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\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{9}{Hodge theory 9: Elliptic 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{Hodge $*$ operator}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Using the Riemannian metric on a manifold
$M$, we can construct a natural metric on all 
tensor powers of the tangent and cotangent bundle,
in particular, on $\Lambda^*(M)$. We normalize the
metric on  $\Lambda^k(M)$ in such a way that
the basis \[ \xi_{i_1} \wedge \xi_{i_2} \wedge ... \wedge \xi_{i_k}, \ \ \ 
i_1 < i_2 < ... < i_k
\] 
is orthonormal, for an orthonormal frame $\xi_1, ..., \xi_n\in T^*M$.

\definition
Let $M$ be an oriented Riemannian manifold, 
and $\xi_1, ..., \xi_n$ an oriented orthonormal frame in $T^*M$.
{\bf The Riemannian volume form} $\Vol$ is 
$\xi_1 \wedge \xi_2 \wedge ... \wedge \xi_n$.
Further on, all Riemannian manifolds are assumed
oriented and equipped with Riemannian volume. 
\ed

\exercise
Let  $\eta\in \Lambda^k M$ be a differential form, and
$* \eta\in \Lambda^{n-k}M$ a form which satisfies 
$(\eta, \xi)\Vol M = *\eta \wedge \xi$ for any 
 $\xi\in \Lambda^k M$.
\enum
\ite
Prove that $*\eta$ is uniquely determined by this 
relation.
\ite Prove that $*\eta$ exists for any $\eta$.
\ite Prove that in the basis
$\xi_{i_1} \wedge \xi_{i_2} \wedge ... \wedge \xi_{i_k}$
defined above, the operator $*$ is written as follows:
\[
* \xi_{i_1} \wedge \xi_{i_2} \wedge ... \wedge \xi_{i_k}=
\sigma\xi_{j_1} \wedge \xi_{j_2} \wedge ... \wedge \xi_{j_{n-k}}
\]
where $\{j_1, j_2, ... j_{n-k}\}:= 
\{ 1, ..., n\}\backslash \{ i_1, i_2, ..., i_k\}$,
ordered by $j_1 < j_2 < ...$, and $\sigma=\pm 1$
is the signature of the permutation
$(j_1, j_2, ... j_{n-k}, i_1, i_2, ..., i_k)$.
\ee
\ez

\exercise
Let $\xi$ be a $k$-form, and $\eta$ an $(n-k)$-form.
Prove that
\enum
\ite  $(*\eta, \xi)= (-1)^{k (n-k)} (\eta, *\xi)$
\ite $*(*\eta) = (-1)^{k (n-k)}\eta$.
\ee
\ez

\exercise[!] Find the eigenvalues of $*$ on 
$\Lambda^{\frac n 2}M$ for even-dimensional $M$,
and dimension of the eigenspaces.
\ez

\exercise[*]
Let $M$ be a pseudo-Riemannian manifold with the
metric of signature $(n-s,s)$. Prove that
$*(*\eta)=(-1)^{k (n-k)+s}\eta$.
\ez

\exercise[!]
Define $d^*:\; \Lambda^k(M) \arrow \Lambda^{k-1}(M)$
using the formula $d^*: = (-1)^{(n+1)(k+1)} * d *$.
Prove that for any form $\xi$ with compact support,
one has \[
\int_M (d^*\eta, \xi)\Vol M = \int_M (\eta, d \xi)\Vol M.
\]
\ez

\hint Use the Stokes' formula:
\begin{align*}
\int_M(d^*\eta, \xi)\Vol M 
= & \int_M (d^*\eta, \xi)\Vol M = -\int_M d *\eta \wedge \xi\\ =&
\int_M *\eta \wedge d \xi = \int_M(\eta, d\xi)\Vol M
\end{align*}
\eh

\remark This implies that $d$ and $d^*$ are 
adjoint.
\er

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Laplace operator on differential forms}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Define the Laplace operator $\Delta$ on differential forms
using $\Delta \eta := dd^* \eta + d^* d\eta.$
\ed

\exercise
Let  \[ \Lambda^k M \otimes \Lambda^1 M
   \stackrel i\arrow \Lambda^{k-1}M
\]
be the ``interior multiplication'' map taking a tensor
$\eta \otimes \theta\in \Lambda^k M \otimes \Lambda^1$ to 
$\eta \cntrct \theta^\sharp$, where $\theta^\sharp$ 
is a vector field dual to $\theta$. Prove that
\[ 
   \iota(\eta \otimes \theta) = (-1)^{(n+1)(k+1)} *(*\eta \wedge \theta).
\]
\ez

\exercise
Consider the exterior multiplication map
\[ \Lambda^k M \otimes \Lambda^1 M
   \stackrel e\arrow \Lambda^{k+1}M.
\]
Let $\nabla$ be a Levi-Civita connection on a Riemannian
manifold $M$.
\enum
\ite Prove that $d\eta = e (\nabla \eta)$.
\ite Prove that $d^*\eta = \iota (\nabla \eta)$.
\ee
\ez

\hint
Prove that $d\eta = e (\nabla \eta)$
is equivalent to $\nabla$ being torsion-free.
\eh


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Elliptic operators}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $B$ be a vector bundle on $M$, and $\Tot(B)$ its total
space. {\bf Zero section} is the set of all zero vectors
in $\Tot B$.
\ed



\definition
Let $F, G$ be vector bundles on $M$. 
Consider the space $S^i(F,G):= \Diff^i(F,G)/\Diff^{i-1}(F,G)=
\Sym^i(TM)\otimes \Hom(F, G)$. {\bf Symbol} of a differential
operator is its class $\sigma(D)\in S^i(F,G)$.
The operator $D$ is called {\bf elliptic}
if $\rk F = \rk G$, and for any non-zero point $\theta\in \Tot(T^*_xM)$
the symbol $\sigma(D)$ evaluated at 
$\underbrace{\theta\otimes... \otimes \theta}_{\text{$i$ times}}$
is a non-degenerate as an element of $\Hom(F, G)\restrict x$.
\ed

\exercise
Let $D_1:\ F \arrow G$, $D_2:\; G \arrow H$
be differential operators, with $F, G, H$
vector bundles of the same rank. 
\enum
\ite 
Prove that the
composition $D_1 \circ D_2$ is elliptic if $D_2, D_2$ are elliptic.
\ite Prove that $D_1$ and $D_2$ are elliptic if $D_1\circ D_2$ is elliptic.
\ee
\ez

\exercise
Let $\Delta:\; C^\infty M \arrow C^\infty M$ be the
Laplace operator on a Riemannian manifold. Prove that it
is elliptic.
\ez

 \exercise[!]
Let $D:\; F \arrow G$ be an elliptic operator,
and $D^*$ is Hermitian adjoint. Prove that $D^*$
is also elliptic.
\ez

\exercise[!]
Let $\nabla:\; B \arrow B\otimes \Lambda^1 M$
be the connection, and 
$\nabla^*:\; B\otimes \Lambda^1 M\arrow B$ its Hermitian
adjoint.
Prove that $\nabla^*\nabla$ is elliptic, and its symbol
is equal to $-g\otimes \Id_B$, where $g\in \Sym^2 TM$
is the metric tensor. 
\ez

\exercise[!]
Consider the operator
$d+d^*:\; \Lambda^* M \arrow
\Lambda^* M$. Prove that it is elliptic.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Elliptic operators of second order}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Let $D:\; C^\infty M \arrow C^\infty M$ be an elliptic
operator of second order. Prove that $\sigma(D)\in \Sym^2(TM)$ is
a positive definite or negative definite bilinear symmetric form.
\ez

\remark From now till the end of this handout, 
$D:\; C^\infty M \arrow C^\infty M$ is a differential operator of second order,
the symbol of $D$ is positive definite, and $D(1)=0$.
\er

\exercise
Let $f\in C^\infty M$ be a function which has a local maximum 
in $z$. Prove that $D(f)(z) \leq 0$.
\ez

\exercise[!]
Let $f\in C^\infty M$, and $D(f)>0$.
Prove that $f$ cannot have a local maximum anywhere on $M$.
\ez

\exercise
Let
\begin{equation}\label{_elli_gene_Equation_}
D(f) = \sum_{i, j} A^{ij} \frac{d^2f}{dx_i dx_j} +
\sum_i B^i \frac{df}{dx_i},
\end{equation}
be an elliptic operator, $D(f)\geq 0$, and $\lambda>0$
a number which satisfies $\lambda A^{1,1}> -B^1$.
Let  $\phi_\epsilon:=\epsilon e^{\lambda x_1}$.
Prove that $D(f+ \phi_\epsilon)>0$ 
for any $\epsilon >0$.
\ez

\exercise
Let $D:\; C^\infty \R^n \arrow C^\infty\R^n$
be an elliptic operator defined as in
\eqref{_elli_gene_Equation_},
$\Omega\subset \R^n$ an open subset with
compact closure, and $f$ a function which reaches
its maximum in $\Omega$. 
\enum
\ite
Prove that one may chose $\lambda>0$ in such a way that 
$\lambda A^{1,1}> - B^1$ on $\Omega$.
\ite 
Let $\delta:= \sup_\Omega f - \sup_{\6\Omega} f>0$.
Chose $\epsilon$ in such a way that
$\sup_\Omega \phi_\epsilon< \frac \delta 2$.
Prove that $f+ \phi_\epsilon$ reaches its maximum
inside $\Omega$. 
\ite Prove that $D(f + \phi_\epsilon)> D(f)$.
\ee
\ez

\exercise[!]
(weak maximum principle for elliptic operators).\\
Let $D:\; C^\infty M \arrow C^\infty M$ be an elliptic
operator of second order, with $D(1)=0$ and positive
definite symbol, and $f\in C^\infty M$ a function
which satisfies $D(f)\geq 0$. Prove that $f$ cannot have
local maxima.
\ez

\hint Use the previous exercise
\eh

\exercise[*] (Strong Maximum Principle) \\
In the assumptions of the previous exercise,
prove that $f(m)< \sup_M f$ for any $m\in M$.
\ez



\end{document}