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%version 1.1,\ \   07.04.2018, $p$ is even in 10.1 (Ivan Frolov)
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\pagestyle{fancy} 
\lhead{\tiny Hogde theory, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{10}{Hodge theory 10: Green operators}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\scriptsize
  {\bf Rules:} You may choose to solve only 
``hard'' exercises (marked with !, * and **) 
or ``ordinary'' ones (marked with ! or unmarked),
or both, if you want to have extra stuff to work.
To have a perfect score, a student must obtain
(in average) a score of 10 points per week.

If you have got credit for 2/3 of ordinary problems
or 2/3 of ``hard'' problems, you receive  
$6t$ points, where $t$ is a number depending on the
date when it is done. Passing all ``hard'' 
or all ``ordinary'' problems brings you $10t$ points.
Solving of ``**'' (extra hard) problems is not
obligatory, but each such problem gives you a credit
for 2 ``*'' or ``!'' problems in the ``hard'' set.

The first 3 weeks after giving a handout, $t=1.5$,
between 21 and 35 days, $t=1$, and afterwards, $t=0.7$.
The scores are not cumulative, only the
best score for each handout counts.
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$L^2_p$-metrics and differential operators}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this handout, the base manifold $M$ is
tacitly assumed compact.


\exercise
Let $|\cdot|^2_p$ be the usual $L^2_p$-metric on a 
torus $T^n$.
\enum
\ite[*] 
Let $D:= \sum_{P_\alpha} P_\alpha^2$ is the sum
of all differential monomials of degree $\leq p$.
Prove that the metric $|f|^2_p$ is equivalent
to the metric $|f|^2_\bullet:= \int_{T^n}(D(f), f)^2\Vol.$
\ite[!] Prove that this metric is equivalent to the metric
\[ 
 |f|^2_\circ := \int_{T^n}
\left(1+\sum_{i=1}^n\left|\frac{d^pf}{dt_i^p}\right|^2\right)\Vol
\]
where $t_1, ..., t_n$ are coordinates on $T^n$.
\ite[!] Prove that the map $f\arrow \sum_i \frac{d^pf}{dt_i^p}$
from $L^2_p(T^n)$ to $L^2(T^n)$ is Fredholm, if $p$ is even.
\ite[!] Let $D_1:\; C^\infty T^n \arrow C^\infty T^n$ be a 
differential operator which has the same symbol as $\sum_i \frac{d^p}{dt_i^p}$.
Prove that $D_1$ defines a Fredholm map from
$L^2_p(T^n)$ to $L^2(T^n)$, if $p$ is even.
\ite Let $\Delta:\; C^\infty T^n \arrow C^\infty T^n$ 
be the Laplace operator, $\Delta=\sum_i \frac{d^2}{dt_i^2}$.
Prove that $\Delta$  defines a Fredholm map from
$L^2_2(T^n)$ to $L^2(T^n)$.
\ee
\ez

\exercise[!]
Let $D:\; B \arrow B$ be a differential operator of
order $p$ such that the map 
$D:\; L^2_{p+i}(B)\arrow L^2_i(B)$ is Fredholm, and
$D_1$ an operator of order $<p$.
Prove that $D+D_1$ is also Fredholm.
\ez



\hint Use the Rellich lemma.
\eh


\exercise\label{_nabla^i_iso_embe_Exercise_}
Let $M$ be a compact Riemannian manifold, equipped with a connection
$\nabla$, and $B$ a vector bundle with metric and connection.
Consider the iterated connection 
$B \stackrel{\nabla^p}\arrow B \otimes \Lambda^1(M)^{\otimes p}$.
Prove that $\oplus_{i=0}^p\nabla^i$ defines an isometric embedding of vector spaces
$L^2_p(B) \arrow \bigoplus_{i=0}^p L^2(B \otimes \Lambda^1(M)^{\otimes i})$.
\ez

\exercise\label{_nabla^*nabla_symbol_Exercise_}
\enum
\ite Consider the differential operator $b\arrow (\nabla)^*\nabla b$.
Prove that it has the same symbol as the Laplace operator $\Delta$.
\ite[!]
In assumptions of the previous exercise,
let $D:\; L^2_{2p}(B)\arrow L^2(B)$ 
denote the differential operator $b\arrow (\nabla^p)^*\nabla^pb$.
Prove that the symbol of $D$ is the same as the symbol of $\Delta^p$.
\ee
\ez


\exercise
Let $D:\; B \arrow B$ be a differential operator 
of order $2p$ such that its symbol is the same as
of $p$-th power of the Laplace operator on a Riemannian
manifold $M$. Prove that
$D:\; L^2_{2p}(B)\arrow L^2(B)$ is Fredholm
\enum
\ite when $(B,\nabla)$ is trivial and $M$ is a torus with
  flat metric
\ite[!] when $(B,\nabla)$ is trivial and $M$ is a torus with
arbitrary metric
\ite[!] on arbitrary Riemannian manifold, for any $(B,
  \nabla)$
\ee
\ez

\hint Use Exercise 
\ref{_nabla^i_iso_embe_Exercise_}
and \ref{_nabla^*nabla_symbol_Exercise_}.
\eh

\exercise[!]
Let $D:\; B \arrow B$ be a differential operator of order $2p$
with the same symbol as $(\nabla^p)^*\nabla^p$.
Prove that $D:\; L^2_{2p+i}(B)\arrow L^2_i(B)$ 
is Fredholm for all $i$.
\ez

\exercise[*]
Let $D:\; B \arrow B$ be an elliptic operator of order $p$.
Prove that $D:\; L^2_{p+i}(B)\arrow L^2_i(B)$ 
is Fredholm for all $i$.
\ez




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Green operator}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Let $(B, \nabla)$ be a bundle with connection over a
Riemannian manifold $M$, and
 $\Delta:\; B \arrow B$ an operator with the same
symbol as $\nabla^*\nabla$, self-adjoint with respect to the $L^2$-metric.
\enum
\ite Prove that $\ker\Delta$ is finite-dimensional,
and $\im\Delta = \ker \Delta^\bot$ (orthogonal is
taken with respect to the $L^2$-metric)
\ite Prove that $\Delta\restrict {\im \Delta}$ is 
invertible in $L^2_p$-topology, for all $p$.
\ite Prove thta $L^2(B) = \im \Delta \oplus \ker \Delta$.
\ite[!] Prove that there exists 
{\bf the Green operator} \[ G_\Delta:\; L^2_{p}(B)\arrow
L^2_p(B)
\]
which is inverse to $\Delta$ on $\im \Delta$
and zero on $\ker \Delta$. 
\ite[!]
Prove that $G_\Delta:\; L^2(B)\arrow L^2(B)$ 
is compact and self-adjoint.
\ite[!]
Prove that $G_\Delta:\; L^2(B)\arrow L^2(B)$ 
can be diagonalized in an orthonormal Hilbert basis on $L^2_p(B)$.
\ee
\ez

\exercise
Identify $L^2_p(B)$ with a subspace in $L^{2}_{p-i}(B)$,
for any $i\geq 0$, using the continuous injective 
map $\Id:\; L^2_p(B)\arrow L^{2}_{p-i}(B)$.
\enum
\ite[!]
Prove that $\bigcap_{p=0}^\infty L^2_p(B)$ is identified with
the space $C^\infty B$ of smooth sections of $B$.
\ite[*] Consider the topology on $\bigcap_p L^2_p(B)$
induced from all $L^2_p$ (that is, a sequence
$b_i\in \bigcap_p L^2_p(B)$ converges if it converges
in $L^2_p$ for all $p$). Denote this topology by $L^2_\infty$.
Prove that  $b_i\in \bigcap_p L^2_p(B)=C^\infty B$
converges in $L^2_\infty$ if and only if the sequence
$\nabla^p b_i$ uniformly converges for any given $p$.
\ee
\ez

\exercise[!]
Let $(B, \nabla)$ be a bundle with connection over a
Riemannian manifold $M$, and
 $\Delta:\; B \arrow B$ an operator with the same
symbol as $\nabla^*\nabla$, self-adjoint with respect to the $L^2$-metric.
Prove that there exists an orthonormal basis in $L^2_0(B)$
diagonalizing $B$. Prove that all eigenvectors of $\Delta$
are smooth functions.
\ez
\hint
Use the previous exercise.
\eh


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Harmonic forms}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise[!]
Let $M$ be a Riemannian manifold, and
$\Delta:= dd^* + d^* d$ the Laplacian operators
on the differential forms. Prove that $\Delta$
has the same symbol as $\nabla^*\nabla$,
where $\nabla:\; \Lambda^* M \arrow \Lambda^* M \otimes \Lambda^1 M$
is the connection on the bundle of all differential forms.
\ez

\definition
A differential form is called {\bf harmonic}
if it lies in $\ker \Delta$.
\ed

\exercise
Prove that $\ker \Delta= \ker d \cap \ker d^*$.
\ez

\hint
\[
(\Delta\eta, \eta)= (dd^*\eta, \eta) + (d^*d\eta, \eta) = 
(d\eta, d\eta)+(d^*\eta, d^*\eta)
\]
\eh

\exercise
Prove that $\im \Delta= \im d + \im d^*$.
\ez

\exercise[!]
Prove that the image of $d$ and $d^*$ is
closed in $L^2$-topology on $\Lambda^*(M)$.
Prove that $\ker d^*=(\im d)^\bot$ and
$\ker d=(\im d^*)^\bot$
\ez

\exercise
Prove that $\Lambda^*(M)= \im d \oplus \im d^* \oplus \ker
\Delta$.
\ez

\exercise[!] Prove that
$\ker d = \im d \oplus \ker \Delta$.
\ez

\remark
From the previous exercise it follows that de Rham
cohomology of a (compact) manifold are identified with the space
of hermonic forms.
\er

\exercise[*]
Let $M$ be a Riemannian manifold (not necessarily
compact), and $\Lambda^i_0(M)$ be the sheaf of harmonic
$i$-form. Let $\R_M$ denote the constant sheaf. 
Prove that the complex of sheaves
\[
0 \arrow \R_M \hookrightarrow \Lambda^0_0(M) 
\stackrel d \arrow \Lambda^1_0(M) \stackrel d \arrow
\Lambda^2_0(M)
 \stackrel d \arrow
\]
is exact.
\ez

\exercise
Let $M$ be a compact Riemannian manifold with boundary.
\enum
\ite[*]
Prove that $\Delta:\; C^\infty M \arrow C^\infty M$ is
surjective.
\ite[**] Prove that $\Delta:\; \Lambda^i M \arrow \Lambda^i
M$ is surjective
for all $i$, or find a counterexample.
\ee
\ez



\end{document}