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periodic manifolds, covering manifolds, band-structure, laplacian
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%-------------------------------------------------------------------
% gaps.tex          Periodic Manifolds with Spectral Gaps
%                   Olaf Post
% 2000-10-31        Institut f"ur Reine und Angewandte Mathematik
%                   RWTH Aachen, Germany
% last change       email: post@iram.rwth-aachen.de 
% 2002-07-12
%--------------------------------------------------------------------

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\begin{document}

\title{Periodic Manifolds with Spectral Gaps}


\author{Olaf Post}

\address{Institut f\"ur Reine und Angewandte Mathematik,
       Rheinisch-Westf\"alische Technische Hochschule Aachen,
       Templergraben 55,
       52062 Aachen,
       Germany}
\email{post@iram.rwth-aachen.de}

\date{05.12.2001}

%------------------------------------------------------------
% Abstract. 
%------------------------------------------------------------

\begin{abstract}
  We investigate spectral properties of the Laplace operator on a
  class of non-compact Riemannian manifolds. For a given number $N$ we
  construct periodic manifolds such that the essential spectrum of the
  corresponding Laplacian has at least $N$ open gaps.  We use two different
  methods.  First, we construct a periodic manifold starting from an infinite
  number of copies of a compact manifold, connected by small cylinders. In the
  second construction we begin with a periodic manifold which will be
  conformally deformed. In both constructions, a decoupling of the different
  period cells is responsible for the gaps.
\end{abstract}

%------------------------------------------------------------
% Keywords
%------------------------------------------------------------

\keywords{Laplacian on a Riemannian manifold, spectral gaps, periodic
  manifolds, operation of a discrete group on a manifold}

\maketitle

%------------------------------------------------------------
% The main part
%------------------------------------------------------------

\section{Introduction}

The spectra of periodic Schr\"odinger or divergence type operators have been
extensively studied. In particular, it is well-known that the spectrum of a
periodic elliptic operator on $L_2(\R^d)$ with smooth coefficients is the
locally finite union of compact intervals, called \emph{(spectral) bands}. We
are mainly interested in the question whether these bands are separated by
\emph{spectral gaps} or not.  By a \emph{gap} in the (essential) spectrum of a
positive operator $H$ we mean an interval $(a,b)$ such that
\begin{displaymath}
  (a,b) \cap \spec H = \emptyset.
\end{displaymath}
To exclude trivial cases we assume that $a$ is greater than the infimum of the
essential spectrum of $H$. The number of gaps is given by the number of
components of the intersection of the resolvent set $\C \setminus \spec H$
with $\R$. Results on spectral gaps where $H$ is a Schr\"odinger or divergence
type operator can be found for example in~\cite{hempel:92b, hempel-herbst:95a,
  hempel-herbst:95b, hempel-lienau:00} (see also the references therein).

In this paper, we want to give examples of (non-compact) periodic manifolds
$\Mper$ such that the corresponding Laplacian $\laplacian \Mper$
\emph{without} potential has spectral gaps. Here, \emph{periodicity} means
that a finitely generated abelian group $\Gamma$ acts isometrically and
properly discontinuously on $\Mper$ (cf.\ for example~\cite{atiyah:76,
  bruening:92, chavel:93, donnelly:81, sunada:89}, periodic manifold are also
called \emph{covering manifolds}). Therefore we obtain the same qualitative
results \emph{only} by the periodic geometry.

As in the Schr\"odinger operator case a decoupling of the different period
cells is responsible for the gaps.  In the Schr\"odinger operator case, the
decoupling is achieved by a high potential barrier separating each period cell
from the others. In the geometric case, decoupling means that the junction
between two period cells is small. Here, a period cell is the closure of a
fundamental domain (see the next section).

From a physical point of view the Laplacian on a manifold is the Hamiltonian
of an electron confined to this manifold (at least in a semi-classical limit,
cf.~for example~\cite{froese-herbst:00, mitchell:01}). Periodic structures
like a periodically curved cylinder or a quantum wire could have applications
in solid state physics. A quantum wire (or quantum wave guide) is a planar
strip, cf.~for example~\cite{exner-seba:89}. The knowledge of the
band-gap structure of $\spec H$ is important for the conductive properties of
the periodic material described by the Hamiltonian $H$. In particular, the
existence and size of the first gap decide whether the material is a conductor
or an insulator.

\subsection*{Basic ideas and results}
The construction of $\Gamma$-periodic manifolds with spectral gaps will be
given later on in detail (cf.~Section~\ref{sec:constr}). Here, we sketch the
\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \begin{picture}(0,0)
       \includegraphics{figure-1}
    \end{picture}
    \setlength{\unitlength}{4144sp}%
    \begin{picture}(4839,1797)(34,-1006)
      \put(421, -1){$\Xeps$}
      \put(1636,-171){$X$}
      \put(3066,  4){ $\Xeps$}
      \put(4366,-106){$\Meps$}
      \put(4279,659){$A_\eps^2$}
      \put(2656,659){$A_\eps^1$}
      \put(4501,-961){$\Mepsper$}
      \put(4816,254){$\eps$}
    \end{picture}
    \caption{Construction of the period cell $\Meps$ and the periodic manifold
     $\Mepsper$ with $\Gamma=\Z$.}
    \label{fig:gaps1}
  \end{center}
\end{figure}
ideas and fix the notation. We start our construction from a compact
Riemannian manifold $X$ of dimension $d \ge 2$ (for simplicity without
boundary). Here, $\Gamma$ is an abelian group with $r$ generators. We choose
$2r$ different points $x_1,\dots,x_{2r} \in X$ and endow each point $x_i$ with
a cylindrical end $A_\eps^i$ (the boundary being isometric to a sphere of
radius $\eps>0$). We call the resulting manifold $\Meps$.
By glueing together $\Gamma$ copies of the period cell $\Meps$ we obtain a
$\Gamma$-periodic manifold $\Mepsper$ (see Figure~\ref{fig:gaps1} for $r=1$
generators). Our first result is the following:
\begin{theorem}
  \label{thm:gaps.constr}
  Each spectral band of the periodic Laplacian $\laplacian \Mepsper$ on
  $\Mepsper$ tends to an eigenvalue of the Laplacian $\laplacian X$ on $X$ as
  $\eps \to 0$. In particular, for each $N \in \N$ there exist at least $N$
  gaps in the spectrum of $\laplacian \Mepsper$ provided $\eps$ is small
  enough.
\end{theorem}
We also prove a similar result in the case where the cells $\Meps$ are joined
by long thin cylinders of fixed length. Let $\Neps$ be the manifold obtained
from $\Meps$ by identifying $\bd A_\eps^{2i-1}$ with a component of the
boundary of a cylinder $C_\eps^i$ of length $L_i \ge 0$ and radius $\eps>0$.
The periodic manifold $\Nepsper$ is obtained in the same way by glueing
together $\Gamma$ copies of $\Neps$.
\begin{theorem}
  \label{thm:gaps.cyl}
  Each spectral band of the periodic Laplacian $\laplacian \Nepsper$ on
  $\Nepsper$ tends to an eigenvalue of the Laplacian $\laplacian X$ on $X$ or
  to an eigenvalue of the Laplacian with Dirichlet boundary conditions on
  $[0,L_i]$ for an $i=1,\dots,r$ with $L_i>0$ as $\eps \to 0$. In particular,
  for each $N \in \N$ there exist at least $N$ gaps in the spectrum of
  $\laplacian \Nepsper$ provided $\eps$ is small enough.
\end{theorem}
The proofs of these two results will be given in Section~\ref{sec:constr}.
Both results are related to articles of Chavel and
Feldman~\cite{chavel-feldman:81} and Ann\'e~\cite{anne:87} where compact
manifolds with small handles resp.\ compact manifolds joined by small
cylinders are analysed. Note that the joined manifolds in~\cite{anne:87} are
not smooth in contrast to our construction.

The second construction is in some sense the reverse of the first construction.  Starting with a given periodic
manifold $\Mper$ of dimension $d \ge 2$ we deform the (periodic) metric $g$ by
a (periodic) conformal factor $\rho_\eps$ to obtain spectral gaps in the
spectrum of the Laplacian. The idea is to let the conformal factor converge to
the indicator function of a closed periodic set $\Xper = \bigcup_{\gamma \in
  \Gamma} \gamma X$ where $X \subset \Mper$ is a closed subset disjoint from
all translates $\gamma X$, $\gamma \ne 0$. This convergence is of course not
uniform because of the discontinuity of the indicator function. We denote the
manifold $\Mper$ with metric $\rho_\eps^2 g$ by $\Mepsper$. We have the
following result:
\begin{theorem}
  \label{thm:gaps.conf}
  Suppose that the dimension of $\Mper$ is greater or equal to $3$.  Then each
  band of the periodic Laplacian $\laplacian \Mepsper$ on the conformally
  deformed periodic manifold $\Mepsper$ tends to an eigenvalue of the Neumann
  Laplacian $\laplacianN X$ on $X$ as $\eps \to 0$. In particular, for each $N
  \in \N$ there exist at least $N$ gaps in the spectrum of $\laplacian
  \Mepsper$ provided $\eps$ is small enough.
\end{theorem}
The assumption $d \ge 3$ is essential here, since in dimension $2$ it is no
longer true that the first band of $\laplacian \Mepsper$ tends to a single
point as $\eps \to 0$. Even in the limit case $\eps=0$, a nontrivial interval
remains due to the special structure of the conformal Laplacian, cf.\ 
Equation~\eqref{eq:quot.dim2}.  Nevertheless, we can prove the existence of
gaps in a simple example by direct calculations (see Example~\ref{ex:dim.2}).
The proof of Theorem~\ref{thm:gaps.conf} and the example in dimension $2$ can
be found in Section~\ref{sec:conf}.

The proofs of our theorems basically use the variational max-min
characterisation of the eigenvalues of an operator with purely discrete
spectrum (called Min-max Principle) to compare the eigenvalues of operators
defined on parameter-depending Hilbert spaces (see the Main
Lemma~\ref{lem:main.lemma}).  The basic idea is taken from \cite{anne:87} and
\cite{fukaya:87} even if the Main Lemma is more general.  Furthermore, Floquet
Theory allows to analyse the spectrum of a periodic operator. Details are
given in the next section.


Davies and Harrell~II~\cite{davies-harrell:87} proved the existence of at
least one gap in the periodic conformally flat case (transforming the
conformal Laplacian in a corresponding Schr\"odinger operator). This result is
a special case of Theorem~\ref{thm:gaps.conf}. Using similar methods, Green
showed in~\cite{green:97} the existence of a finite number of gaps in the
$2$-dimensional conformally flat case. Furthermore,
Yoshitomi~\cite{yoshitomi:98} proved the existence of spectral gaps for the
(Dirichlet) Laplacian on periodically curved quantum wave guides. In all these
three papers the existence of gaps is established basically by analysing a
one-dimensional problem. Here, in contrast, we directly study the
multi-dimensional problem.

Green conjectured that a necessary requirement for a large number of gaps is
for the curvature to be large in absolute value at some points. In the
examples given here the same phenomenon occurs (see
Remarks~\ref{rem:constr.curv} and~\ref{rem:conf.curv}).  Therefore the results
of Fukaya~\cite{fukaya:87} cannot be applied here. Fukaya showed the
continuity of the $k$-th eigenvalue of the Laplacian on $\Meps$ where $\Meps$
is a convergent family of manifolds (in a certain sense) with given bound on
the curvature.


There are also results on periodic operators on manifolds with
\emph{non-commutative} groups $\Gamma$. For example, Br\"uning and Sunada
proved in~\cite{bruening:92} that the Laplacian on a periodic manifold still
has band structure even for certain non-commutative groups $\Gamma$.
Furthermore, Sunada~\cite{sunada:89} showed that --- in contrast to the
Schr\"odinger operator case on $\R^d$ (cf.~\cite{reed-simon-4}) --- there
exist (non-compact!) periodic manifolds with commutative group $\Gamma$ such
that the corresponding Laplacian has an eigenvalue (possibly embedded in
another band).

Finally note that Lott~\cite{lott:01} has constructed a (non-periodic)
$2$-di\-mensio\-nal complete non-compact finite-volume manifold such that the
corresponding Laplacian has an infinite number of gaps.  In the periodic case
in contrast we would expect that the Generalized Bethe-Sommerfeld conjecture
is true, i.e. that there are only finitely many gaps in the spectrum of
$\laplacian \Mper$ if $d \ge 2$.  Skriganov~\cite{skriganov:79} proved this
conjecture for periodic Schr\"odinger operators in Euclidean space.
Furthermore, an asymptotic upper bound on the number of gaps have been
established in~\cite{bruening:92} (not implying the Generalized
Bethe-Sommerfeld conjecture). Note that our results do not say anything
whether the number of gaps is finite or not.

%------------------------------------------------------------
\section{Preliminaries}

%------------------------------------------------------------
\subsection*{Laplacian on a manifold}

Throughout this article we study manifolds of dimension $d \ge 2$. For a
Riemannian manifold $M$ (compact or not) without boundary we denote by $\Lsqr
M$ the usual $L_2$-space of square integrable functions on $M$ with respect to
the volume measure on $M$. Locally in a chart, the volume measure has the
density $(\det g)^\frac12$ with respect to the Lebesgue measure, where $\det
g$ is the determinant of the metric tensor $(g_{ij})$ in this chart. The norm
of $\Lsqr M$ will be denoted by $\norm[M] \cdot$. For $u \in \Cci M$, the
space of compactly supported smooth functions, we set
\begin{displaymath}
  \check q_M(u):=\int_{M} |\dd u|^2.
\end{displaymath}
\sloppy Here $\dd u$ denotes the exterior derivate of $u$, which is a section
in the cotangent bundle $T^* \! M$ over $M$. Locally in a chart, $| \dd u |^2=
\sum_{i,j} g^{ij} \partial_i u \, \partial_j \overline u$ where $(g^{ij})$ is
the inverse of $(g_{ij})$.

We denote the closure of the non-negative quadratic form $\check q_M$ by
$q_M$.  Note that the domain $\dom q_M$ of the closed quadratic form $q_M$
consists of those functions in $L_2(M)$ such that the weak derivative $\dd u$
is also square integrable (i.e., $q_M(u) < \infty$).

We define the \emph{Laplacian} $\laplacian M$ (for a manifold without
boundary) as the unique self-adjoint and non-negative operator associated with
the closed quadratic form $q_M$, i.e., operator and quadratic form are related
by
\begin{displaymath}
  q_M(u)=\iprod {\laplacian M u} u
\end{displaymath}
for $u \in \Cci M$ (for details on quadratic forms see
e.g.~\cite[Chapter~VI]{kato:66}, \cite{reed-simon-1} or \cite{davies:96}).

If $M$ is a compact manifold with (piecewise) smooth boundary $\bd M \ne
\emptyset$ we can define the Laplacian with \emph{Dirichlet} resp.\ 
\emph{Neumann boundary conditions} in the same way. Here, we start from the
(closure of the) quadratic form $\check q_M$ defined on $\Cci M$, the space of
smooth functions with support \emph{away} from the boundary, resp.\ on $\Ci
M$, the space of functions smooth up to the boundary. We denote the closure of
the quadratic form by $q_M^\Dir$ resp.\ $q_M^\Neu$ and the corresponding
operator by $\laplacianD M$ resp.\ $\laplacianN M$.

If $M$ is compact the spectrum of $\laplacian M$ (with any boundary condition
if $\bd M \ne \emptyset$) is purely discrete. We denote the corresponding
eigenvalues by $\EW k M$ (resp.\ $\EWD k M$ or $\EWN k M$ in the Dirichlet or
Neumann case) written in increasing order and repeated according to
multiplicity.


%------------------------------------------------------------
\subsection*{Periodic manifolds and Floquet Theory}

Let $\Gamma$ be an abelian group of infinite order with $r$ generators and
neutral element $1$. Such groups are isomorphic to $\Z^{r_0} \times
\Z_{p_1}^{r_1} \times \dots \times \Z_{p_a}^{r_a}$ with $r_0>0$ and $r_0 +
\dots + r_a=r$.  Here, $\Z_p$ denotes the abelian group of order $p$.  A
$d$-dimensional (non-compact) Riemannian manifold $\Mper$ will be called
\emph{$\Gamma$-periodic} or a \emph{covering manifold} if $\Gamma$ acts
properly discontinuously, isometrically and cocompactly, i.e., the quotient
$\Mper / \Gamma$ is a $d$-dimensional compact Riemannian manifold such that
the quotient map is a local isometry (cf.~e.g.~\cite{atiyah:76, bruening:92,
  chavel:93, donnelly:81, sunada:89}). For simplicity we assume that $\Mper$
has no boundary.

A compact subset $M$ of $\Mper$ is called a \emph{period cell} if $M$ is the
closure of a fundamental domain $D$, i.e., $M=\overline D$, $D$ is open and
connected, $D$ is disjoint from any translate $\gamma D$ for all $\gamma \in
\Gamma$, $\gamma \ne 1$, and the union of all translates $\gamma M$ is equal
to $\Mper$.

Now we want to analyse the spectrum of periodic elliptic operators on $\Mper$.
Here, \emph{periodicity} means that the operator commutes with all translation
operators on $\Lsqr \Mper$ induced by the group action
(cf.~e.g.~\cite{bruening:92, sunada:89}). In particular the
Laplacian on $\Mper$ is periodic. From Floquet theory it suffices to analyse
the spectra of the periodic operator restricted to a period cell $M$ with
quasi-periodic boundary conditions (cf.~e.g.~\cite{donnelly:81,
  reed-simon-4}).  In order to do this, we define $\theta$-periodic boundary
conditions. Let $\theta$ be an element of the dual group $\hat \Gamma =
\Hom(\Gamma,\Torus^1)$ of $\Gamma$, which is isomorphic to a subgroup of the
$r$-dimensional torus $\Torus^r = \{ \theta \in \C^r; \, |\theta_i|=1 \text{
  for all $i$} \}$. We denote by $q_M^\theta$ the closure of the quadratic
form $\check q_M$ defined on the space of those functions $u \in \Ci M$ that
satisfy
\begin{displaymath}
  u(\gamma x)= \overline {\theta(\gamma)} \, u(x)
\end{displaymath}
for all $x \in \bd M$ and all $\gamma \in \Gamma$ such that $\gamma x \in \bd
M$. The corresponding operator is denoted by $\laplacianT M$. Again,
$\laplacianT M$ has purely discrete spectrum denoted by $\EWT k M$. The
eigenvalues depend continuously on $\theta$. Furthermore $\EWT k M$ depends
even analytically on $\theta$ if we exclude those $\theta \in \hat \Gamma$ for
which $\EWT k M$ is a multiple eigenvalue (cf.~e.g.~\cite{bratteli:99,
  reed-simon-4}).  From Floquet theory we obtain
\begin{displaymath}
  \spec \laplacian \Mper = \bigcup_{\theta \in \hat \Gamma} \spec \laplacianT M
                         = \bigcup_{k \in \N} B_k(\Mper)
\end{displaymath}
where $B_k=B_k(\Mper) = \{ \EWT k M ; \, \theta \in \hat \Gamma\}$ is a
compact interval, called \emph{$k$-th band} (cf.~e.g.~\cite{reed-simon-4}).
In general, we do not know whether the intervals $B_k$ overlap or not. But we
can show the existence of gaps by proving that $\EWT k M$ does not vary too
much in $\theta$.

\begin{remark}
  \label{rem:abs.cont}
  Note that the first band cannot be trivial (i.e. $B_1=\{0\}$) since the
  first eigenvalue $\EWT 1 M$ is $0$ if and only if we are in the case of
  periodic boundary conditions, i.e., $\theta=1$. There are no constant
  \emph{and} $\theta$-periodic functions if $M$ is connected. This means that
  the first band consists of absolutely continuous spectrum provided the first
  and second band do not overlap or more precisely, that $B_1 \cap B_2^c$ is
  absolutely continuous. In general it is not true that all the spectrum is
  absolutely continuous. In~\cite{sunada:89} one can find an example where a
  band reduces to a point (not necessarily being an isolated eigenvalue).
\end{remark}


%------------------------------------------------------------
\subsection*{Main Lemma and Min-max principle}
Here we state a formal result on how to deal with parameter-depending Hilbert
spaces and operators with purely discrete spectrum on these spaces. In
particular, we are interested in the dependence of the eigenvalues on the
parameter. Such Hilbert spaces occur in the next section when we construct a
period cell depending on a parameter $\eps$. The basic idea of the Main
Lemma~\ref{lem:main.lemma} is taken from \cite{anne:87} and \cite{fukaya:87}.
Nevertheless the Main Lemma is more general: here we allow even non-uniform
convergence, i.e., the convergence assumed in Conditions~\eqref{eq:cond.norm}
and~\eqref{eq:cond.quad} could depend on $(u_\eps)$. Furthermore, we can
choose alternatively between the convergence or the inequality in the
assumptions.

We first quote the \emph{Min-max Principle}. Suppose that $q$ is a closed,
non-negative qua\-dra\-tic form on the separable Hilbert space $\HS$ such that
the corresponding operator $Q$ has purely discrete spectrum denoted by $\spec
Q = \{\lambda_k \,|\, k \in \N \}$. Note that $\lambda_k \ge 0$ for all $k$.
Throughout this article we assume that the sequence of eigenvalues
$(\lambda_k)$ is written in increasing order and repeated according to
multiplicity. We then have
\begin{equation}
  \label{eq:max.min}
  \lambda_k = \inf_{L_k} \sup_{u \in L_k, u \ne 0} \frac {q(u)}{\normsqr u}
\end{equation}
where the infimum is taken over all $k$-dimensional subspaces $L_k$ of $\dom
q$ (for this version of the Min-max principle see e.g.~\cite{davies:96}).

Suppose now that for each $\eps>0$ we have separable Hilbert spaces $\HS_\eps$
and $\HS'_\eps$. Furthermore suppose that $q_\eps$ and $q'_\eps$ are
non-negative, closed quadratic forms on $\HS_\eps$ and $\HS'_\eps$. Finally
suppose that the corresponding operators have purely discrete spectrum
denoted by $\lambda_k(\eps)$ and $\lambda'_k(\eps)$, $k \in \N$ (written in
increasing order and repeated according to multiplicity). A corresponding
orthonormal basis of eigenfunctions of $Q_\eps$ is denoted by
$(\phi_k^\eps)_k$ and the linear span of the first $k$ eigenvalues by
$L_k(\eps)$.
\begin{lemma}[Main Lemma]
  \label{lem:main.lemma}
  Suppose that for each $\eps>0$ a linear map $\Phi_\eps \colon \dom q_\eps
  \longrightarrow \dom q'_\eps$ is given such that for all $u_\eps \in
  L_k(\eps)$ Conditions~\eqref{eq:cond.norm} and~\eqref{eq:cond.quad} are
  satisfied:
  \begin{align}
  \label{eq:cond.norm}
    \lim_{\eps \to 0} ( \normsqr[\HS'_\eps]{\Phi_\eps u_\eps}  -
                        \normsqr[\HS _\eps]{          u_\eps}) &= 0
    & \text{or} &&
                        \normsqr[\HS _\eps]{          u_\eps}  &\le
                        \normsqr[\HS'_\eps]{\Phi_\eps u_\eps}  \\
  \label{eq:cond.quad}
    \lim_{\eps \to 0} ( q'_\eps(\Phi_\eps u_\eps) -
                        q _\eps(          u_\eps) ) &= 0
    & \text{or} &&
                        q _\eps(          u_\eps)   &\ge
                        q'_\eps(\Phi_\eps u_\eps).
  \end{align}
  Furthermore, we assume that for each $k \in \N$ there exist a constant
  $c_k>0$ such that
  \begin{equation}
  \label{eq:cond.ew}
    \lambda_k(\eps) \le c_k \qquad \text{for all $\eps>0$.}
  \end{equation}
  Then for each $k \in \N$ there exists a function $\delta_k(\eps) \ge 0$
  converging to $0$ as $\eps \to 0$ such that
  \begin{equation}
    \label{eq:ew.est}
    \lambda'_k(\eps) \le \lambda_k(\eps) + \delta_k(\eps)
  \end{equation}
  for small enough $\eps > 0$.
\end{lemma}
\begin{proof}
  For $u=u_\eps = \sum_{i=1}^k \alpha_i^\eps \phi_i^\eps$ with complex numbers
  $\alpha_i=\alpha_i^\eps$ we have
  \begin{displaymath}
    \frac{q'_\eps(\Phi_\eps u)} {\normsqr{\Phi_\eps u}} -
    \frac{q _\eps(          u)} {\normsqr{          u}}    =
    \frac 1 {\normsqr{\Phi_\eps u}}
    \Bigl(
      \frac{q _\eps(          u)} {\normsqr{        u}}
      \bigl(
        \normsqr{u} - \normsqr{\Phi_\eps u}
      \bigr) +
      \bigl(
        q'_\eps(\Phi_\eps u) - q_\eps(u)
      \bigr) 
    \Bigr).
  \end{displaymath}
  Furthermore, we estimate
  \begin{multline}
  \label{eq:est.norm}
     \normsqr{u} - \normsqr{\Phi_\eps u} =
     \sum_{i,j=1}^k \alpha_i \overline {\alpha_j}
     \bigl(
       \delta_{ij} - \iprod {\Phi_\eps \phi_i^\eps} {\Phi_\eps \phi_j^\eps}
     \bigr) \\ \le
     \delta'_k(\eps) \sum_{j=1}^k |\alpha_j|^2 = 
     \delta'_k(\eps) \, \normsqr {u}
  \end{multline}
  where
  \begin{displaymath}
    \delta'_k(\eps) := k \max_{i,j = 1, \dots, k} 
      |\delta_{ij} - \iprod {\Phi_\eps \phi_i^\eps } {\Phi_\eps \phi_j^\eps }|
  \end{displaymath}
  by the Cauchy-Schwarz Inequality. The Polarisation Identity
  together with Condition~\eqref{eq:cond.norm} yields $\delta'_k(\eps) \to 0$
  as $\eps \to 0$. If we are in the second alternative of
  Condition~\eqref{eq:cond.norm} we simply set $\delta'_k(\eps)=0$.  By a
  similar argument we can show the existence of a function $\delta''_k(\eps)
  \ge 0$ converging to $0$ as $\eps \to 0$ such that
  \begin{equation}
  \label{eq:eq:est.quad}
     q'_\eps(\Phi_\eps u) - q_\eps(u) \le
     \delta''_k(\eps) \normsqr{u}.
  \end{equation}
  From Estimate~\eqref{eq:est.norm} we also conclude
  \begin{equation}
  \label{eq:est.norm2}
    \normsqr{u} \le \frac 1 {1-\delta'_k(\eps)} \normsqr{\Phi_\eps u}
  \end{equation}
  provided $\eps$ is small enough. Applying Condition~\eqref{eq:cond.ew} we
  obtain the estimate
  \begin{equation}
  \label{eq:est.ew}
    \frac{q'_\eps(\Phi_\eps u)} {\normsqr{\Phi_\eps u}} -
    \frac{q _\eps(          u)} {\normsqr{          u}}    \le
    \delta_k(\eps):= \frac 1 {1-\delta'(\eps)} 
      \bigl( 
         c_k \delta'(\eps) + \delta''(\eps)
      \bigr).
  \end{equation}
  Estimate~\eqref{eq:est.norm2} also yields the injectivity of
  $\Phi_\eps \restr {L_k(\eps)}$, i.e., $\Phi_\eps(L_k(\eps))$ is a
  $k$-dimen\-sional subspace of $\dom q'_\eps$. Finally, the Min-max
  Principle~\eqref{eq:max.min} implies the desired estimate on the
  eigenvalues.
\end{proof}



%------------------------------------------------------------
\section{Construction of a Periodic Manifold}
\label{sec:constr}

Suppose that $X$ is a compact oriented and connected Riemannian manifold of
dimension $d \ge 2$ (for simplicity without boundary). We want to construct a
$\Gamma$-periodic manifold where $\Gamma$ is an abelian group with $r$
generators $e_1, \dots, e_r$. We choose $2r$ distinct points
$x_1,\dots,x_{2r}$. For each point $x_i$, denote by $B^i_\eps$ the open
geodesic ball around $x_i$ of radius $\eps>0$.  Suppose further that
$B^i_{\eps_0}$ are pairwise disjoint where $\eps_0>0$ denotes the injectivity
radius of $X$. Denote by $B_\eps$ the union of all balls $B^i_\eps$, $i=1,
\dots, 2r$. Let $\Xeps := X \setminus B_{2\eps}$ for $0 < 2\eps < \eps_0$ with
metric inherited from $X$.

On $B^i_{\eps_0}$, the metric of $X$ is given in polar coordinates
$(s,\sigma) \in ]0,\eps_0[ \times \Sphere^{d-1}$ by
\begin{equation}
  \label{eq:met.pol.coord}
  g = \dd s^2 + h^i_s
\end{equation}
where $h^i_s$ denotes a metric on $\{s\} \times \Sphere^{d-1}$ (see
Figure~\ref{fig:gaps2}). Here, $\Sphere^{d-1}$ denotes the $(d-1)$-dimensional
sphere with standard metric $\dd \sigma^2$.

Let $r_\eps$ be a smooth monotone function with $r_\eps(s)=\eps$ for
$0 \le s \le \eps/2$ and $r_\eps(s)=s$ for $2\eps \le s \le \eps_0$.
Furthermore, let $\chi_\eps$ be a smooth cut-off function having values
between $0$ and $1$ and satifying $\chi_\eps(s)=0$ if $s \le \eps$ and
$\chi_\eps(s)=1$ if $s \ge 2\eps$. Now we let the modified metric
$h^i_{\eps,s}$ be a convex combination of the original metric and the
spherical metric $(r_\eps(s))^2 \dd \sigma^2$, i.e.,
\begin{equation}
  \label{eq:conv.comb}
  h^i_{\eps,s} := 
  \chi_\eps(s) h^i_s + \bigl( 1-\chi_\eps(s) \bigr) 
                       \bigl( r_\eps(s) \bigr)^2 \dd \sigma^2.
\end{equation}
We denote the completion of $X \setminus \{x_1, \dots, x_{2r}\}$ together with
the modified metric
\begin{displaymath}
  g^i_\eps := \dd s^2 + h^i_{\eps,s} 
\end{displaymath}
on $B^i_{\eps_0}$ by $\Meps$ (see Figure~\ref{fig:gaps2}).  Since
$g^i_\eps(s,\sigma) = g(s,\sigma)$ for $s \ge 2 \eps$, the punched manifold
$\Xeps$ is embedded in $\Meps$. Furthermore, since $g^i_\eps(s,\sigma) = \dd
s^2 + \eps^2 \dd \sigma^2$ for $s \le \eps/2$, there exists a neighbourhood of
$\bd \Meps \cap B^i_\eps$ given in coordinates by $[0,\eps/2] \times
\Sphere^{d-1}$ which is isometric to a cylinders of radius $\eps$ and length
$\eps/2$.
\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \begin{picture}(0,0)(-8,0)%10,100)(162,107)
      \includegraphics{figure-2}
    \end{picture}
    \setlength{\unitlength}{4144sp}
%    \begin{picture}(5094,1654)(195,-884)
    \begin{picture}(5094,1654)(195,-884)
      \put(4816,-826){$\Meps$}
      \put(4341,614){$\eps$}
      \put(1801,-826){$X$}
      \put(1261,299){$x_i$}
      \put(361,-781){$s$}
      \put(3196,-781){$s$}
      \put(521,-421){$\eps_0$}
      \put(3411,-421){$\eps_0$}
      \put(856,-71){$2\eps$}
      \put(3721,-66){$2\eps$}
      \put(3946,149){$\eps$}
      \put(1101,104){$\eps$}
      \put(4146,299){$\eps/2$}
      \put(2639,-115){$\Xeps$}
      \put(3129,365){$A_\eps^i$}
    \end{picture}
    \caption{Modification of the metric $g$ of the $2$-dimensional manifold
      $X$ near the point $x_i$.}
    \label{fig:gaps2}
  \end{center}
\end{figure}
Let $A^i_\eps$ be the \emph{cylindrical end} of the manifold $\Meps$ near
$x_i$ given in coordinates by $[0,2\eps] \times \Sphere^{d-1}$. Next, let
$A^i_{\eps,x}$ be the sphere with distance $s$ from the boundary given in
coordinates by $\{s\} \times \Sphere^{d-1}$.  Finally, let $\Aeps$ be the
union of all cylindrical ends $A^i_\eps$, $i = 1, \dots, 2r$.

Finally, we construct the corresponding periodic manifold $\Mepsper$: Let
$\gamma \Meps$ be an isometric copy of $\Meps$ with identification $x \mapsto
\gamma x$ for each $\gamma \in \Gamma$. We construct a new (non-compact)
manifold $\Mepsper$ by identifying $\gamma \bd A^{2i-1}_\eps$ with $e_i \gamma
\bd A^{2i}_\eps$ for each $\gamma \in \Gamma$ and $i=1, \dots, r$. Remember
that $e_i$ denotes the $i$-th generator of $\Gamma$. Since in a neighbourhood
of $\bd A^i_\eps$ the manifold is isometric to a cylinder of radius $\eps$, we
can choose a smooth atlas and a smooth metric on the glued manifold
$\Mepsper$. We therefore obtain a (non-compact) $\Gamma$-periodic manifold
$\Mepsper$ and $\Meps$ is a period cell for $\Mepsper$.

Now we are able to state the following theorem (Theorem~\ref{thm:gaps.constr}
follows via Floquet Theory):
\begin{theorem}
  \label{thm:conv.constr}
  As $\eps \to 0$ we have $\EWT k \Meps \to \EW k X$ uniformly in $\theta \in
  \hat \Gamma$.
\end{theorem}
Therefore, the $k$-th band $B_k(\Mepsper)$ reduces to the point $\{ \EW k X
\}$ as $\eps \to 0$.  Note that the convergence is \emph{not} uniform in $k$
since there are topological obstructions (see the discussion in
\cite{chavel-feldman:81}). We therefore could not expect that an
\emph{infinite} number of gaps occur.

Before we prove Theorem~\ref{thm:conv.constr}, we need two lemmas. The idea
is to compare the $\theta$-periodic eigenvalues on $\Meps$ with Dirichlet and
Neumann eigenvalues on $\Xeps$. The crucial point is, that the corresponding
$\theta$-periodic eigenfunctions on $\Meps$ do not concentrate on $\Aeps$,
i.e., on the cylindrical ends. This will be shown in the second lemma. First
we need to compare the density of the $(d-1)$-dimensional volume of
$A^i_{\eps, s}$ with the volume of the sphere of radius $r_\eps(s)$:
\begin{lemma}
  \label{lem:mod.met.est}
  There exists a constant $c \ge 1$ such that
  \begin{displaymath}
    \frac 1c \, r_\eps(s)^{d-1} \le 
    (\det h^i_{\eps,s})^\frac12 \le
    c        \, r_\eps(s)^{d-1}
  \end{displaymath}
  for all $0 \le s \le \eps_0$ and all $i$.
\end{lemma}
\begin{proof}
  The metric $g= \dd s^2 + h^i_s$ on $B^i_{\eps_0}$ can be compared with the
  flat metric $\dd s^2 + s^2 \dd \sigma^2$ (pointwise in the sense of
  sesquilinear forms), i.e., there exists a constant $c'\ge 1$ such that
  \begin{displaymath}
    \frac1{c'} \, s^2 \dd \sigma \le
    h^i_s \le
    c'         \, s^2 \dd \sigma.
  \end{displaymath}
  By our assumptions on $r_\eps$ the same estimate is true with $s^2$ replaced
  by $r_\eps(s)^2$ and $h^i_s$ replaced by the convexe
  combination~\eqref{eq:conv.comb}. The result follows from the monotonicity
  of $\det$.
\end{proof}

Now we prove the non-concentration of the eigenfunctions on the cylindrical
ends as $\eps \to 0$:
\begin{lemma}
  \label{lem:no.conc}
  There exists a positive function $\omega(\eps)$ converging to $0$ as $\eps
  \to 0$ such that
  \begin{equation}
    \label{eq:no.conc}
    \int_{\Aeps} |u|^2 \le
    \omega(\eps) \int_{\Meps} \bigl( |u|^2 + | \dd u|^2 \bigr),
  \end{equation}
  for all $u$ out of the domain of the quadratic form with
  $\theta$-periodic boundary conditions on $\Meps$.
\end{lemma}
Note that $\omega(\eps)$ only depends on the geometry of $X$ near $x_i$.
\begin{proof}
  Without loss of generality, we can assume that $u \in \Ci \Meps$. Suppose
  furthermore that $u(\eps_0,\sigma)=0$ for all $\sigma \in \Sphere^{d-1}$.
  First we show an $L_2$-estimate over $A^i_{\eps,s}$ with its induced metric
  $h^i_{\eps,s}$.
  
  Applying the Cauchy-Schwarz Inequality and Lemma~\ref{lem:mod.met.est}
  yields
  \begin{multline*}
    |u(s,\sigma)|^2 =
    \Big| \int_s^{\eps_0} \partial_t u(t,\sigma)\, \dd t\Big|^2 \\ \le
       c \int_s^{\eps_0} \hspace{-1ex} r_\eps(t)^{1-d}  \dd t \, 
         \int_s^{\eps_0} |\partial_t u (t,\sigma)|^2 
                 (\det h^i_{\eps,t}(\sigma))^\frac12 \dd t.
  \end{multline*}
  If we integrate over $\sigma \in \Sphere^{d-1}$ and apply
  Lemma~\ref{lem:mod.met.est} once more we obtain
  \begin{align}
    \int_{A^i_{\eps,s}} |u|^2  &= 
    \int_{\Sphere^{d-1}} |u(s,\sigma)|^2 
     (\det h^i_{\eps,s}(\sigma))^\frac12 \nonumber\\ 
     \label{eq:int.est}  & \le
     c^2 r_\eps(s)^{d-1} \int_s^{\eps_0} r_\eps(t)^{1-d}  \dd t
                     \int_{\Meps} |\dd u|_{T^* \! \Meps}^2.
  \end{align}
  If $0 \le s \le 2\eps$ we have $r(s)^{d-1} \le (2\eps)^{d-1}$. Furthermore,
  the integral over $t$ can be split into an integral over $0 \le t \le 2\eps$
  and $2\eps \le t \le \eps_0$. The first integral can be estimated by
  $\eps^{2-d}$, the second by $\int_{2\eps}^{\eps_0} t^{1-d} \dd t$. Therefore
  we have an estimate of the order $O(\eps)$ if $d \ge 3$ resp.\ $O(\eps|\ln
  \eps|)$ if $d=2$. Finally, if we integrate the integral on the LHS
  of~\eqref{eq:int.est} over $s \in [0, 2\eps]$ we obtain the desired
  Estimate~\eqref{eq:no.conc}.
  If $u(\eps_0,\sigma)\ne 0$ we choose a cut-off function.
\end{proof}

The argument in the proof is due to \cite{anne:87}. The following lemma is
proven in \cite{chavel-feldman:78} resp.\ \cite{anne:87}.

\begin{lemma}
  \label{lem:anne.chavel}
  We have $\EWD k \Xeps \to \EW k X$ resp.\ $\EWN k \Xeps \to \EW k X$.
\end{lemma}

Now we can show Theorem~\ref{thm:conv.constr}:
\begin{proof}
  From the Min-max Principle~\eqref{eq:max.min} we conclude
  \begin{equation}
    \label{eq:upper.est}
    \EWT k \Meps \le  \EWD k \Xeps
  \end{equation}
  since the domains of the quadratic forms obey the opposite inclusions.  In
  particular, $\EWT k \Meps$ is bounded in $\theta$ and $\eps$ by some
  constant $c_k>0$. To prove the opposite inequality we apply our Main
  Lemma~\ref{lem:main.lemma} with $\HS_\eps := \Lsqr \Meps$, $q_\eps :=
  q^\theta_{\Meps}$, $\HS'_\eps := \Lsqr \Xeps$, $q'_\eps := q^\Neu_{\Xeps}$
  and $\Phi_\eps u := u \restr \Xeps$ being the restriction operator.
  
  \sloppy
  Condition~\eqref{eq:cond.norm} is satisfied by Lemma~\ref{lem:no.conc},
  the inequality in Condition~\eqref{eq:cond.quad} is trivially
  satisfied. Finally, Condition~\eqref{eq:cond.ew} is satisfied by the upper
  bound~\eqref{eq:upper.est} and Lemma~\ref{lem:anne.chavel}. The Main Lemma
  therefore yields
  \begin{displaymath}
    \EWN k \Xeps \le \EWT k \Meps + \delta_k(\eps)
  \end{displaymath}
  with $\delta_k(\eps) \to 0$ as $\eps \to 0$. Together with
  Estimate~\eqref{eq:upper.est} and Lemma~\ref{lem:anne.chavel} we are done.
\end{proof}

\begin{remark}
  \label{rem:constr.curv}
  If we assume that the metric $g$ of the original manifold $X$ is flat on
  $B^i_{\eps_0}$, i.e., $h^i_s=s^2 \dd \sigma^2$ in
  Equation~\eqref{eq:met.pol.coord}, we can calculate the sectional curvature
  of the cylindrical end $\Aeps$ of $\Meps$.  Let $\partial_s$,
  $\partial_{\sigma_1}, \dots, \partial_{\sigma_{d-1}}$ be orthonormal tangent
  vectors corresponding to $(s,\sigma)$ on $A^i_\eps$.  Then the sectional
  curvature is
    \begin{displaymath}
      K(\partial_s,\partial_{\sigma_j}) = -\frac{\ddot r_\eps} {r_\eps}
          \qquad \text{and} \qquad
      K(\partial_{\sigma_j},\partial_{\sigma_k}) = 
         \frac{1 - \dot r_\eps^2} { r_\eps^2}
    \end{displaymath}
    for $j \ne k$. With our assumptions on $r_\eps$,
    $K(\partial_s,\partial_{\sigma_j})$ is a negative number of order
    $O(\eps^{-2})$ and $K(\partial_{\sigma_j},\partial_{\sigma_k})$ is a
    positive number of the same orderas $\eps \to 0$. In particular, the Ricci
    and the scalar curvature are also of order $O(\eps^{-2})$. Therefore, all
    curvature terms are unbounded in $\eps$ as Green conjectured in
    \cite{green:97}.
\end{remark}

%------------------------------------------------------------
\subsection*{Periodic manifold joined by cylinders}
In the previous example $\Mepsper$ one may think that it is essential for
obtaining spectral gaps that the parts which break down reduce to a point. To
avoid this impression we will confront another periodic manifold with spectral
gaps where the collapsing parts reduce to intervals instead of points. It only
seems to be important that the period cells are seperated by very short closed
geodesics (in dimension $2$) or more generally by very small submanifolds of
codimension $1$. The construction in this section is closely related to the
work of Ann\'e~\cite{anne:87}.

For simplicitywe will only assume that $\Gamma=\Z$. Let $I$ be the interval
$[0,L]$ and let $\Ceps$ be the cylinder $I \times \Sphere^{d-1}$ of radius
$\eps$ and length $L>0$ with metric $\dd s^2 + \eps^2 \dd \sigma^2$ (if $L=0$
we are in the case of the previous section) . We denote by $\Neps$ the period
cell $\Meps$ where we have glued the cylinder $\Ceps$ by identifying $\{0\}
\times \Sphere^{d-1}$ with $\bd A^1_\eps$. In the same way as above we obtain
a periodic manifold $\Nepsper$ with period cell $\Neps$ by glueing together
$\Z$ copies of $\Neps$. As above we can prove:
\begin{theorem}
  \label{thm:conv.cyl}
  As $\eps \to 0$ we have $\EWT k \Neps \to \EWD k {X \dcup I}$ uniformly in
  $\theta \in \hat \Gamma$.
\end{theorem}
Here, $\EWD k {X \dcup I}$ denotes the eigenvalues of the operator $\laplacian
X \oplus \laplacianD I$ written in increasing order and repeated according to
multiplicity. Note that $\spec (\laplacian X \oplus \laplacianD I) = \spec
\laplacian X \cup \spec \laplacian I$, i.e., $\EWD k {X \dcup I}$ is a
reordering of $\EWD {k_1} X$ and $\EWD {k_2} I$. Again, by Floquet Theory
Theorem~\ref{thm:gaps.cyl} follows.

\begin{proof}
  We only sketch the proof since it is similar to the proof of
  Theorem~\ref{thm:conv.constr}. Again, we have an upper estimate of $\EWT k
  \Neps$ by the $k$-th Dirichlet eigenvalue $\EWD k {\Xeps \dcup \Ceps}$ on
  $\Xeps \dcup \Ceps$. For the estimate from below we apply the Main Lemma
  once more with $\HS_\eps := \Lsqr \Neps$, $q_\eps := q^\theta_{\Neps}$,
  $\HS'_\eps := \Lsqr \Xeps \oplus \Lsqr \Ceps$ and $q'_\eps := q^\Neu_{\Xeps}
  \oplus q^\Dir_{\Ceps}$. Furthermore, for $u \in \dom q^\theta_{\Neps}$, we
  set
  \begin{displaymath}
    \Phi_\eps u := 
    u \restr \Xeps \oplus (u \restr \Ceps -h)   
  \end{displaymath}
  where $h=h_\eps$ is the (unique) function satisfying
  \begin{align*}
    \laplacian \Ceps h &= 0 &\text{and}&&
    h \restr {\bd \Ceps} &= u \restr {\bd \Ceps}
  \end{align*}
  (see~\cite{anne:87} and \cite{post:00}).  To verify
  Condition~\ref{eq:cond.norm}, we estimate
  \begin{multline*}
    \bigl| \normsqr[\Xeps \dcup \Ceps]{\Phi_\eps u} - 
           \normsqr[\Neps] u 
    \bigr| \le
    \int_{\Aeps} \hspace{-1ex} |u|^2 + 
    \int_{\Ceps} \hspace{-1ex} \bigl( |u-h|^2 - |u|^2 \bigr) \\ \le
    \normsqr[\Aeps] u + 2 \norm[\Ceps] u \norm[\Ceps] h + \normsqr[\Ceps] h.
  \end{multline*}
  By Lemma~\ref{lem:no.conc} we only need to show that the harmonic extenstion
  $h$ converges to $0$ in $\Lsqr \Ceps$ uniformly in $\theta$ (note that
  $h=h^\theta_\eps$ depends also on $\theta$ since $u=u^\theta_\eps \in \dom
  q^\theta_\Meps$ does). The convergence does not seem to be very surprising
  since the harmonic function $h$ is given on the boundary $\bd \Ceps$ by $u$
  which is small (in $L_2$-sense) by Lemma~\ref{lem:no.conc}. Nevertheless a
  little more work is necessary which will be omitted here (see~\cite{anne:87}
  and \cite{post:00}).

  Condition~\ref{eq:cond.quad} is satisfied since
  \begin{displaymath}
    q_\eps(u) = 
   \int_\Neps | \dd u|^2 \ge 
   \int_\Xeps | \dd u|^2 + \int_{\Ceps} | \dd h|^2 =
    q'_\eps(\Phi_\eps u).
  \end{displaymath}
  Note that the harmonic function $h$ minimizes the energy integral
  $q_\Ceps(u)$.  Since 
  \begin{equation}
    \label{eq:harmonic}
   \int_\Ceps \iprod {\dd (u-h)} {\dd h} = 0 
  \end{equation}
  by the Gauss-Green Formula we have $q_\Ceps(u) = q_\Ceps(h)+q_\Ceps(u-h) \ge
  q_\Ceps(h)$. Note that $u-h$ satisfies Dirichlet boundary conditions.
  
  From the Main Lemma we obtain a lower bound (up to an error term) given by
  the $k$-th eigenvalue $\EWND k {\Xeps \dcup \Ceps}$ with Neumann boundary
  conditions on $\Xeps$ and Dirichlet boundary conditions on $\Ceps$. We only
  have to add on that $\EWND k {\Xeps \dcup \Ceps}$ resp.\ $\EWD k {\Xeps
    \dcup \Ceps}$ converge to $\EWD k {X \dcup I}$ if the cylinder $\Ceps$
  collapes to the interval $I$ as $\eps \to 0$.
\end{proof}




%------------------------------------------------------------
\section{Conformal Deformation}
\label{sec:conf}

Suppose $\Mper$ is a $\Gamma$-periodic Riemannian manifold of dimension $d \ge
2$ with metric $g$. Let $X \subset \Mper$ be a compact subset with smooth
boundary such that $\gamma X \cap X \ne \emptyset$ implies $\gamma=0$. Then a
period cell $M$ with $\dist(\bd X, \bd M)>0$ exists.  We introduce
\emph{normal} or \emph{Fermi coordinates} $(r,y)$ with respect to $Y:=\bd X$
(for details cf.\ \cite{chavel:93}). Here, $r \in (-r_0,r_0)$ parametrises
the normal direction and $y \in Y$ parametrises the tangential direction;
$r<0$ corresponds to the interior of $X$ and $r=0$ corresponds to $Y$.

Furthermore, we assume that normal coordinates also exist on $\overline{M
  \setminus X}$, i.e., we suppose that $\overline{M \setminus X}$ can be
parametrised by $(r,y)$ with $r \in I_y$ and $y \in Y$. Here, $I_y$ is a
compact subset of $\R$ containing $[0,r_0]$.  The existence of normal
coordinates on $\overline{M \setminus X}$ is a geometrical restriction on $X$.
In some situations this condition means that $X$ is close enough to $\bd M$.
For example, this condition is satisfied for a centered ball in a cube.

Suppose we have for each $\eps>0$ a smooth $\Gamma$-periodic function
$\rho_\eps \colon \Mper \longrightarrow (0,1]$ with the following
properties:
\begin{align}
  \label{eq:cond.one}
  \rho_\eps(x) &=1    
               &&\text{for all $x \in X$,} \\
  \label{eq:cond.small}
  \rho_\eps(x) &=\eps 
               &&\text{for all $x \in M$ with $\dist(x,X) \ge \eps^d$.}
\end{align}
For simplicity, we also assume that $\rho_\eps(x)$ is only a function on $r$
in normal coordinates on $\overline{M \setminus X}$.  Note that the function
$\rho_\eps$ converges pointwise to the indicator function of the set
$\Xper=\bigcup_{\gamma \in \Gamma} X$. We define $g_\eps := \rho_\eps^2 g$ and
denote the resulting Riemannian manifolds with metric $g_\eps$ by $\Mepsper$
resp.\ $\Meps$. We therefore obtain a conformally deformed $\Gamma$-periodic
manifold $\Mepsper$ with periodic metric $g_\eps$ and period cell $\Meps$.  In
particular, the squared norm and the quadratic form on the deformed manifold
$\Meps$ are given by
\begin{align}
  \label{eq:conf.fact}
  \normsqr[\Meps] u &= \int_M |u|^2 \rho_\eps^d       
                    &\text{and}&&
  q_\Meps (u)       &= \int_M |\dd u|_{T^*\!M}^2 \,
                                        \rho_\eps^{d-2}.
\end{align}
Here we can see that the case $d=2$ is in some sense particular, since the
quadratic form does not depend on $\eps$ any more (but the norm does).

First, let us calculate the curvature of the conformally deformed case:
\begin{remark}
  \label{rem:conf.curv}
  We denote the sectional curvatures of $\Meps$ by the subscript $\eps$ and
  the sectional curvatures of $M$ without subscript. Let $\partial_r$,
  $\partial_{y_1}, \dots, \partial_{y_{d-1}}$ be orthonormal basis tangent
  vectors corresponding to the coordinates $(r,y)$ near $Y=\bd X$.  Then the
  sectional curvatures are given by
    \begin{align*}
      K_\eps(\partial_r,\partial_{y_j}) & = 
       \rho_\eps^5 
       \bigl(
           -\ddot\rho_\eps + \partial_r g_{y_j y_j}\dot \rho_\eps/2
           + \rho_\eps K(\partial_r,\partial_{y_j}) 
       \bigr)\\
      K_\eps(\partial_{y_j},\partial_{y_k}) & = 
       \rho_\eps^6
       \bigl( 
            -\dot \rho_\eps^2 - 
               (\partial_r g_{y_j y_j} + \partial_r g_{y_k y_k})
                   \dot \rho_\eps \rho_\eps/2 +
            \rho_\eps^2 K(\partial_{y_j},\partial_{y_k})
       \bigr)
    \end{align*}
    for $j \ne k$. With our assumptions on $\rho_\eps$,
    $K(\partial_r,\partial_{y_j}) = O(\eps^{-2d-5})$ with changing sign and
    $K(\partial_{\sigma_j},\partial_{\sigma_k})=O(\eps^{-2d-6})$ provided
    $\eps > 0$ is small enough. Calculating the Ricci and scalar curvature one
    can see that all curvature terms are not bounded in $\eps$ as Green
    conjectured in \cite{green:97}.
\end{remark}
%------------------------------------------------------------
\subsection*{The Higher Dimensional Case}

Now we are able to state the following theorem (again,
Theorem~\ref{thm:gaps.conf} follows via Floquet Theory):
\begin{theorem}
  \label{thm:conv.conf}
  Suppose that $d \ge 3$. Then $\EWT k \Meps \to \EWN k X$ as $\eps \to 0$
  uniformly in $\theta \in \hat \Gamma$.
\end{theorem}
Again, the $k$-th band $B_k(\Mepsper)$ reduces to the point $\{ \EWN k X
\}$ as $\eps \to 0$ and the convergence is \emph{not} uniform in $k$.

As in the previous section, we need the following lemma which shows that the
$\theta$-periodic eigenfunctions on $\Meps$ do not concentrate on the
metrically shrunken set $\Meps \setminus X$ as $\eps \to 0$:
\begin{lemma}
  \label{lem:no.conc.conf}
  There exists a positive function $\omega(\eps)$ converging to $0$ as $\eps
  \to 0$ such that
  \begin{equation}
    \label{eq:no.conc.conf}
    \int_{\Meps \setminus X} |u|^2 \le
    \omega(\eps) \int_M 
        \bigl( |u|^2 + | \dd u|_{T^*\!\Meps}^2 \bigr),
  \end{equation}
  for all $u$ out of the domain of the quadratic form with
  $\theta$-periodic boundary conditions on $\Meps$.
\end{lemma}
Again, $\omega(\eps)$ only depends on the geometry of $M \setminus X$.
\begin{proof}
  We proceed in the same way as in the proof of Lemma~\ref{lem:no.conc}. We
  introduce normal coordinates.  For notational simplicity only, we assume
  that $I_y=[0,r_y]$ for some number $r_0 \le r_y$. Suppose that $u \in \Ci
  \Meps$ with $u(r,y)=0$ for all $y \in Y$ and $r \le -r_0$. As in
  \eqref{eq:met.pol.coord} we have the orthogonal splitting
  \begin{displaymath}
    g_\eps = \rho_\eps^2 \, g_\eps = \rho_\eps^2 \, (\dd r^2 + h_r)
  \end{displaymath}
  in normal coordinates where $h_r$ is a parameter-dependent metric on $Y$.
  By the Cauchy-Schwarz Inequality we have
  \begin{multline*}
    |u(s,y)|^2 =
    \Big| 
      \int_{-r_0}^s \partial_r u(r,y)\, \dd r 
    \Big|^2  \\ \le
    \int_{-r_0}^s \bigl( \det g(r,y) \bigr)^{-\frac12} \,\dd r
          \cdot 
    \int_{-r_0}^s \bigl( 
                        |\partial_r u|^2 (\det g)^\frac12 
                  \bigr)(r,y) \, \dd r
  \end{multline*}
  for $0 \le s \le r_y$. Since $Y$ is compact we can estimate the first
  integral by $c>0$. Therefore integrating over $s \in I_y$ and $y \in Y$
  yields
  \begin{multline*}
    \int_{\Meps \setminus X} |u|^2 =
    \int_{y \in Y} \int_{s=0}^{r_y}  
         \bigl( 
               |u|^2  (\det g)^\frac12 \rho_\eps^d 
         \bigr)(s,y)  \,\, \dd s \,\dd y 
            \\ \le
    c \int_{y \in Y} \int_{s=0}^{r_y}  (\det g)^\frac12(s,y) \rho_\eps^d(s)
        \int_{r=-r_0}^s 
           \bigl( 
               | \partial_r u|^2 (\det g)^\frac12 
           \bigr)\!(r,y)\,\, \dd r\,\dd s\,\dd y.
  \end{multline*}
  We can estimate the $s$-dependent terms as follows: for $\eps^d \le s \le
  r_y$ we have $\rho_\eps(s)=\eps$ by Assumption~\eqref{eq:cond.small}.
  Furthermore, there exists a constant $c'>0$ such that $(\det g)^\frac12(s,y)
  \le c'$ for all $y \in Y$ and $0 \le s \le r_y$ since $\overline{M \setminus
    X}$ is compact. Therefore the integral over $0 \le s \le \eps^d$ and
  $\eps^d \le s \le r_y$ can be estimated by $c' \eps^d$. We conclude
  \begin{align*}
    \int_{\Meps \setminus X} |u|^2 
    & \le
    c \, c' \eps^d \int_{y \in Y} \int_{r=-r_0}^{r_y} 
         \bigl( 
            | \partial_r u|^2 (\det g)^\frac12 
         \bigr)\!(r,y)\,\, \dd r\, \dd y 
          \\ 
    & \le
    c \, c' \eps^2 \int_{y \in Y} \int_{r=-r_0}^{r_y} 
         \bigl( 
            | \partial_r u|^2 (\det g)^\frac12 \rho_\eps^{d-2} 
         \bigr)\!(r,y)\,\,\dd r\,\dd y \\
    & \le 
    c \, c' \eps^2 \int_M |\dd u|_{T^*\!\Meps}^2
  \end{align*}
  where we have used $\rho_\eps \ge \eps$ in the second line.
  
  If $u(r,y) \ne 0$ for some $y \in Y$ and $r< -r_0$ we multiply $u$ with a
  cut-off function $\chi$ such that $\chi(r) = 1$ for $r \ge -r_0/2$ and
  $\chi(r) = 0$ for $r \le -r_0$. Note that $\supp \chi \subset X$, i.e., on
  $\supp \chi$, there is no conformal deformation. If $u \in \dom
  q^\theta_\Meps$ we apply an approximation argument.
\end{proof}

In the same way we have proven Theorem~\ref{thm:conv.constr} we can show
Theorem~\ref{thm:conv.conf}:
\begin{proof}
  First, we prove an upper bound. For this we apply the Main
  Lemma~\ref{lem:main.lemma} with $\HS := \Lsqr X$, $q := q^\Neu_X$ (not
  depending on $\eps$), $\HS'_\eps := \Lsqr \Meps$ and $q'_\eps :=
  q^\theta_\Meps$. Furthermore, let $\Phi u$ be an extension of $u \in \dom
  q^\Neu_X$ onto $M$ such that $u=0$ in a neighbourhood of $\bd M$. In
  particular, $u \in \dom q^\theta_\Meps$. The inequality of
  Condition~\eqref{eq:cond.norm} is trivially satisfied,
  Condition~\eqref{eq:cond.quad} follows because of
  \begin{displaymath}
    \bigl| q'_\eps(\Phi u) - q(u) \bigr| =
    \int_{\Meps \setminus X} |\dd \Phi u|_{T^*\!\Meps}^2 =
    \int_{\Meps \setminus X} |\dd \Phi u|_{T^*\!\Meps}^2 \rho_\eps^{d-2} \,
                                                     \to 0
  \end{displaymath}
  by the Lebesgue convergence theorem and Assumption~\eqref{eq:cond.small}.
  Here, the assumption $d \ge 3$ is essential.  In the $\eps$-independent
  case, Condition~\eqref{eq:cond.ew} is obsolete. The Main Lemma yields
  \begin{equation}
    \label{eq:upper.est.conf}
    \EWT k \Meps \le \EWN k X + \delta_k(\eps) 
  \end{equation}
  with $\delta_k(\eps) \to 0$ as $\eps \to 0$.
  
  To prove the opposite inequality we apply the Main Lemma once more, this
  time with $\HS_\eps := \Lsqr \Meps$, $q_\eps := q^\theta_{\Meps}$, $\HS' :=
  \Lsqr X$, $q' := q^\Neu_X$ and $\Phi_\eps u := u \restr X$ being the
  restriction operator.
  
  Again, Condition~\eqref{eq:cond.norm} is satisfied by
  Lemma~\ref{lem:no.conc.conf}, and the inequality in
  Condition~\eqref{eq:cond.quad} is trivially satisfied. Finally,
  Condition~\eqref{eq:cond.ew} is satisfied by the upper
  bound~\eqref{eq:upper.est.conf}. The Main Lemma therefore yields
  \begin{displaymath}
    \EWN k X \le \EWT k \Meps + \delta_k(\eps)
  \end{displaymath}
  with $\delta_k(\eps) \to 0$ as $\eps \to 0$. Together
  with Estimate~\eqref{eq:upper.est.conf} we are done.
\end{proof}


%------------------------------------------------------------
\subsection*{The Two-Dimensional Case}
In dimension $2$, the special form of the quotient
$q^\theta_\Meps(u)/\normsqr[\Meps] u$ causes a different behaviour. The
$\theta$-periodic eigenvalue of the Laplacian on a period cell $\Meps$ still
converges, but the limit depends on $\theta$, i.e., the $k$-th band
$B_k(\Mepsper)$ in general does not reduce to a point.  In particular, if we
assume that $\rho_\eps$ is monotonely decreasing (as $\eps \searrow 0$), the
first $\theta$-periodic eigenvalue is monotonely increasing (as $\eps \searrow
0$) since we have
\begin{equation}
  \label{eq:quot.dim2}
  \EWT k \Meps =  \inf_ u \frac{q^\theta_\Meps(u)}
                               {\normsqr[\Meps] u}
               =  \inf_ u \frac{\int_M |\dd u|^2}
                               {\int_M |u|^2 \rho_\eps^2s}
\end{equation}
due to the Min-max Principle and~\eqref{eq:conf.fact}. Here, the
infimum is taken over all $u \in \dom q^\theta_\Meps$ such that $u \ne 0$.
Note that $\dom q^\theta_\Meps$ is independent of $\eps$ as vector space.
Since the first band $B_1(\Mepsper)$ of a connected periodic manifold
$\Mepsper$ has always nontrivial interior (see Remark~\ref{rem:abs.cont}),
$B_1(\Mepsper)$ cannot reduce to a point as $\eps \to 0$.

As in the higher dimensional case, the norm on $\Meps$ converges to the norm
on $X$, i.e., the limit quadratic form lives in the Hilbert space $\Lsqr X$.
But there is no reason why the limit form should only be an integral over $X$
since the quadratic form $q^\theta_\Meps$ does not depend on $\eps$ any more
(see Equation~\eqref{eq:quot.dim2}).

Indeed, the following candidate for the limit form is the right one (a more
detailed motivation can be found in~\cite{post:00}, note that harmonic
functions minimize the energy integral, i.e., the integral over $|\dd u|^2$).
For $u \in \dom q^\Neu_X$ let $h = H^\theta u \in \dom q^\theta_M$ be the
$\theta$-periodic harmonic extension of $u$ on $M$. In particular, $u=h$ on
$X$ and $\laplacian {M \setminus X} h=0$ on $M \setminus X$ such that $h$ and
$\dd h$ are $\theta$-periodic, i.e., $h(x)=\theta(\gamma) h(\gamma x)$ resp.\ 
$\dd h(x)=\theta(\gamma) \dd h(\gamma x)$ for all $x \in M$ and $\gamma \in
\Gamma$ such that $\gamma x \in M$.  Then we set
\begin{displaymath}
  q^\theta_0(u):= \int_M | \dd (H^\theta u)|^2
\end{displaymath}
for all $u \in \dom q^\theta_0 := \dom q^\Neu_X$. Note that $q^\Neu_X(u) \le
q^\theta_0 (u)$ for all $u$. In particular, the corresponding operator to
$q^\theta_0$ has purely discrete spectrum denoted by $\EWT k 0$
(written in increasing order and repeated according to multiplicity).
Furthermore, since $\dom q^\theta_0 \supset \dom q^\Dir_X$ the Min-max
Principle yields
\begin{displaymath}
  \EWN k X \le \EWT k 0 \le \EWD k X.
\end{displaymath}

Now we show the convergence of the $\theta$-periodic eigenvalues on $\Meps$ to
the eigenvalues $\lambda^\theta_k(0)$:
\begin{theorem}
  \label{thm:conv.conf2}
  Suppose that $d=2$. Then $\EWT k \Meps \to \EWT k 0$ as $\eps \to 0$ for all
  $\theta \in \hat \Gamma$.
\end{theorem}

\begin{proof}
  First, we prove an upper bound. For this note that $\norm[X] u \le
  \norm[\Meps] {H^\theta u}$ and $q^\theta_0(u) = q^\theta_\Meps(H^\theta u)$.
  By the Min-max Principle (or formally one could also apply the Main Lemma)
  we obtain
  \begin{equation}
    \label{eq:upper.est.conf2}
    \EWT k \Meps \le \EWT k 0. 
  \end{equation}
  
  To prove the opposite inequality we apply the Main
  Lemma~\ref{lem:main.lemma} once more, this time with $\HS_\eps := \Lsqr
  \Meps$, $q_\eps := q^\theta_{\Meps}$, $\HS' := \Lsqr X$, $q' := q^\theta_0$
  and $\Phi_\eps u := u \restr X$ being the restriction operator.
  
  Again, Condition~\eqref{eq:cond.norm} is satisfied by
  Lemma~\ref{lem:no.conc.conf}. The inequality in
  Condition~\eqref{eq:cond.quad} is satisfied since 
  \begin{align*}
    q_\eps(u) &= 
    \int_X |\dd u|^2 + \int_{M \setminus X} |\dd u   |^2
    \\ &\ge
    \int_X |\dd u|^2 + 
             \int_{M \setminus X} |\dd (H^\theta (u \restr X))|^2  =
    q^\theta_0(\Phi_\eps u).
  \end{align*}
  Note that the harmonic function $H^\theta (u\restr X)$ minimizes the second
  integral, see Equation~\eqref{eq:harmonic}. Finally,
  Condition~\eqref{eq:cond.ew} is satisfied since $\EWT k \Meps \le \EWD k X$.
  The Main Lemma therefore yields
  \begin{displaymath}
    \EWT k 0 \le \EWT k \Meps + \delta_k(\eps)
  \end{displaymath}
  with $\delta_k(\eps) \to 0$ as $\eps \to 0$. With regard to
  Estimate~\eqref{eq:upper.est.conf2} the proof is finished.
\end{proof}

Next we characerise the domain of the operator $Q^\theta_0$ corresponding to
$q^\theta_0$. Note that $\dom Q^\theta_0$ consists of those $u \in \dom
q^\theta_0$ such that there exists a (unique) element $v \in L_2(X)$
satisfying
\begin{displaymath}
  q^\theta_0(u,w)= \iprod v w
\end{displaymath}
for all $w \in \dom q^\theta_0$. In particular, $Q^\theta_0 u = v$ (see
e.g.~\cite[Theorem~VI.2.1]{kato:66}). Here, we can give a more explicit
characterisation of the limit operator:
\begin{lemma}
  \label{lem:dom.op}
  The domain of the operator $Q_0^\theta$ corresponding to the limit quadratic
  form $q_0^\theta$ is given by
  \begin{equation}
    \label{eq:dom.op}
    \dom Q_0^\theta=\bigl\{ u \in \Sob[2] X \, \bigl| \bigr. \,\, 
         \normder u = \normder H^\theta u \text{ on $\bd X$} \bigr\}.
  \end{equation}
  Furthermore, $Q_0^\theta u = \Delta_X u$ for $u \in \dom Q_0^\theta$.
\end{lemma}
Here, $\normder u$ denotes the normal (outer) derivate with respect to $X$.
Furthermore, $\Sob[2] X$ denotes the Sobolev space of square integrable weak
derivatives up to second order.
\begin{proof}
  The lemma follows from the Gauss-Green Formula and the characterisation of
  $\dom Q^\theta_0$. Note that the integral over $\bd M$ vanishes since
  $h=H^\theta u$ and $\dd h$ are both $\theta$-periodic. Furthermore, $u,
  \Delta u \in L_2(X)$ imply $u \in \Sob[2] X$ by regularity theory.
\end{proof}
 
Since the limit operator is quite complicated, we are only able to construct
an example of a conformally deformed $2$-dimensional manifold with gaps in the
spectrum of its Laplacian:

\begin{example}
  \label{ex:dim.2}
  Let $\Mper :=\R \times \Sphere^1$ be a cylinder with $\Gamma=\Z$ acting on
  $\Mper$ by $\gamma (x,\sigma)=(\gamma+x,\sigma)$.  The periodic metric is
  given by $g=\dd x^2 + r^2 \dd \sigma^2$ for some fixed $r>0$. We choose
  $M=[0,1] \times \Sphere^1$ as period cell.
  
  Let $0<a<b<1$ and let $X=[a,b] \times \Sphere^1$ be the undisturbed region
  of $M$.  Note that normal coordinates exist on $\overline{M \setminus X}$.
  Let $\theta \in \hat \Gamma \cong \Torus^1$. In this context we prefer to
  view $\theta$ as $\e^{\im \theta} \in \Torus^1$.
  
  We first have to calculate the $\theta$-periodic harmonic extension
  $h=H^\theta u$ of a function $u \in \Ci {X}$ given by
  $u(x,\sigma)=v(x)\e^{\im n \sigma}$ for some $n \in \Z$, i.e., we have to
  solve the boundary value problem
  \begin{displaymath}
    \Delta_{M \setminus X} h =
     -\partial_{xx} h - \frac1{r^2} \partial_{\sigma \sigma} h= 0, 
     \qquad \text{with} \qquad
    \begin{aligned}
      h(a,\cdot) &= u(a,\cdot) \\ %v(a) \e^{\im n \sigma},\\
      h(b,\cdot) &= u(b,\cdot) \\ % v(b) \e^{\im n \sigma},\\
      h(1,\cdot) &= \e^{i\theta} h(0,\cdot)\\
      \partial_x h(1,\cdot) &= \e^{i\theta} \partial_x h(0,\cdot)
    \end{aligned}
  \end{displaymath}
  which has a unique solution. Next, we search for eigenvalues $\lambda=\EWT k
  0 \ge 0$ with eigenfunctions $u$. Again, by separating the variables we can
  calculate them expicitely. Since eigenfunctions $u$ have to be in the domain
  of $Q^\theta_0$, the normal derivatives of $u$ and the harmonic extension
  $h$ agree on $\bd X$ by the preceding lemma. Therefore we have to solve
  \begin{displaymath}
    \Delta_ X u =
     -\partial_{xx} u - \frac1{r^2} \partial_{\sigma \sigma} u = \lambda u, 
     \qquad \text{with} \qquad
    \begin{aligned}
      \partial_x u(a,\cdot) &= \partial_x h(a,\cdot) \\ 
      \partial_x u(b,\cdot) &= \partial_x h(b,\cdot).
    \end{aligned}
  \end{displaymath}
  This gives a restriction on the possible values of $\lambda$. A long, but
  straightforward calculation yields
  \begin{align}
    \label{eq:sol.0}
    2 \bigl( \cos (L \omega) - \cos \theta \bigr) &= 
      \ell \omega \sin (L \omega)\\
    \label{eq:sol.not.0}
    \Bigl( \omega^2 - \frac{n^2}{r^2} \Bigr) \sin (L \omega) &= 
    2 \omega \frac n r \cdot
    \frac{\cosh (\ell \frac n r) \cos (L \omega) - \cos \theta}
         {\sinh (\ell \frac n r)}
  \end{align}
  with $\lambda = \omega^2 +n^2/r^2$ and $\ell:=1-L=1-b+a$ where the first
  equation is valid for $n=0$ and $\omega \ge 0$ and the second equation for
  $n \ne 0$ and $\omega > 0$ (see~\cite{post:00}).  Note that $\ell$ is the
  length of the perturbed cylinder $M \setminus X$ and that $L$ is the length
  of the unperturbed cylinder $X$.
  
  If $n=0$ we obtain smooth functions $\theta \mapsto \omega_m(\theta)$ for
  each $m \in \N_0$ solving Equation~\eqref{eq:sol.0} (see
  Figure~\ref{fig:gaps3}). 
  \begin{figure}[h]
    \begin{center}
      \leavevmode
      \begin{picture}(0,0)(-7.5,0)
        \includegraphics{figure-3}
      \end{picture}
      \setlength{\unitlength}{4144sp}
      \begin{picture}(2437,2453)(181,-1647)
        \put(213,-846){$\frac \pi L$}
        \put(181,-190){$\frac{2 \pi} L$}
        \put(2441,520){$\omega_4$}
        \put(2441, 67){$\eta_{1,1}$}
        \put(2441,-190){$\omega_3$}
        \put(2441,-648){$\omega_2$}
        \put(2441,-1352){$\omega_1$}
        \put(1343,-1605){$\pi$}
        \put(310,-1605){$0$}
        \put(2333,-1605){$2\pi$}
        \put(2618,-1482){$\theta$}
        \put(181,425){$\frac{3 \pi} L$}
        \put(407,401){$\sqrt{B_4}$}
        \put(407,165){$\sqrt{B_{1,1}}$}
        \put(407,-88){$\sqrt{B_3}$}
        \put(407,-788){$\sqrt{B_2}$}
        \put(407,-1234){$\sqrt{B_1}$}
        \put(407,665){$\sqrt \lambda$}
        \put(236,  1){$\frac 1 r$}
      \end{picture}
      \caption{The square root of the eigenvalues of the limit operator
          $Q_0^\theta$  plotted for $L=0.5$ and $r=1/13$. Here, at least $m=2$
          gaps occure.}
      \label{fig:gaps3}
    \end{center}
  \end{figure}
  In this case, we have $\lambda=\sqrt{\omega_m(\theta)}$.  Furthermore, note
  that the compact intervals $B_m:=\{\omega^2_m(\theta); \, 0 \le \theta \le
  2\pi \}$ are all disjoint: One can prove that
  \begin{align*}
    \inf B_m &= \Bigl(\frac{m\pi} L \Bigr)^2 & \text{and} &&
    \sup B_m &= \Bigl(\frac{(m+1)\pi} L \Bigr)^2 - \eps_0
  \end{align*}
  for all $m \in \N_0$ if $\eps_0=\eps_0(L)$ is small enough.  Finally
  note that $(m\pi/L)^2$ are the Neumann eigenvalues of the interval $[0,L]$.
  
  If $n \ne 0$ we have 
  \begin{displaymath}
    \eta := 
    \sqrt {\lambda} = 
    \sqrt{\omega^2 + n^2/r^2} >
    \frac {|n|} r \ge 
    \frac 1 r.
  \end{displaymath}
  If we replace $\omega$ by $\sqrt{\eta^2 - n^2/r^2}$ in
  Equation~\eqref{eq:sol.not.0} we obtain solutions $\theta \mapsto
  \eta_{n,p}(\theta)$ for $n,p \in \N$ (see Figure~\ref{fig:gaps3}).  Note
  that we do not expect that the intervals $B_{n,p}:=\{\eta^2_{n,p}(\theta) ;
  \, 0 \le \theta \le 2\pi \}$ are disjoint, we rather expect that the
  intervals $B_{n,p}$ cover the gaps between the intervals $B_m$ when $m,n$ or
  $p$ are large. But we still have $\inf B_{n,p} \ge 1/r^2$, i.e., if $r \le
  \frac L {m \pi}$, the intervals $B_0, \dots, B_m$ remain disjoint.
  Therefore we have proven the following:
  \begin{theorem}
    \label{thm:dim.2}
    Suppose that $\Mepsper$ is a conformally perturbed cylinder of radius
    $r>0$ with conformal factor satisfying Conditions~\eqref{eq:cond.one}
    and~\eqref{eq:cond.small}. Suppose further that the unperturbed area is a
    cylinder $X$ of length $0 < L < 1$ (periodically continued with period
    $1$). Then the corresponding Laplacian has at least $m$ gaps if $r \le
    \frac L {m \pi}$ and if $\eps>0$ is small enough.
  \end{theorem}
\end{example}

\section{Conclusions and outlook}
So far we have proven the existence of two different classes of periodic
manifolds with spectral gaps. But we still have not found a satisfying answer
(in terms of geometrical properties) whether a given periodic manifold has
gaps or not. We define the $\nu$-isoperimetric constant of a periodic manifold
$\Mper$ as
\begin{displaymath}
  I_\nu(\Mper) := 
     \inf_M \inf_\Omega \frac{(\vol_{d-1} \bd \Omega)^\nu}
                             {(\vol_d \Omega)^{\nu-1}}
\end{displaymath}
where $\vol_k$ denotes $k$-dimensional Riemannian volume and $\nu>1$. The
infimum is taken over all period cells $M$ of $\Mper$ and all open
submanifolds $\Omega \subset M$ with $\bd M \cap \bd \Omega = \emptyset$.  In
both classes of examples given in Sections~\ref{sec:constr} and~\ref{sec:conf}
we have $\vol_{d-1}(\bd \Meps) \to 0$ whereas $\vol_d(\Meps)$ is bounded from
below by some positive constant. Therefore $I_\nu(\Mepsper) \to 0$ as $\eps
\to 0$ for all $\nu>1$.  Are the isoperimetric constants some hint for the
analytic decoupling? Here, by analytic decoupling we mean that the $k$-th
$\theta$-periodic eigenvalue of $\Meps$ converges uniformly to some
$\theta$-independent constant $\lambda_k$ (as in
Theorems~\ref{thm:conv.constr},~\ref{thm:conv.cyl} and~\ref{thm:conv.conf}).
Note, for example that the isoperimetric constants do not all converge to $0$
for a straight cylinder $\Sphere^{d-1} \times \R$ of diameter of order $\eps$
and length of a period cell of order $\eps^\alpha$, $\alpha>0$. Clearly, the
spectrum of the straight cylinder has no gaps.

What r\^ole does the curvature play? In the construction in
Section~\ref{sec:constr} as well as the conformal deformation in
Section~\ref{sec:conf} the curvatures are neither bounded from below nor
bounded from above. There is a result of Li \cite{li:94} showing that the
essential spectrum of a complete non-compact Riemannian manifold $\Mper$ with
non-negative Ricci curvature and a pole is equal to $[0,\infty)$. (A pole is a
point $x_0$ where the exponential map is a diffeomorphism from $T_{x_0}\Mper$
onto $\Mper$, which is a strong geometric condition.) If furthermore, $\Mper$
is periodic then the bound on the curvature prevents the existence of spectral
gaps. In contrast, we would expect to have a great number of gaps if the
curvature has great absolute values.


%------------------------------------------------------------
% Acknowledgements (Optional)
%------------------------------------------------------------

\section*{Acknowledgment}
  I am indebted to my thesis advisor Rainer Hempel for his kind attention.
  Furthermore I thank Colette Ann\'e for the helpful discussion concerning her
  article. After all I thank Fernando Lled\'o for final comments.

%\pagebreak
%------------------------------------------------------------
% References
%------------------------------------------------------------

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% file "gaps.bbl" here included
%--------------------------------
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 1388 2301 l gs col0 s gr 
% Polyline
n 2398 2269 m
 2398 2299 l gs col0 s gr 
% Polyline
 [15 30] 30 sd
n 379 750 m
 2413 750 l gs col0 s gr  [] 0 sd
% Polyline
n 341 1649 m
 371 1649 l gs col0 s gr 
% Polyline
n 341 1009 m
 371 1009 l gs col0 s gr 
% Polyline
n 341 384 m
 371 384 l gs col0 s gr 
% Polyline
n 359 1446 m 391 1446 l 391 1650 l 359 1650 l
 cp gs 0.00 setgray ef gr gs col0 s gr 
% Polyline
n 359 744 m 391 744 l 391 759 l 359 759 l
 cp gs 0.00 setgray ef gr gs col0 s gr 
% Polyline
n 359 1908 m 391 1908 l 391 2272 l 359 2272 l
 cp gs 0.00 setgray ef gr gs col0 s gr 
% Polyline
n 359 884 m 391 884 l 391 1009 l 359 1009 l
 cp gs 0.00 setgray ef gr gs col0 s gr 
% Polyline
n 359 287 m 391 287 l 391 383 l 359 383 l
 cp gs 0.00 setgray ef gr gs col0 s gr 
% Polyline
n 341 958 m
 371 958 l gs col0 s gr 
% Polyline
 [15 15] 15 sd
n 301 800 m
 342 957 l gs col0 s gr  [] 0 sd
% Polyline
n 376 2270 m 379 2270 l 385 2270 l 394 2269 l 405 2269 l 418 2267 l
 432 2265 l 448 2262 l 466 2257 l 486 2251 l 505 2243 l
 519 2237 l 528 2232 l 533 2229 l 536 2226 l 538 2223 l
 542 2220 l 550 2214 l 562 2206 l 577 2197 l 594 2186 l
 605 2179 l 609 2175 l 611 2173 l 613 2172 l 617 2169 l
 625 2163 l 638 2154 l 652 2144 l 657 2141 l 654 2141 l
 652 2142 l 660 2137 l 684 2126 l 701 2119 l 716 2112 l
 727 2108 l 735 2105 l 740 2103 l 744 2101 l 748 2100 l
 755 2098 l 765 2094 l 781 2088 l 803 2079 l 831 2068 l
 856 2058 l 878 2049 l 896 2042 l 908 2036 l 916 2033 l
 921 2030 l 925 2028 l 929 2025 l 935 2023 l 944 2019 l
 958 2013 l 978 2006 l 1005 1996 l 1036 1986 l 1068 1976 l
 1096 1967 l 1119 1960 l 1135 1955 l 1147 1951 l 1155 1949 l
 1161 1947 l 1167 1945 l 1175 1942 l 1187 1940 l 1203 1936 l
 1225 1931 l 1252 1926 l 1283 1921 l 1317 1916 l 1344 1914 l
 1363 1912 l 1374 1912 l 1379 1912 l 1382 1913 l 1384 1914 l
 1389 1915 l 1399 1916 l 1416 1917 l 1440 1919 l 1470 1921 l
 1500 1924 l 1524 1927 l 1541 1929 l 1552 1930 l 1558 1931 l
 1562 1932 l 1566 1933 l 1572 1934 l 1583 1936 l 1600 1940 l
 1624 1946 l 1653 1954 l 1682 1963 l 1706 1970 l 1723 1976 l
 1733 1980 l 1739 1983 l 1742 1985 l 1746 1987 l 1751 1990 l
 1762 1994 l 1779 2001 l 1803 2011 l 1832 2022 l 1858 2032 l
 1880 2041 l 1898 2048 l 1911 2053 l 1920 2057 l 1927 2060 l
 1932 2062 l 1937 2064 l 1943 2066 l 1951 2070 l 1963 2075 l
 1979 2081 l 1998 2089 l 2019 2097 l 2045 2107 l 2064 2114 l
 2074 2118 l 2079 2119 l 2080 2119 l 2081 2119 l 2085 2120 l
 2092 2123 l 2106 2129 l 2123 2137 l 2143 2148 l 2155 2155 l
 2160 2159 l 2161 2161 l 2161 2163 l 2166 2167 l 2178 2175 l
 2198 2187 l 2215 2197 l 2229 2204 l 2239 2209 l 2244 2212 l
 2248 2214 l 2251 2216 l 2257 2218 l 2267 2223 l 2282 2229 l
 2302 2237 l 2322 2244 l 2337 2249 l 2348 2253 l 2354 2254 l
 2358 2255 l 2362 2256 l 2368 2257 l 2377 2259 l 2390 2262 l
 2405 2266 l 2426 2270 l 2439 2271 l 2446 2271 l 2451 2271 l

 2453 2270 l gs col0 s gr 
% Polyline
n 372 1448 m 375 1448 l 382 1448 l 393 1447 l 406 1447 l 422 1447 l
 439 1447 l 460 1448 l 484 1450 l 512 1452 l 535 1455 l
 555 1457 l 568 1459 l 577 1460 l 581 1461 l 583 1461 l
 586 1462 l 591 1463 l 600 1464 l 614 1467 l 636 1472 l
 663 1477 l 691 1483 l 714 1488 l 732 1492 l 743 1495 l
 750 1496 l 755 1497 l 760 1498 l 767 1500 l 779 1503 l
 798 1508 l 823 1515 l 853 1523 l 883 1532 l 908 1539 l
 926 1544 l 936 1548 l 943 1550 l 947 1552 l 950 1554 l
 957 1556 l 967 1560 l 985 1565 l 1009 1572 l 1039 1581 l
 1069 1589 l 1093 1596 l 1110 1601 l 1121 1604 l 1127 1606 l
 1131 1608 l 1135 1609 l 1141 1611 l 1152 1613 l 1169 1617 l
 1193 1622 l 1222 1627 l 1251 1632 l 1274 1635 l 1291 1637 l
 1300 1639 l 1306 1640 l 1309 1641 l 1312 1641 l 1317 1641 l
 1327 1642 l 1344 1642 l 1368 1642 l 1398 1642 l 1424 1641 l
 1447 1639 l 1465 1638 l 1478 1637 l 1486 1636 l 1492 1635 l
 1496 1634 l 1500 1633 l 1506 1632 l 1514 1630 l 1527 1628 l
 1545 1625 l 1568 1621 l 1595 1616 l 1625 1610 l 1650 1605 l
 1668 1601 l 1679 1598 l 1686 1596 l 1690 1594 l 1694 1593 l
 1701 1591 l 1711 1588 l 1728 1583 l 1751 1577 l 1778 1570 l
 1804 1563 l 1825 1557 l 1839 1553 l 1847 1551 l 1851 1549 l
 1852 1549 l 1853 1548 l 1857 1547 l 1865 1544 l 1879 1540 l
 1901 1534 l 1929 1527 l 1954 1521 l 1977 1515 l 1995 1511 l
 2008 1507 l 2018 1505 l 2024 1503 l 2030 1502 l 2035 1501 l
 2041 1499 l 2051 1497 l 2065 1494 l 2083 1490 l 2107 1485 l
 2133 1480 l 2163 1474 l 2187 1470 l 2204 1467 l 2215 1465 l
 2221 1464 l 2226 1464 l 2229 1463 l 2235 1463 l 2245 1462 l
 2259 1460 l 2279 1457 l 2302 1455 l 2328 1453 l 2349 1451 l
 2365 1451 l 2378 1450 l 2389 1451 l 2397 1451 l 2404 1451 l
 2408 1452 l
 2410 1452 l gs col0 s gr 
% Polyline
n 376 1007 m 378 1007 l 382 1007 l 388 1008 l 397 1008 l 409 1008 l
 422 1008 l 437 1008 l 454 1008 l 473 1008 l 494 1007 l
 518 1005 l 546 1003 l 577 1000 l 609 996 l 636 993 l
 658 990 l 674 988 l 684 986 l 690 985 l 695 984 l
 700 982 l 707 981 l 718 979 l 734 976 l 758 972 l
 788 966 l 824 960 l 855 954 l 884 949 l 906 945 l
 923 941 l 934 939 l 941 937 l 945 935 l 947 934 l
 950 932 l 955 931 l 962 929 l 975 926 l 994 923 l
 1020 919 l 1052 915 l 1090 910 l 1125 906 l 1158 903 l
 1186 900 l 1209 898 l 1228 896 l 1242 895 l 1252 894 l
 1261 893 l 1268 892 l 1275 891 l 1283 890 l 1294 890 l
 1308 889 l 1327 889 l 1351 888 l 1381 888 l 1415 888 l
 1452 888 l 1493 889 l 1528 891 l 1557 893 l 1578 894 l
 1592 896 l 1601 897 l 1606 899 l 1610 900 l 1613 901 l
 1618 903 l 1627 905 l 1640 907 l 1661 910 l 1688 914 l
 1722 919 l 1760 924 l 1794 929 l 1826 933 l 1853 937 l
 1875 941 l 1893 944 l 1906 946 l 1915 948 l 1923 949 l
 1929 950 l 1935 951 l 1942 953 l 1951 954 l 1963 957 l
 1979 959 l 1999 962 l 2023 966 l 2051 971 l 2080 975 l
 2115 980 l 2144 985 l 2166 988 l 2180 990 l 2189 992 l
 2194 992 l 2196 993 l 2199 993 l 2203 994 l 2209 995 l
 2221 996 l 2237 998 l 2258 1001 l 2281 1003 l 2312 1006 l
 2337 1007 l 2356 1008 l 2372 1008 l 2385 1008 l 2395 1008 l
 2403 1007 l 2408 1007 l
 2410 1007 l gs col0 s gr 
% Polyline
n 376 287 m 378 287 l 383 287 l 390 287 l 401 288 l 416 288 l
 433 289 l 452 290 l 473 291 l 496 292 l 520 293 l
 546 295 l 575 297 l 606 299 l 640 302 l 677 305 l
 713 308 l 745 312 l 771 314 l 791 317 l 805 319 l
 814 320 l 820 321 l 824 322 l 829 323 l 835 324 l
 844 325 l 857 327 l 876 330 l 902 333 l 933 337 l
 968 341 l 1003 345 l 1034 349 l 1058 352 l 1077 355 l
 1090 357 l 1098 358 l 1103 359 l 1107 361 l 1111 361 l
 1116 363 l 1124 364 l 1137 365 l 1156 367 l 1181 369 l
 1212 372 l 1247 374 l 1282 376 l 1314 377 l 1340 378 l
 1360 379 l 1374 379 l 1383 379 l 1389 380 l 1394 380 l
 1399 380 l 1405 380 l 1415 379 l 1429 379 l 1449 378 l
 1476 377 l 1508 376 l 1545 374 l 1582 372 l 1614 369 l
 1641 367 l 1661 364 l 1675 363 l 1685 361 l 1691 360 l
 1695 359 l 1700 358 l 1706 356 l 1715 354 l 1729 352 l
 1750 350 l 1777 346 l 1810 342 l 1847 338 l 1884 334 l
 1918 330 l 1945 327 l 1966 325 l 1982 323 l 1992 321 l
 1999 320 l 2005 320 l 2010 319 l 2018 318 l 2028 317 l
 2042 315 l 2062 313 l 2089 311 l 2120 308 l 2155 305 l
 2194 302 l 2229 299 l 2260 297 l 2287 296 l 2311 294 l
 2332 293 l 2351 293 l 2368 292 l 2383 292 l 2395 291 l
 2403 291 l 2408 291 l
 2410 291 l gs col0 s gr 
$F2psEnd
rs

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