\nabla s,t) + q(s, \nabla t).
\dop q(s,t) = q(\nabla s,t) + q(s, \nabla t).
\end{align*}
\begin{expl*}
......@@ -310,9 +314,9 @@ With respect to these, we have
& = \int_M \left<\nabla^{f^{*}\T N}\nu, \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, \tau(f)\right> \dop V_g,
\end{align*}
where $(*)$ follows by a calculation in local co-ordinates.
= - \int_M \left< \nu, \tau(f)\right> \dop V_g,
\end{align*}
where $(*)$ follows by a calculation in local co-ordinates.
\end{proof}
\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^{
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:
%%% mode: latex
%%% TeX-master: "skript-rtg-lectures-ws1617"
......
......@@ -60,6 +60,8 @@
\DeclareMathOperator{\ad}{ad} % ad
\DeclareMathOperator{\Der}{Der} % Derivationen
\DeclareMathOperator{\Hess}{Hess} % Hessian
% abstract simplicial complexes
\DeclareMathOperator{\st}{st} % star
\DeclareMathOperator{\cl}{cl} % closure
......@@ -85,7 +87,10 @@
\DeclareMathOperator{\Aut}{Aut} % automorphisms
\DeclareMathOperator{\Sym}{Sym} % symmetric group
\DeclareMathOperator{\End}{End} % endomorphisms
\DeclareMathOperator{\Isom}{Isom} % isometries
\DeclareMathOperator{\Isom}{Isom} % isometriess
\DeclareMathOperator{\Imm}{Imm} % immersions
\DeclareMathOperator{\Vol}{Vol} % volume
\DeclareMathOperator{\Ext}{Ext} % Ext term
\DeclareMathOperator{\Tor}{Tor} % Tor term
......
......@@ -23,6 +23,20 @@
\usepackage{setspace}
\renewcommand{\baselinestretch}{1.1}
\usepackage{todonotes}
% get rid of rounded corners in todonotes
\makeatletter
\tikzstyle{notestyleraw}=[
draw=\@todonotes@currentbordercolor,
fill=\@todonotes@currentbackgroundcolor,
line width=0.6pt,
text width=\@todonotes@textwidth-1.6ex-1pt,
inner sep=0.8ex
]
\makeatother
% todonotes setup
\colorlet{draftnotesbgcolor}{orange}
\presetkeys{todonotes}{linecolor=draftnotesbgcolor, backgroundcolor=draftnotesbgcolor, bordercolor=draftnotesbgcolor, figcolor=white}{}
\usepackage{lecture-notes}
......
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