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\begin{document}
\title[]{Cheeger-Gromov Theory and Applications to General Relativity}
\author[]{Michael T. Anderson}
\thanks{Partially supported by NSF Grant DMS 0072591}
\maketitle
\tableofcontents
\setcounter{section}{0}
This paper surveys aspects of the convergence and degeneration of
Riemannian metrics on a given manifold $M$, and some recent
applications of this theory to general relativity. The basic point of
view of convergence/degeneration described here originates in the work
of Gromov, cf. [27]-[29], with important prior work of Cheeger [14],
leading to the joint work of [16].
This Cheeger-Gromov theory assumes $L^{\infty}$ bounds on the full
curvature tensor. For reasons discussed below, we focus mainly on the
generalizations of this theory to spaces with $L^{\infty},$ (or
$L^{p})$ bounds on the Ricci curvature. Although versions of the
results described hold in any dimension, for the most part we restrict
the discussion to 3 and 4 dimensions, where stronger results hold and
the applications to general relativity are most direct.
I am grateful to many of the participants of the Carg\`ese meeting for
their comments and suggestions, and in particular to Piotr Chru\'sciel
for organizing such a fine meeting.
\section{Background: Examples and Definitions.}
\setcounter{equation}{0}
The space ${\mathbb M}$ of Riemannian metrics on a given manifold $M$
is an infinite dimensional cone, (in the vector space of symmetric
bilinear forms on $M$), and so is highly non-compact. Arbitrary
sequences of Riemannian metrics can degenerate in very complicated
ways.
On the other hand, there are two rather trivial but nevertheless
important sources of non-compactness.
\medskip
$\bullet$ {\it Diffeomorphisms}. The group $\mathcal{D}$ of
diffeomorphisms of $M$ is non-compact and acts properly on ${\mathbb
M}$ by pullback. Hence, if $g$ is any metric in ${\mathbb M}$ and
$\phi_{i}$ is any divergent sequence of diffeomorphisms, then $g_{i} =
\phi_{i}^{*}g$ is a divergent sequence in ${\mathbb M}$. However, all
the metrics $g_{i}$ are isometric, and so are indistinguishable
metrically. In terms of a local coordinate representation, the metrics
$g_{i}$ locally are just different representatives of the fixed metric
$g$.
Thus, for most problems, one considers only equivalence classes of
metrics $[g]$ in the moduli space
$$\mathcal{M} = {\mathbb M} /\mathcal{D}.$$
(A notable exception is the Yamabe problem, which is not well-defined
on $\mathcal{M}$).
$\bullet$ {\it Scaling}. For a given metric $g$ and parameter $\lambda
> $ 0, let $g_{\lambda} = \lambda^{2}g$ so that all distances are
rescaled by a factor of $\lambda .$ If $\lambda \rightarrow \infty ,$
or $\lambda \rightarrow $ 0, the metrics $g_{\lambda}$ diverge. In the
former case, the manifold $(M, g_{\lambda})$, say compact, becomes
arbitrarily large, in that global invariants such as diameter, volume,
etc. diverge to infinity; there is obviously no limit metric. In the
latter case, $(M, g_{\lambda})$ converges, as a family of metric
spaces, to a single point. Again, there is no limiting Riemannian
metric on $M$.
\medskip
Although one has divergence in both cases described above, they can be
combined in natural ways to obtain convergence. Thus, for $g_{\lambda}$
as above, suppose $\lambda \rightarrow \infty ,$ and choose any fixed
point $p\in M.$ For any fixed $k > $ 0, consider the geodesic ball
$B_{p} = B_{p}(k/\lambda ),$ so the $g$-radius of this ball is
$k/\lambda \rightarrow $ 0, as $\lambda \rightarrow \infty$. On the
other hand, in the metric $g_{\lambda},$ the ball $B_{p}$ is a geodesic
ball of fixed radius $k$. Since $k/\lambda $ is small, one may choose a
local coordinate system $\mathcal{U} = \{u_{i}\}$ for $B_{p},$ with $p$
mapped to the origin in ${\mathbb R}^{n}.$ Let $u_{i}^{\lambda} =
\lambda u_{i} = \phi_{\lambda} \circ u_{i}$, where $\phi_{\lambda}(x) =
\lambda x$. Thus $\phi_{\lambda}$ is a divergent sequence of
diffeomorphisms of $\mathbb{R}^{n}$, and $\mathcal{U}_{\lambda} =
\{u_{i}^{\lambda}\}$ is a new collection of charts. One then easily
sees that
\begin{equation} \label{e1.1}
g_{\lambda}(\partial /\partial u_{i}^{\lambda}, \partial /\partial
u_{j}^{\lambda}) = g(\partial /\partial u_{i}, \partial /\partial
u_{j}) = g_{ij}.
\end{equation}
As $\lambda \rightarrow \infty ,$ the ball $B_{p}$ shrinks to the
point $p$ and the coefficients $g_{ij}$ tend to the constants
$g_{ij}(p).$ On the other hand, the metrics $g_{\lambda}$ are defined
on the intrinsic geodesic ball of radius $k$. Since $k$ is arbitrary,
the metrics $\phi_{\lambda}^{*}g_{\lambda}$ converge smoothly to the
limit flat metric $g_{0}$ on the tangent space $T_{p}(M),$ induced by
the inner product $g_{p}$ on $T_{p}(M),$
\begin{equation} \label{e1.2}
(M, \phi_{\lambda}^{*}g_{\lambda}) \rightarrow (T_{p}M, g_{0}).
\end{equation}
This process is called ``{\sf blowing up}'', since one restricts
attention to smaller and smaller balls, and blows them up to a definite
size. Note that the part of $M$ at any definite $g$-distance to $p$
escapes to infinity, and is not detected in the limit $g_{0}.$ Thus, it
is important to attach base points to the blow-up construction;
different base points may give rise to different limits, (although in
this situation all pointed limits are isometric).
\medskip
There is an analogous, although more subtle blowing up process for
Lorentzian metrics due to Penrose, where the limits are non-flat plane
gravitational waves, cf. [37].
\bigskip
If $(M, g)$ is complete and non-compact, one can carry out a similar
procedure with $\lambda \rightarrow 0$, called ``{\sf blowing down}'',
where geodesic balls, (about a given point), of large radius
$B_{p}(k/\lambda )$ are rescaled down to unit size, i.e. size $k$. This
is of importance in understanding the large scale or asymptotic
behavior of the metric and will arise in later sections.
\medskip
This discussion leads to the following definition for convergence of
metrics. Let $\Omega $ be a domain in ${\mathbb R}^{n}$ and let
$C^{k,\alpha}$ denote the usual H\"{o}lder space of $C^{k}$ functions
on $\Omega$ with $\alpha$-H\"{o}lder continuous $k^{\rm th}$ partial
derivatives. Similarly, let $L^{k,p}$ denote the Sobolev space of
functions with $k$ weak derivatives in $L^{p}.$ Since one works only
locally, we are only interested in the local spaces
$C^{k,\alpha}_{loc}$ and $L^{k,p}_{loc}$ and corresponding local norms
and topology.
\begin{definition} \label{d 1.1}
{\rm A sequence of metrics $g_{i}$ on $n$-manifolds $M_{i}$ is said to
{\sf converge in the $L^{k,p}$ topology} to a limit metric $g$ on the
$n$-manifold $M$ if there is a locally finite collection of charts
$\{\phi_{k}\}$ covering $M$, and a sequence of diffeomorphisms $F_{i}:
M \rightarrow M_{i},$ such that
\begin{equation} \label{e1.3}
(F_{i}^{*}g_{i})_{\alpha\beta} \rightarrow g_{\alpha\beta},
\end{equation}
in the $L^{k,p}_{loc}$ topology. Here $(F_{i}^{*}g_{i})_{\alpha\beta}$
and $g_{\alpha\beta}$ are the local component functions of the metrics
$F_{i}^{*}g_{i}$ and $g$ in the charts $\phi_{k}.$ }
\end{definition}
The same definition holds for convergence in the $C^{k,\alpha}$
topology, as well as the weak $L^{k,p}$ topology. (Recall that a
sequence of functions $f_{i}\in L^{p}(\Omega )$ converges weakly in
$L^{p}$ to a limit $f \in L^{p}(\Omega )$ iff $\int f_{i}g \rightarrow
\int fg,$ for all $g\in L^{q}(\Omega ),$ where $p^{-1}+q^{-1} =$ 1).
It is easily seen that this definition of convergence is independent
of the choice of charts $\{\phi_{k}\}$ covering $M$. The manifolds $M$
and $M_{i}$ are not required to be closed.
\bigskip
In order to obtain local control on a metric, or sequence of metrics,
one assumes curvature bounds. The theory described by Cheeger-Gromov
requires a bound on the full Riemann curvature tensor
\begin{equation} \label{e1.4}
|Riem| \leq K,
\end{equation}
for some $K < \infty .$ Since the number of components of the Riemann
curvature is much larger than that of the metric tensor itself, (in
dimensions $\geq 4$), this corresponds to an overdetermined set of
constraints on the metric and so is overly restrictive. It is much more
natural to impose bounds on the Ricci curvature
\begin{equation} \label{e1.5}
|Ric| \leq k,
\end{equation}
since the Ricci curvature is a symmetric bilinear form, just as the
metric is. Of course, assuming bounds on Ricci is natural in general
relativity, via the Einstein equations. Thus throughout the paper, we
emphasize (1.5) over (1.4) whenever possible.
\bigskip
The Cheeger-Gromov theory may be viewed as a vast generalization of
the basic features of Teichm\"{u}ller theory to higher dimensions and
variable curvature, (although it was not originally phrased in this
way). Recall that Teichm\"{u}ller theory describes the moduli space
$\mathcal{M}_{c}$ of constant curvature metrics on surfaces. On closed
surfaces, one has a {\it basic trichotomy} for the behavior of
sequences of such metrics, normalized to unit area:
\smallskip
$\bullet$ {\it Compactness/Convergence}. A sequence $g_{i}\in
\mathcal{M}_{c}$ has a subsequence converging smoothly, ($C^{\infty}$),
to a limit metric $g\in \mathcal{M}_{c}.$ As in the definition above,
the convergence is understood to be modulo diffeomorphisms. For
instance this is always the case on $S^{2},$ since the moduli space
$\mathcal{M}_{c}$ is a point for $S^{2}$.
\smallskip
$\bullet$ {\it Collapse}. The sequence $g_{i}\in \mathcal{M}_{c}$
collapses everywhere, in that
\begin{equation} \label{e1.6}
inj_{g_{i}}(x) \rightarrow 0,
\end{equation}
at every $x$, where $inj_{g_{i}}$ is the injectivity radius w.r.t.
$g_{i}$. This collapse occurs only on the torus $T^{2}$ and such
metrics become very long and very thin. There is no limit metric on
$T^{2}.$ Instead, by choosing (arbitrary) base points $x_{i},$ one may
consider based sequences $(T^{2}, g_{i}, x_{i}),$ whose limits are then
the collapsed space $({\mathbb R} , g_{\infty}, x_{\infty}).$
\smallskip
$\bullet$ {\it Cusp Formation}. This is a mixture of the two previous
cases, and occurs only for hyperbolic metrics, i.e. on surfaces
$\Sigma_{g}$ of genus $g \geq $ 2. In this case, there are based
sequences $(\Sigma_{g}, g_{i}, x_{i})$ which converge to a limit
$(\Sigma , g_{\infty}, x_{\infty})$ which is a complete non-compact
hyperbolic surface of finite volume, hence with a finite number of cusp
ends $S^{1}\times {\mathbb R}^{+}$. The convergence is smooth, and
uniform on compact subsets. As one goes to infinity in any cusp end
$S^{1}\times {\mathbb R}^{+}$, the limit metric collapses in the sense
that $inj_{g_{\infty}}(z_{k}) \rightarrow 0$, as $z_{k} \rightarrow
\infty$. There are other based sequences $(\Sigma , g_{i}, y_{i})$
which collapse, i.e. (1.6) holds on domains of arbitrarily large but
bounded diameter about $y_{i}$. As before, limits of such sequences are
of the form $({\mathbb R} , g_{\infty}, y_{\infty}).$
\section{Convergence/Compactness.}
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To prove the (pre)-compactness of a family of metrics, or the
convergence of a sequence of metrics, the main point is to establish a
lower bound on the radius of balls on which one has apriori control of
the metric in a given topology, say $C^{k,\alpha}$ or $L^{k,p}.$ Given
such uniform local control, it is then usually straightforward to
obtain global control, via suitable global assumptions on the volume or
diameter. (Alternately, one may work instead on domains of bounded
diameter).
To obtain such local control, the first issue is to choose a good
"gauge", i.e. representation of the metric in local coordinates. For
this, it is natural to look at coordinates built from the geometry of
the metric itself. In the early stages of development of the theory,
geodesic normal coordinates were used. Later, Gromov [27] used suitable
distance coordinates. However, both these coordinate systems entail
loss of derivatives - two in the former case, one in the latter. It is
now well-known that Riemannian metrics have optimal regularity
properties in harmonic coordinates, cf. [21]; this is due to the
special form of the Ricci curvature in harmonic coordinates, known to
relativists long ago.
Given the choice of harmonic gauge, it is natural to associate a
harmonic radius $r_{h}: M \rightarrow {\mathbb R}^{+}$, which measures
the size of balls on which one has harmonic coordinates in which the
metric is well controlled. The precise definition, cf. [1], is as
follows.
\begin{definition} \label{d2.1}
{\rm Fix a function topology, say $L^{k,p}$, and a constant $c_{o} >
1$. Given $x\in (M, g)$, define the $L^{k,p}$ harmonic radius to be the
largest radius $r_{h}(x) = r_{h}^{k,p}(x)$ such that on the ball
$B_{x}(r_{h}(x))$ one has a harmonic coordinate chart $U =
\{u_{\alpha}\}$ in which the metric $g = g_{\alpha\beta}$ is controlled
in $L^{k,p}$ norm: thus,
\begin{equation} \label{e2.1}
c_{o}^{-1}\delta_{\alpha\beta} \leq g_{\alpha\beta} \leq
c_{o}\delta_{\alpha\beta},
\ \ {\rm (as \ bilinear \ forms)},
\end{equation}
\begin{equation} \label{e2.2}
[r_{h}(x)]^{kp-n}\int_{B_{x}(r_{h}(x))}|\partial^{k}g_{\alpha\beta}|^{p}
dV \leq c_{o}-1.
\end{equation}}
\end{definition}
Here, it always assumed that $kp > n = dim M$, so that $L^{k,p}$ embeds
in $C^{0},$ via Sobolev embedding. The precise value of $c_{o}$ is
usually unimportant, but is understood to be fixed once and for all.
Both estimates in (2.1)-(2.2) are scale invariant, (when the harmonic
coordinates are rescaled as in (1.1)), and hence the harmonic radius
scales as a distance.
\medskip
Note that if $r_{h}(x)$ is large, then the metric is close to the
flat metric on large balls about $x$, while if $r_{h}(x)$ is small,
then the derivatives of $g_{\alpha \beta}$ up to order $k$ are large in
$L^{p}$ on small balls about $x$. Thus, the harmonic radius serves as a
measure of the degree of concentration of $g_{\alpha \beta}$ in the
$L^{k,p}$ norm.
It is important to observe that the harmonic radius is continuous with
respect to the (strong) $L^{k,p}$ topology on the space of metrics, cf.
[1], [3]. In general, it is not continuous in the weak $L^{k,p}$
topology.
One may define such harmonic radii w.r.t. other topologies, for
instance $C^{k,\alpha}$ in a completely analogous way; these have the
same properties.
\medskip
Suppose $g_{k}$ is a sequence of metrics on a manifold $M$, (possibly
open), with a uniform lower bound on $r_{h}.$ On each ball, one then
has $L^{k,p}$ control of the metric components. The well-known
Banach-Alaoglu theorem, (bounded sequences are weakly compact in Banach
spaces), then implies that the metrics on the ball have a weakly
convergent subsequence in $L^{k,p},$ so one obtains a limit metric on
each ball. It is straightforward to verify that the overlaps of these
charts are in $L^{k+1,p}$, and so one has a limit $L^{k,p}$ metric on
$M$. The convergence to limit is in the weak $L^{k,p}$ topology and
uniform on compact subsets. Strictly speaking, one also has to prove
that the harmonic coordinate charts for $g_{k}$ also converge, or more
precisely may be replaced by a fixed coordinate chart, but this also is
not difficult, cf. [1], [3] for details.
The same type of arguments hold w.r.t. the $C^{k,\alpha}$ topology,
via the Arzela-Ascoli theorem; here weak $L^{k,p}$ convergence is
replaced by convergence in the $C^{k,\alpha'}$ topology, for $\alpha'
< \alpha$.
\medskip
Thus, the main issue in obtaining a convergence result is to obtain a
lower bound on a suitable harmonic radius $r_{h}$ under geometric
bounds. The following result is one typical example.
\begin{theorem} \label{t 2.2} {\bf (Convergence I).}
Let $M$ be a closed $n$-manifold and let $\mathcal{M} (\lambda , i_{o},
D)$ be the space of Riemannian metrics such that
\begin{equation} \label{e2.3}
|Ric| \leq k, \ inj \geq i_{o}, \ diam \leq D.
\end{equation}
Then $\mathcal{M} (\lambda , i_{o},$ D) is precompact in the
$C^{1,\alpha}$ and weak $L^{2,p}$ topologies, for any $\alpha < $ 1
and $p < \infty$.
\end{theorem}
Thus, for any sequence, there is a subsequence which converges, in
these topologies, to a limit $C^{1,\alpha}\cap L^{2,p}$ metric
$g_{\infty}$ on M.
\medskip
\noindent
{\bf Sketch of Proof}: As discussed above, it suffices to prove a
uniform lower bound on the $L^{2,p}$ harmonic radius $r_{h} =
r_{h}^{2,p},$ i.e.
\begin{equation} \label{e2.4}
r_{h}(x) \geq r_{o} = r_{o}(k, i_{o}, D),
\end{equation}
under the bounds (2.3).
Overall, the proof of (2.4) is by contradiction. Thus, if (2.4) is
false, there is a sequence of metrics $g_{i}$ on $M$, satisfying the
bounds (2.3), but for which $r_{h}(x_{i}) \rightarrow $ 0, for some
points $x_{i}\in M$. Without loss of generality, (since $M$ is closed),
assume that the base points $x_{i}$ realize the minimal value of
$r_{h}$ on $(M, g_{i}).$ Then rescale the metrics $g_{i}$ by this
minimal harmonic radius, i.e. set
\begin{equation} \label{e2.5}
\bar g_{i} = r_{h}(x_{i})^{-2}\cdot g_{i}.
\end{equation}
If $\bar r_{h}$ denotes the harmonic radius w.r.t. $\bar g,$ by scaling
properties one has
\begin{equation} \label{e2.6}
\bar r_{h}(x_{i}) = 1, \ \ {\rm and} \ \ \bar r_{h}(y_{i}) \geq 1,
\end{equation}
for all $y_{i}\in (M, \bar g_{i}).$ By the remarks preceeding the
proof, the pointed Riemannian manifolds $(M, \bar g_{i}, x_{i})$ have a
subsequence converging in the {\it weak} $L^{2,p}$ topology to a
limit $L^{2,p}$ Riemannian manifold $(N, \bar g_{\infty}, x_{\infty}).$
(Again, this convergence is understood to be modulo diffeomorphisms, as
in Definition 1.1). Of course $diam_{\bar g_{i}}M \rightarrow \infty
,$ so that the complete open manifold $N$ is distinct from the original
compact manifold $M$. The convergence is uniform on compact subsets.
So far, nothing essential has been done - the construction above more
or less amounts to just renormalizations. There are two basic
ingredients in obtaining further control however, one geometric and one
analytic.
We begin with the geometric argument. The limit space $(N, \bar
g_{\infty})$ is Ricci-flat, since the bound (2.3) on the Ricci
curvature of $g_{i}$ becomes in the scale $\bar g_{i},$
\begin{equation} \label{e2.7}
|Ric_{\bar g_{i}}| \leq k\cdot r_{h}(x_{i}) \rightarrow 0, \ \ {\rm
as} \ \ i \rightarrow \infty .
\end{equation}
Actually, it is Ricci-flat in a weak sense, since the convergence is
only in weak $L^{2,p}.$ However, it is easy to see, (cf. also below),
that weak $L^{2,p}$ solutions of the (Riemannian) Einstein equations
are real-analytic, and so the limit is in fact a smooth Ricci-flat
metric.
Next, by (2.3), the injectivity radius of $\bar g_{i}$ satisfies
\begin{equation} \label{e2.8}
inj_{\bar g_{i}} \geq i_{o}\cdot r_{h}(x_{i})^{-1} \rightarrow
\infty , \ \ {\rm as} \ \ i \rightarrow \infty ,
\end{equation}
so that, roughly speaking, the limit $(N, \bar g_{\infty})$ has
infinite injectivity radius at every point. More importantly, the bound
(2.8) implies that $(M, \bar g_{i})$ contains arbitrarily long,
(depending on $i$), minimizing geodesics in any given direction through
the center point $x_{i}.$ It follows that the limit $(N, \bar
g_{\infty})$ has infinitely long minimizing geodesics in every
direction through the base point $x_{\infty}.$ This means that $(N,
\bar g_{\infty})$ contains a line in every direction through
$x_{\infty}.$
Now the well-known Cheeger-Gromoll splitting theorem [15] states that
a complete manifold with non-negative Ricci curvature splits
isometrically along any line. It follows that $(N, \bar g_{\infty})$
splits isometrically in every direction through $x_{\infty},$ and hence
$(N, \bar g_{\infty}) = ({\mathbb R}^{n}, g_{0}),$ where $g_{0}$ is the
flat metric on ${\mathbb R}^{n}.$
Now of course $({\mathbb R}^{n}, g_{0})$ has infinite harmonic radius.
If the convergence of $(N, \bar g_{i})$ to the limit $({\mathbb R}^{n},
g_{0})$ can be shown to be in the {\sf strong} $L^{2,p}$ topology, then
the continuity of $r_{h}$ in this topology immediately gives a
contradiction, since by (2.6), the limit $(N, \bar g_{\infty})$ has
$r_{h}(x_{\infty}) = 1$.
\medskip
The second or analytic part of the argument is to prove strong
$L^{2,p}$ convergence to the limit. The idea here is to use elliptic
regularity to bootstrap or improve the smoothness of the convergence.
In harmonic coordinates, the Ricci curvature of a metric $g$ has the
following especially simple form:
\begin{equation} \label{e2.9}
-\frac{1}{2}\Delta g_{\alpha\beta} + Q_{\alpha\beta}(g, \partial g)=
Ric_{\alpha\beta},
\end{equation}
where $\Delta = g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}$ is
the Laplacian w.r.t. the metric $g$ and $Q$ is quadratic in $g$, its
inverse, and $\partial g.$ In particular, if $r_{h}(x) =$ 1 and
$r_{h}(y) \geq r_{o} > $ 0, for all $y\in\partial B_{x}(1),$ then one
has a uniform $L^{1,p}$ bound on $Q$ and uniform $L^{2,p}$ bounds on
the coefficients for the Laplacian within $B_{x}(1+\frac{1}{2}r_{o}).$
If now $Ric$ is uniformly bounded in $L^{\infty},$ then standard
elliptic regularity applied to (2.9) implies that $g_{\alpha\beta}$ is
uniformly controlled in $L^{2,q},$ for any $q < \infty$, (in
particular for $q > p$). More importantly, if $g_{i}$ is a sequence of
metrics for which $(Ric_{g_{i}})_{\alpha\beta}$ converges strongly in
$L^{p}$ to a limit $(Ric_{g_{\infty}})_{\alpha\beta},$ then elliptic
regularity again implies that the metrics $(g_{i})_{\alpha\beta}$
converge strongly in $L^{2,p}$ to the limit
$(g_{\infty})_{\alpha\beta}.$ For the metrics $\bar g_{i}$, (2.7)
implies that $Ric \rightarrow 0$ in $L^{\infty}$, and so $Ric
\rightarrow 0$ strongly in $L^{q}$, for any $q < \infty$.
These remarks essentially prove that the $L^{2,p}$ harmonic radius is
continuous w.r.t. the strong $L^{2,p}$ topology. Further, when applied
to the sequence $\bar g_{i}$ and using (2.6), they imply that the
metrics $\bar g_{i}$ converge strongly in $L^{2,p}$ to the limit $\bar
g_{\infty}.$ This completes the proof.
\bigskip
It is easy to see from the proof that the lower bound on the
injectivity radius in (2.3) can be considerably weakened. For instance,
define the 1-cross $Cro_{1}(x)$ of $(M, g)$ at $x$ to be the length of
the longest minimizing geodesic in $(M, g)$ with center point $x$ and
set
$$Cro_{1}(M,g) = \inf_{x}Cro_{1}(x).$$
We introduce this notion partly because it has a natural analogue in
Lorentzian geometry, when a minimizing geodesic is replaced by a
maximizing time-like geodesic, cf. \S 5. Then one has the following
result on 4-manifolds, cf. [4].
\begin{theorem} \label{t 2.3} {\bf (Convergence II).}
Let $M$ be a 4-manifold. Then the conclusions of Theorem 2.1 hold under
the bounds
\begin{equation} \label{e2.10}
|Ric| \leq k, \ Cro_{1} \geq c_{o}, \ vol \geq v_{o}, \ diam \leq
D.
\end{equation}
\end{theorem}
The proof is the same as that of Theorem 2.2. The lower bound on
$Cro_{1}$ implies that on the blow-up limit $(N, \bar g_{\infty},
x_{\infty})$ above, one has {\it a} line. Hence, the splitting theorem
implies that $N = N' \times {\mathbb R}$. It follows that $N'$ is
Ricci-flat and hence, since $dim N' = 3$, $N'$ is flat. Using the
volume bound in (2.10), it follows that $(N, \bar g_{\infty}) =
({\mathbb R}^{4}, g_{0})$, cf. (2.12)-(2.13) below. (The volume bound
rules out the possibility that $N'$ is a non-trivial flat manifold of
the form ${\mathbb R}^{3}/\Gamma$). This gives the same contradiction
as before.
\medskip
Of course, in dimension 3 any Ricci-flat manifold is necessarily flat,
and so the same proof shows that one has $C^{1,\alpha}$ and $L^{2,p}$
precompactness within the class of metrics on 3-manifolds satisfying
\begin{equation} \label{e2.11}
|Ric| \leq k, \ vol \geq v_{o}, \ diam \leq D.
\end{equation}
\begin{remark} \label{r 2.4.} {\bf (i).}
{\rm Although (2.4) gives the existence of a lower bound on $r_{h}$ in
terms of the bounds $k$, $i_{o}$ and $D$, currently there is no proof
of an effective or computable bound. Equivalently, there is no direct
proof of Theorem 2.1, which does not involve a passage to limits and
invoking a contradiction. This is closely related to the fact there is
currently no {\it quantitative} or {\it finite} version of the
Cheeger-Gromoll splitting theorem, where one can deduce definite bounds
on the metric in the presence of (a collection of) minimizing geodesics
of a finite but definite length.
If however the bound on $|Ric|$ in (2.3) is strengthened to a bound
on $|Riem|$, as in (1.4), then it is not difficult to obtain an
effective or computable lower bound on $r_{h}$, cf. [32].}
{\bf (ii).} {\rm The proof above can be easily adapted to give a
similar result if the $L^{\infty}$ bound on $Ric$ is replaced by an
$L^{q}$ bound, for some $q > n/2;$ one then obtains convergence in
weak $L^{2,q}.$
In the opposite direction, the convergence can be improved if one has
bounds on the derivatives of the Ricci curvature. This will be the case
if $Ric$ satisfies an elliptic system of PDE, for instance the Einstein
equations. In this case, one obtains $C^{\infty}$ convergence to the
limit. }
{\bf (iii).} {\rm The assumption that $M$ is closed in Theorem 2.2 is
merely for convenience, and an analogous result holds for open
manifolds, away from the boundary. }
\end{remark}
The bounds on injectivity radius in (2.3), or even the 1-cross in
(2.10), are rather strong and one would like to replace them with
merely a lower volume bound, as in (2.11).
An elementary but important result, the volume comparison theorem of
Bishop-Gromov [27], states that if $Ric \geq (n-1)k$, for some $k$, on
$(M, g)$, $n = dim M$, then the ratio
\begin{equation} \label{e2.12}
\frac{volB_{x}(r)}{volB_{k}(r)} \downarrow
\end{equation}
is monotone non-increasing in $r$; here $volB_{k}(r)$ is the volume of
the geodesic $r$-ball in the $n$-dimensional space form of constant
curvature $k$. In particular, if the bounds (2.11) hold, in dimension
$n$, then (2.12) gives a lower bound on the volumes of balls on {\it
all} scales:
\begin{equation} \label{e2.13}
volB_{x}(r) \geq \frac{volM}{volB_{k}(D)}\cdot volB_{k}(r).
\end{equation}
Note that the estimate (2.13) also implies that, for any fixed $r > 0$,
if $volB_{x}(r) \geq v_{0} > 0$, then $volB_{y}(r) \geq v_{1} > 0$,
where $v_{1}$ depends only on $v_{0}$ and $dist_{g}(x, y)$. Thus, the
ratio of the volumes of unit balls cannot become arbitrarily large or
small on domains of bounded diameter.
Now a classical result of Cheeger [14] implies that if (2.11) is
strengthened to
\begin{equation} \label{e2.14}
K_{P} \geq - K, \ vol \geq v_{o}, \ diam \leq D,
\end{equation}
where $K_{P}$ is the sectional curvature of any plane $P$ in the
tangent bundle $TM$, then one has a lower bound on the injectivity
radius, $inj_{g}(M) \geq i_{o}(K, v_{o}, D)$. However, this estimate
fails under the bounds (2.11), cf. [2]. It is worthwhile to exhibit a
simple concrete example illustrating this.
\begin{example} \label{ex 2.5}
{\rm Let $g_{\lambda}$ be the family of Eguchi-Hanson metrics on the
tangent bundle $TS^{2}$ of $S^{2}.$ The metrics $g_{\lambda}$ are given
explicitly by
\begin{equation} \label{e2.15}
g_{\lambda} = [1-(\frac{\lambda}{r})^{4}]^{-1}dr^{2} +
r^{2}[1-(\frac{\lambda}{r})^{4}]\theta_{1}^{2} +
r^{2}(\theta_{2}^{2}+\theta_{3}^{2}).
\end{equation}
Here $\theta_{1}, \theta_{2}, \theta_{3}$ are the standard
left-invariant coframing of $SO(3) = {\mathbb R}{\mathbb P}^{3}$, (the
sphere bundles in $TS^{2})$ and $r \geq \lambda .$ The locus
$r=\lambda $ is the image of the 0-section and is a totally geodesic
round $S^{2}(\lambda )$ of radius $\lambda .$
The metrics $g_{\lambda}$ are Ricci-flat, and are all homothetic, i.e.
are rescalings (via diffeomorphisms) of a fixed metric; in fact,
\begin{equation} \label{e2.16}
g_{\lambda} = \lambda^{2}\cdot \psi_{\lambda}^{*}(g_{1}),
\end{equation}
where $\psi_{\lambda}(r) = \lambda r,$ and $\psi_{\lambda}$ acts
trivially on the $SO(3)$ factor. As $\lambda \rightarrow $ 0, i.e. as
one blows down the metrics, $g_{\lambda}$ converges to the metric
$g_{0},$ the flat metric on the cone $C({\mathbb R}{\mathbb P}^{3}).$
The convergence is smooth in the region $r \geq r_{o},$ for any fixed
$r_{o} > 0$, but is not smooth at $r =$ 0. Since $S^{2}(\lambda )$ is
totally geodesic, the injectivity radius at any point of $S^{2}(\lambda
)$ is $2\pi\lambda ,$ which tends to 0. On the other hand, the volumes
of unit balls, or balls of any definite radius, remain uniformly
bounded below. }
\end{example}
One sees here that the metrics $(TS^{2}, g_{\lambda})$ converge as
$\lambda \rightarrow $ 0 to a limit metric on a singular space
$C({\mathbb R}{\mathbb P}^{3}).$ The limit is an orbifold ${\mathbb
R}^{4}/{\mathbb Z}_{2},$ where ${\mathbb Z}_{2}$ acts by reflection in
the origin.
\medskip
The Eguchi-Hanson metric is the first and simplest example of a large
class of Ricci-flat ALE (asymptotically locally Euclidean) spaces,
whose metrics are asymptotic to cones $C(S^{3}/\Gamma )$, $\Gamma
\subset SO(4)$, on spherical space forms. This is the family of ALE
gravitational instantons, studied in detail by Gibbons and Hawking, cf.
[26] and references therein, in connection with Hawking's Euclidean
quantum gravity program.
\medskip
It is straightforward to modify the construction in Example 2.5 to
obtain orbifold degenerations on compact 4-manifolds satisfying the
bounds (2.11). Thus, one does not have $C^{1,\alpha}$ or even $C^{0}$
(pre)-compactness of the space of metrics on $M$ under the bounds
(2.11). Singularities can form in passing to limits, although the
singularities are of a relatively simple kind. The next result from [1]
shows that this is the only kind of possible degeneration or
singularity formation.
\begin{theorem} \label{t 2.6} {\bf (Convergence III).}
Let $\{g_{i}\}$ be a sequence of metrics on a 4-manifold, satisfying
the bounds
\begin{equation} \label{e2.17}
|Ric| \leq k, \ vol \geq v_{o}, \ diam \leq D.
\end{equation}
Then a subsequence converges, (in the Gromov-Hausdorff topology), to an
orbifold $(V, g)$, with a finite number of singular points $\{q_{j}\}.$
Each singular point $q$ has a neighborhood homeomorphic to a cone
$C(S^{3}/\Gamma ),$ for $\Gamma $ a finite subgroup of $SO(4)$.
The metric $g$ is $C^{1,\alpha}$ or $L^{2,p}$ on the regular set
$$V_{0} = V \setminus \cup\{q_{j}\},$$
and extends in a local uniformization of a singular point to a $C^{0}$
Riemannian metric. Further, there are embeddings
$$F_{i}: V_{0} \rightarrow M$$
such that $F_{i}^{*}(g_{i})$ converges in the $C^{1,\alpha}$ topology
to the metric $g$.
\end{theorem}
Here, convergence in the Gromov-Hausforff topology means convergence
as metric spaces, cf. [27]. We mention only a few important issues in
the proof of Theorem 2.6. First, the Chern-Gauss-Bonnet formula implies
that for metrics with bounded Ricci curvature and volume on
$4$-manifolds, one has an apriori bound on the $L^{2}$ norm of the full
curvature tensor:
$${\textstyle \frac{1}{8\pi^{2}}}\int_{M}|R|^{2}dV \leq \chi (M) + C(k,
V_{o}), $$
where $C(k, V_{o})$ is a constant depending only on $k$ from (2.17) and
an upper bound $V_{o}$ on $vol_{g}M$: $\chi (M)$ is the Euler
characteristic of $M$. Second, with each singular point $q \in V$,
there is a associated a sequence of rescalings $\bar g_{i} =
\lambda_{i}^{2}g_{i}, \lambda_{i} \rightarrow \infty ,$ and base
points $x_{i}\in M$, $x_{i} \rightarrow q$, such that a subsequence of
$(M, \bar g_{i}, x_{i})$ converges in $C^{1,\alpha} \cap L^{2,p}$ to a
non-trivial Ricci-flat ALE space $(N, \bar g_{\infty})$ as above. It is
not difficult to see that any such ALE space has a definite amount of
curvature in $L^{2}.$ This implies basically that there are only a
finite number of such singular points. Further, the ALE spaces $N$ are
embedded in $M$, in a topologically essential way.
\section{Collapse/Formation of Cusps.}
\setcounter{equation}{0}
In this section, we consider what happens when
$$vol \rightarrow 0 \ \ {\rm or} \ \ diam \rightarrow \infty$$
in the bounds (2.11). This involves the notion of Cheeger-Gromov
collapse, or collapse with bounded curvature.
For simplicity, we restrict the discussion to dimension 3. While there
is a corresponding theory in higher dimensions, cf. [16], there are
special and advantageous features that hold only in dimension 3.
Further, the relations with general relativity are most direct in
dimension 3, in that the discussion can be applied to the behavior of
space-like hypersurfaces in a given space-time.
The simplest non-trivial example of collapse is the Berger collapse of
the 3-sphere along $S^{1}$ fibers of the Hopf fibration. Thus, consider
the family of metrics on $S^{3}$ given by
\begin{equation} \label{e3.1}
g_{\lambda} = \lambda^{2}\theta_{1}^{2} +
(\theta_{2}^{2}+\theta_{3}^{2}),
\end{equation}
where $\theta_{1}, \theta_{2}, \theta_{3}$ are the standard
left-invariant coframing of $S^{3}.$ The metrics $g_{\lambda}$ have an
isometric $S^{1}$ action, with Killing field $K$ dual to $\theta_{1},$
with length of the $S^{1}$ orbits given by $2\pi\lambda .$ Thus, in
letting $\lambda \rightarrow $ 0, one is blowing down the metric in
{\it one} direction. (This is exactly what occurs on approach to the
horizon of the Taub-NUT metric, cf. [31]). A simple calculation shows
that the curvature of $g_{\lambda}$ remains uniformly bounded as
$\lambda \rightarrow 0$. Clearly $vol_{g_{\lambda}}S^{3} \sim \lambda
\rightarrow 0$. The metrics $g_{\lambda}$ collapse $S^{3}$ to a limit
space, in this case $S^{2}.$
This same procedure may be carried out, with the same results, on any
3-manifold (or $n$-manifold) which has a free or locally free isometric
$S^{1}$ action; locally free means that the isotropy group of any orbit
is a finite subgroup of $S^{1},$ i.e. there are no fixed points of the
action. Similarly, one may collapse along the orbits, as in (3.1), of a
locally free $T^{k}$-action, where $T^{k}$ is the $k$-torus.
Remarkably, Gromov [28] showed that more generally one may collapse
along the orbits of an isometric nilpotent group action, and
furthermore, such groups are {\it only} groups which allow such a
collapse with bounded curvature. Thus for instance collapsing along the
orbits of an isometric $G$-action, where $G$ is semi-simple and
non-abelian, increases the curvature without bound.
\medskip
A 3-manifold which admits a locally free $S^{1}$ action is called a
{\sf Seifert fibered space}. Such a space admits a fibration over a
surface $V$, with $S^{1}$ fibers. Where the action is free, this
fibration is a circle bundle. There may exist an isolated collection of
non-free orbits, corresponding to isolated points in $V$.
Topologically, a neighborhood of such an orbit is of the form $D^{2}
\times S^{1}$, where the $S^{1}$ acts by rotation on the $S^{1}$ factor
and by rotation through a rational angle about $\{0\}$ in $D^{2}$.
The collection of Seifert fibered spaces falls naturally into 6
classes, according to the topology of the base surface $V$, i.e. $V =
S^{2}$, $T^{2}$, or $\Sigma_{g}$, $g \geq 2$, and according to whether
the $S^{1}$ bundle is trivial or not trivial. These account for 6 of
the 8 possible geometries of 3-manifolds in the sense of Thurston [39];
these geometries are: $S^{2} \times {\mathbb R}$, ${\mathbb R}^{3}$,
${\mathbb H}^{2} \times {\mathbb R}$, $S^{3}$, $Nil$, and
$\widetilde{SL(2, {\mathbb R})}$, respectively. The two remaining
geometries are $Sol$, corresponding to non-trivial torus bundles over
$S^{1}$, and the hyperbolic geometry ${\mathbb H}^{3}$.
\medskip
Now suppose $N$ is a compact Seifert fibered space with boundary. The
boundary is a finite collection of tori, on which one has a free
$S^{1}$ action. In a neighborhood of the boundary, this $S^{1}$ action
then embeds in the standard free $T^{2}$ action on $T^{2}\times I.$
Given a collection of such spaces $N_{i},$ one may then glue the toral
boundaries together by automorphisms of the torus, i.e. by elements in
$SL(2, {\mathbb Z})$. For example, the glueing may interchange the
fiber and base circles.
\begin{definition} \label{d 3.1}
A {\sf graph manifold} G is a 3-manifold obtained by glueing Seifert
fibered spaces by toral automorphisms of the boundary tori.
\end{definition}
Thus, a graph manifold has a decomposition into two types of regions,
\begin{equation} \label{e3.2}
G = S \cup L.
\end{equation}
Each component of $S$ is a Seifert fibered space, while each component
of $L$ is $T^{2}\times I,$ and glues together different boundary
components of elements in $S$. The exceptional case of glueing two
copies of $T^{2}\times I$ by toral automorphisms of the boundary is
also allowed; this defines the class of $Sol$ manifolds, up to finite
covers. The Seifert fibered components have a locally free $S^{1}$
action, the $T^{2}\times I$ components have a free $T^{2}$ action; in
general, these group actions do not extend to actions on topologically
larger domains.
Graph manifolds are an especially simple class of 3-manifolds; one
has a complete understanding of their topological classification [41].
The terminology comes from the fact that one may associate a graph to
$G$, by assigning a vertex to each component of $S$, and an edge to
each component of $L$ which connects a pair of components in $S$.
\medskip
It is not difficult to generalize the construction above to show that
any closed graph manifold $G$ admits a sequence of metrics $g_{i}$
which collapse with uniformly bounded curvature, i.e.
\begin{equation} \label{e3.3}
|Ric_{g_{i}}| \leq k, \ \ vol_{g_{i}}G \rightarrow 0.
\end{equation}
The metrics $g_{i}$ collapse the Seifert fibered pieces along the
$S^{1}$ orbits, while collapsing the toral regions $T^{2} \times I$
along the tori. Thus the collapse is rank 1 along $S$, while rank 2
along $L$. (Of course a bound on the full curvature is the same as a
bound on the Ricci curvature in dimension 3).
If the graph manifold is Seifert fibered, then the collapse (3.3) may
be carried out with bounded diameter,
\begin{equation} \label{e3.4}
diam_{g_{i}}S \leq D, \ {\rm for \ some} \ D < \infty.
\end{equation}
In fact, if $S$ is a $Nil$-manifold, then the collapse may be carried
out so that $diam_{g_{i}}S \rightarrow 0$, cf. [28].
On the other hand, suppose $G$ is non-trivial in that it has both $S$
and $L$ components. If $N$ denotes any $S$ or $L$ component, then
necessarily
\begin{equation} \label{e3.5}
diam_{g_{i}}N \rightarrow \infty
\end{equation}
under the bounds (3.3). (This phenomenon can be viewed as a refinement
of the remark following (2.13)). In particular, the transition from
Seifert fibered domains to toral domains takes longer and longer
distance the more collapsed the metrics are. One obtains different
collapsed ``limits'' depending on choice of base point. This ``pure''
behavior on regions of bounded diameter is special to dimension 3.
The Cheeger-Gromov theory, [16], implies that the converse also holds.
\begin{theorem} \label{t 3.2} {\bf (Collapse).}
If $M$ is a closed 3-manifold which collapses with bounded curvature,
i.e. there is a sequence of metrics such that (3.3) holds, then $M$ is
a graph manifold.
\end{theorem}
In fact, this result holds if $M$ admits a sufficiently collapsed
metric, i.e. $|Ric_{g}| \leq k$ and $vol_{g}M \leq \varepsilon_{o}$,
for some $\varepsilon = \varepsilon_{o}(k)$ sufficiently small. Note of
course that a collapsing sequence of metrics $g_{i}$ is not {\it
necessarily} invariant under the (merely smooth) $S^{1}$ or $T^{2}$
actions associated with the graph manifold structure.
\medskip
In a certain sense, the vast majority of 3-manifolds are not graph
manifolds, and so Theorem 3.2 gives strong topological restrictions on
the existence of sufficiently collapsed metrics.
\medskip
\noindent
{\bf Idea of proof}: First, it is easy to see that
$$vol_{g_{i}}B_{x}(1) \rightarrow 0 \Rightarrow inj_{g_{i}}(x)
\rightarrow 0.$$
At any $x$, rescale the metrics $g_{i}$ to make $inj(x) = 1$, i.e. set
$$\bar g_{i} = [inj_{g_{i}}(x)]^{-2} \cdot g_{i}.$$
Now the bound (3.3) gives $|Ric_{\bar g_{i}}| \sim 0$. Thus, the
metrics $\bar g_{i}$ are close to flat metrics on ${\mathbb
R}^{3}/\Gamma$, where $\Gamma$ is a non-trivial discrete group of
Euclidean isometries, (by Theorem 2.2 for instance). Thus, essentially,
${\mathbb R}^{3}/\Gamma = {\mathbb R}^{2} \times S^{1}$, or ${\mathbb
R} \times S^{1} \times S^{1}$. It follows that the local geometry, i.e.
the geometry on the scale of the injectivity radius, is modeled by {\it
non-trivial, flat} 3-manifolds. One then shows that these local
structures for the geometry and topology can be glued together
consistently to give a global graph manifold structure.
\medskip
If $S$ is a Seifert fibered space, possibly with boundary $\partial
S,$ the orbits of the $S^{1}$ action always inject in $\pi_{1}(S)$, i.e.
$$\pi_{1}(S^{1}) \hookrightarrow \pi_{1}(S),$$
unless $S = S^{3}/\Gamma $ , or in the case of boundary, $S =
D^{2}\times S^{1},$ cf. [36]. Thus, if a graph manifold $G$ is not a
spherical space form, or does not have a solid torus component in its
Seifert fibered decomposition (3.2), then the fibers of the
decomposition, namely circles and tori, always inject in $\pi_{1}$:
\begin{equation} \label{e3.6}
\pi_{1}(fiber) \hookrightarrow \pi_{1}(G).
\end{equation}
Hence, in this situation, one can pass to covering spaces to {\it
unwrap} any collapse. Thus, if $g_{i}$ is a collapsing sequence of
metrics, by passing to larger and larger covering spaces, based
sequences will always have convergent subsequences (in domains of
arbitrary but bounded diameter). In addition, the isometric covering
transformations on the covers have displacement functions converging
uniformly to 0 on compact subsets. Hence, all such limits have a free
isometric $S^{1}$ or $T^{2}$ action, depending on whether the collapse
is rank 1 or 2 on the domains. This means that the limits have an {\sf
extra symmetry} not necessarily present on the initial collapsing
sequence. Again, this feature of being able to unwrap collapse by
passing to covering spaces is special to dimension 3.
\bigskip
Finally, we discuss the third possibility, the formation of cusps.
This case, although the most general, corresponds to a mixture of the
two previous cases convergence/collapse, and so no essentially new
phenomenon occurs. To start, given a complete Riemannian manifold $(M,
g)$, choose $\varepsilon > $ 0 small, and let
\begin{equation} \label{e3.7}
M^{\varepsilon} = \{x\in M: volB_{x}(1) \geq \varepsilon\}, \
M_{\varepsilon} = \{x\in M: volB_{x}(1) \leq \varepsilon\}.
\end{equation}
$M^{\varepsilon}$ is called the $\varepsilon$-thick part of $(M, g)$,
while $M_{\varepsilon}$ is the $\varepsilon$-thin part.
Now suppose $g_{i}$ is a sequence of complete Riemannian metrics on
the manifold $M$.
\noindent
$\bullet$ If $x_{i} \in M^{\varepsilon}$, for some fixed $\varepsilon >
0$, then one has convergence, (in subsequences), in domains of
arbitrary but bounded diameter about $\{x_{i}\}$, cf. (2.13ff).
Essentially, the bounds (2.11) hold on such domains in this case.
\noindent
$\bullet$ If $y_{i} \in M_{\varepsilon_{o}}$, for $\varepsilon_{o}$
sufficiently small, then domains of bounded, depending on
$\varepsilon_{o}$, diameter about $\{y_{i}\}$ are graph manifolds, in
fact Seifert fibered spaces.
\noindent
$\bullet$ If $z_{i} \in M_{\varepsilon_{i}}$, $\varepsilon_{i}
\rightarrow 0$, then domains of arbitrary but bounded diameter about
$\{z_{i}\}$ are collapsing.
\medskip
If $(M_{\varepsilon}, g_{i}) = \emptyset$, for some fixed
$\varepsilon > 0$, then one is in the convergence situation. If
$(M^{\varepsilon}, g_{i}) = \emptyset$, for all $\varepsilon > 0$
sufficiently small, depending on $i$, then one is in the collapsing
situation. The only remaining possibility is that there exist points
$x_{i}$ and $y_{i}$ in $M$ such that, for any fixed $\varepsilon > 0$,
\begin{equation} \label{e3.8}
(M^{\varepsilon}, g_{i}) \neq \emptyset, \ {\rm and} \
(M_{\varepsilon}, g_{i}) \neq \emptyset .
\end{equation}
This is equivalent to the existence of base points $x_{i}$, $y_{i}$,
such that,
\begin{equation} \label{e3.9}
vol B_{x_{i}}(1) \geq \varepsilon_{1}, \ \ vol B_{y_{i}}(1)
\rightarrow 0,
\end{equation}
for some $\varepsilon_{1} > 0$. Observe that the volume comparison
theorem (2.13) implies that $dist_{g_{i}}(x_{i}, y_{i}) \rightarrow
\infty $ as $i \rightarrow \infty ,$ so that these different behaviors
become further and further distant as $i \rightarrow \infty .$
This leads to the following result, cf. [5], [16] for further details.
\begin{theorem} \label{t 3.3} {\bf (Cusp Formation).}
Let $M$ be a 3-manifold and $g_{i}$ a sequence of unit volume metrics
on $M$ satisfying (3.8). Then pointed subsequences (M, $g_{i}, p_{i})$
converge to one of the following:
\noindent
$\bullet$ complete cusps $(N, g_{\infty}, p_{\infty})$. These are
complete, open Riemannian 3-manifolds, of finite volume and with graph
manifold ends, which collapse at infinity. The convergence in
$C^{1,\alpha}$ and weak $L^{2,p}$ topologies, uniform on compact
subsets.
\noindent
$\bullet$ Collapsed graph manifolds of infinite diameter.
\end{theorem}
In contrast to the topological implications of collapse in Theorem
3.2, (i.e collapse implies $M$ is a graph manifold), in general there
are no apriori topological restrictions on $M$ imposed by Theorem 3.3.
To illustrate, let $M$ be an arbitrary closed 3-manifold and let
$\{C_{k}\}$ be a collection of disjoint solid tori $D^{2}\times S^{1}$
embedded in $M$; for example $\{C_{k}\}$ may be a tubular neighborhood
of a (possibly trivial) link in $M$. Then it is not difficult to
construct a sequence of metrics of bounded curvature which converge to
a collection of complete cusps on $M\setminus \cup C_{k}$ and collapse
along the standard graph manifold structure on each $C_{k}.$
The ends of the cusp manifolds $N$ in Theorem 3.3, i.e. the graph
manifolds, necessarily have embedded tori. If such tori are essential
in $M$, i.e. inject on the $\pi_{1}$ level, then Theorem 3.3. does
imply strong topological constraints on the topology of $M$; cf. \S 6
for some further discussion.
\begin{remark} \label{r 3.4.}
{\rm We point out that there are versions of Theorems 3.2 and 3.3 also
in dimension 4, as well as in higher dimensions. The concept of graph
manifold is generalized to manifolds having an ``F-structure'', or an
``N-structure'' (F is for flat, N is for nilpotent), cf. [16], provided
bounds are assumed on the full curvature, as in (1.4). In dimension 4,
this can be relaxed to bounds on the Ricci curvature, as in (1.5),
provided one allows for a finite number of singularities in
F-structure, as in Theorem 2.6. }
\end{remark}
\section{Applications to Static and Stationary Space-Times.}
\setcounter{equation}{0}
In this section, we discuss applications of the results of \S 2-3 to
static and stationary space-times, i.e. space-times ({\bf M, g}) which
admit a time-like Killing field $K$. These space-times are viewed as
being the end or final state of evolution of a (time dependent)
gravitational field. Since they are time-independent in a natural
sense, they may be analysed by methods of Riemannian geometry, which
are not available in general for Lorentzian manifolds.
Throughout this section, we assume that ({\bf M, g}) is chronological,
i.e. ({\bf M, g}) has no closed time-like curves, and that $K$ is a
complete vector field.
\medskip
Let $\Sigma $ be the orbit space of the isometric ${\mathbb R}$-action
generated by the Killing field $K$, and let $\pi: {\bf M} \rightarrow
\Sigma$ be the projection to the orbit space. The 4-metric {\bf g} has
the form
\begin{equation} \label{e4.1}
{\bf g} = - u^{2}(dt+\theta )^{2} + \pi^{*}(g),
\end{equation}
where $K = \partial /\partial t, \theta $ is a connection 1-form for
the bundle $\pi$, $u^{2} = -{\bf g}(K,K) > 0$ and $g = g_{\Sigma}$ is
the metric induced on the orbit space.
The vacuum Einstein equations are equivalent to an elliptic system of
P.D.E's in the data $(\Sigma, g, u, \theta)$. Let $\omega$ be the twist
1-form on $\Sigma$,
given by $2\omega = *(\kappa\wedge d\kappa) = -u^{4}d\theta$, where
$\kappa = - u^{2}(dt+\theta )$ is the 1-form dual to $K$. Then the
equations
on $\Sigma $ are:
\begin{equation} \label{e4.2}
Ric_{g} = u^{-1}D^{2}u + 2u^{-4}(\omega\otimes\omega -
|\omega|^{2}g),
\end{equation}
\begin{equation} \label{e4.3}
\Delta u = - 2u^{-3}|\omega|^{2},
\end{equation}
\begin{equation} \label{e4.4}
d\omega = 0.
\end{equation}
The maximum principle applied to (4.3) immediately implies that if
$\Sigma $ is a closed $3$-manifold, then $(\Sigma, g)$ is flat and $u =
const$, and so
({\bf M, g}) is a (space-like) isometric quotient of empty Minkowski
space
$({\mathbb R}^{4}, \eta)$. Thus, we assume $\Sigma$ is open, possibly
with boundary.
Locally of course there are many solutions to the system (4.2)-(4.4);
to obtain uniqueness, one needs to impose boundary conditions.
\bigskip
We consider first the global situation, and so assume that $(\Sigma,
g)$ is a complete, non-compact Riemannian 3-manifold. Boundary
conditions are then at infinity, i.e. conditions on the asymptotic
behavior of the metric. In this respect, one has the following
classical result, c.f. [33], [22].
\begin{theorem} \label{t 4.1} {\bf (Lichnerowicz).}
The only complete, stationary vacuum space-time ({\bf M, g}) which is
asymptotically flat (AF) is empty Minkowski space-time $({\mathbb
R}^{4}, \eta).$
\end{theorem}
It is most always taken for granted that $\Sigma $ should be AF.
Stationary space-times are meant to model isolated physical systems,
and the only physically realistic models are AF. In fact, from this
physical perspective, the Lichnerowicz theorem may be viewed as a
triviality. Since there is no source for the gravitational field, it
must be the empty Minkowski space-time.
However, mathematically, the Lichnerowicz theorem is not (so) trivial,
and if it were false one would be forced to revise physical intuition.
Moreover, the assumption that ({\bf M, g}) is AF is contrary to the
spirit of general relativity. Such a boundary condition is adhoc, and
its imposition is in fact circular in a certain sense. Apriori, there
might well be complete stationary solutions for which the curvature
does not decay anywhere to 0 at infinity. The following result from [6]
clarifies this issue.
\begin{theorem} \label{t 4.2} {\bf (Generalized Lichnerowicz).}
The only complete stationary vacuum space-time ({\bf M, g}) is empty
Minkowski space-time $({\mathbb R}^{4}, \eta)$, or a discrete isometric
quotient of it.
\end{theorem}
The starting point of the proof of this result is to study first the
moduli space of all complete stationary vacuum solutions. As noted
above, apriori any given solution may, apriori, have unbounded
curvature, i.e. $|Ric_{g}|$ may diverge to infinity on divergent
sequences in $\Sigma .$ Under such a condition, the first step is then
to show, by taking suitable base points and rescalings, that one may
then obtain a new stationary vacuum solution, (i.e. a new point in the
moduli space), with uniformly bounded curvature, and non-zero curvature
at a base point. This step uses the Cheeger-Gromov theory, as described
in \S 2-\S 3, and requires the special features of collapse in
3-dimensions.
The next step in the proof is to recast the problem in the Ernst
formulation. Define the conformally related metric $\widetilde g$ by
\begin{equation} \label{e4.5}
\widetilde g = u^{2}g.
\end{equation}
A simple calculation shows that (4.2) becomes
\begin{equation} \label{e4.6}
Ric_{\widetilde g} =2(d\ln u)^{2} + 2u^{-4}\omega^{2} \geq 0.
\end{equation}
Further, the system (4.2)-(4.4) becomes the Euler-Lagrange equations
for an effective 3-dimensional action given by
$${\mathcal S}_{eff} = \int[R - \frac{1}{2}(\frac{|d\phi|^{2} +
|du^{2}|^{2}}{u^{4}})]dV.$$
Here $\phi$ is the twist potential, given by $d\phi = 2\omega$. (In
general one must pass to the universal cover to obtain the existence of
$\phi$).
This action is exactly 3-dimensional (Riemannian) gravity on
$(\Sigma, \widetilde g)$ coupled to a $\sigma$-model with target the
hyperbolic plane $(H^{2}(-1), g_{-1}).$ Thus, the Ernst map $E = (\phi
, u^{2})$ is a harmonic map
\begin{equation} \label{e4.7}
E: (\Sigma , \widetilde g) \rightarrow (H^{2}(-1), g_{-1}).
\end{equation}
Now it is well-known that harmonic maps $E: (M, g) \rightarrow (N,
h)$ from Riemannian manifolds of non-negative Ricci curvature to
manifolds of non-positive sectional curvature have strong rigidity
properties, via the Bochner-Lichnerowicz formula,
\begin{equation} \label{e4.8}
\frac{1}{2}\Delta|DE|^{2} = |D^{2}E|^{2} + - \sum
(E^{*}R_{h})(e_{i},e_{j},e_{j},e_{i}).
\end{equation}
By analysing (4.8) carefully, one shows that $E$ is a constant map,
from which it follows easily that ({\bf M, g}) is flat.
\begin{remark} \label{r 4.3} {\bf (i).}
{\rm The same result and proof holds for stationary gravitational
fields coupled to $\sigma$-models, whose target spaces are Riemannian
manifolds of non-positive sectional curvature, i.e. $E: (\Sigma,
\widetilde g) \rightarrow (N, g_{N})$ with $Riem_{g_{N}} \leq 0$.
{\bf (ii).} Curiously, the Riemannian analogue of Theorem 4.2 remains
an open problem. Thus, does there exist a complete non-flat Ricci-flat
Riemannian 4-manifold which admits a free isometric $S^{1}$ action?
{\bf (iii).} It is interesting to note that the analogue of Theorem
4.2 is false for stationary Einstein-Maxwell solutions. A
counterexample is provided by the (static) Melvin magnetic universe
[34], cf. also [25]. I am grateful to David Garfinkle for pointing this
out to me. For the stationary Einstein-Maxwell system, the target space
of the Ernst map is $SU(2,1)/S(U(1,1)\times U(1))$, ($SO(2,1)/SO(1,1)$
for static Einstein-Maxwell). Both of these target spaces have
indefinite, (i.e. non-Riemannian), metrics. }
\end{remark}
The rigidity result Theorem 4.2 leads to apriori estimates on the
geometry of general stationary solutions of the Einstein equations, cf.
[6]. Thus, if $\Sigma $ is not complete, it follows that
$\partial\Sigma \neq \emptyset .$ Note that part of $\partial\Sigma
$ may correspond to the horizon $H = \{u =$ 0\} where the Killing field
vanishes.
\begin{theorem} \label{t 4.4} {\bf (Curvature Estimate).}
Let ({\bf M, g}) be a stationary vacuum space-time. Then there is a
constant
$C < \infty ,$ independent of ({\bf M, g}), such that
\begin{equation} \label{e4.9}
|{\bf R}|(x) \leq C/r^{2}[x],
\end{equation}
where $r[x] = dist_{\Sigma}(\pi (x), \partial\Sigma ).$
\end{theorem}
Here, the curvature norm $|{\bf R}|$ may be given by
$$|{\bf R}|^{2} = |R_{\Sigma}|^{2} + |d\ln u|^{2} +
|u^{-2}\omega|^{2}.$$
Note that Theorem 4.2 follows from Theorem 4.4 by letting $r
\rightarrow \infty$. Conversely, it is a general principle for elliptic
geometric variational problems that a global rigidity result as in
Theorem 4.2 leads to apriori local estimates as in Theorem 4.4.
\begin{remark} \label{r 4.5} {\bf (i).}
{\rm Using elliptic regularity, one also has higher order bounds:
\begin{equation} \label{e4.10}
|\nabla^{k}{\bf R}|(x) \leq C_{k}/r^{2+k}[x].
\end{equation}
{\bf (ii).} A version of this result also holds for stationary
space-times with energy-momentum tensor $T$. Thus, for example one has
\begin{equation} \label{e4.11}
|{\bf R}
|(x) \leq C_{\alpha}\cdot |T|_{C^{\alpha}(B_{[x]}(1))},
\end{equation}
for any $\alpha > 0$, where $B_{[x]}(1)$ is the unit ball in $(\Sigma,
g)$ about $[x]$. The proof is the same as that of (4.9) given in [6]. }
\end{remark}
Thus, one can use the Cheeger-Gromov theory to control the local
behavior of stationary space-times, possibly with matter terms, away
from any boundary.
\medskip
The results above can in turn be applied to study the possible
asymptotic behavior of general stationary or static vacuum space-times,
without any apriori AF assumption. For example, (4.9) implies that the
curvature decays at least quadratically in any end $(E, g)$ of
$(\Sigma, g)$. For simplicity, we restrict to static space-times.
Thus, let ({\bf M, g}) be a static space-time with orbit space
$(\Sigma, g)$, with $\partial\Sigma \neq \emptyset$. Define
$\partial\Sigma$ to be {\it pseudo-compact} if there exists $r_{o} >
0$ such that the level set $\{r = r_{o}\}$ in $\Sigma $ is compact;
recall that $r$ is the distance function to the boundary $\partial
\Sigma$. (There are numerous examples of static space-times for which
$\partial\Sigma $ is non-compact, with $\partial \Sigma$
pseudo-compact). Let $S(r) = r^{-1}(r) \subset \Sigma$. If $E$ is an
end of $(\Sigma, g)$, define its mass $m_{E}$ by
\begin{equation} \label{e4.12}
m_{E} = lim_{s\rightarrow\infty}\frac{1}{4\pi}\int_{S(s)}<\nabla \ln u,
\nabla t>dA.
\end{equation}
It is easily seen from the static vacuum equations that the integral is
monotone non-increasing in $s$, and so the limit exists. The mass
$m_{E}$ coincides with the Komar mass in case $E$ is AF. The following
result is from [7].
\begin{theorem} \label{t 4.6} {\bf (Static Asymptotics).}
Let ({\bf M, g}, u) be a static vacuum space-time with pseudo-compact
boundary. Then
({\bf M, g}) has a finite number of ends. Any end $E$ on which
\begin{equation} \label{e4.13}
liminf_{E}u > 0,
\end{equation}
is either:
\begin{center}
AF
\end{center}
or
\begin{equation} \label{e4.14}
{\rm small} \equiv_{def} \int_{1}^{\infty}areaS(r)^{-1}dr < \infty .
\end{equation}
Further, if $m_{E} \neq $ 0 and $sup_{E}u < \infty ,$ then $E$ is AF.
\end{theorem}
This result is sharp in the sense that if any of the hypotheses are
dropped, then the conclusion is false. For instance, if (4.13) fails,
then there are examples of static vacuum solutions with ends neither
small nor AF.
We note that when $E$ is AF, the result implies it is AF in the strong
sense that
\begin{equation} \label{e4.15}
|g - g_{0}| = \frac{2m}{r} + O(r^{-2}), \ |R| = O(r^{-3}), \ {\rm and}
\ |u- 1| = \frac{m}{r} + O(r^{-2}).
\end{equation}
More precise asymptotics can then be obtained by using standard
elliptic estimates on the equations (4.2)-(4.4), or from [12]. Again, a
version of Theorem 4.6 holds for static space-times with matter, cf.
again [7] for further information.
The idea of the proof is to study the asymptotic behavior of an end
$E$ by blowing it down, as described in \S 1. Thus, for $R$ large and
any fixed $k$, consider the metric annuli $A(R, kR)$ about some base
point $x_{o}\in (\Sigma, g)$ and consider the rescalings $g_{R} =
R^{-2}g.$ The annulus $A(R, kR)$ then becomes an annulus of the metric
form $A(1,k)$ w.r.t. $g_{R}.$ Further, the estimate (4.9) implies that
the curvature of $g_{R}$ in $A(1,k)$ is uniformly bounded. Thus, one
may apply the Cheeger-Gromov theory as described in \S 2,\S 3, to a
sequence $(A(1,k), g_{R_{i}})$, with $R_{i} \rightarrow \infty$. One
proves that the convergence case gives rise to AF ends, while the
collapse case gives rise to small ends.
\medskip
Note that in the collapsing situation, one obtains an extra $S^{1}$ or
$T^{2}$ symmetry when the collapse is unwrapped in covering spaces.
Thus, the behavior in this case is described by axisymmetric static
solutions, i.e. the Weyl metrics. Small ends typically have the same
end structure as ${\mathbb R}^{2}\times S^{1},$ where the $S^{1}$
factor has bounded length and so typically have at most quadratic
growth for the area of geodesic spheres.
\medskip
It is worth pointing out that there are static vacuum solutions,
smooth up to a compact horizon, which have a single small end. This is
the family of Myers metrics [35], or periodic Schwarzschild metrics,
(discovered later and independently by Korotkin and Nicolai). The
manifold $\Sigma $ is topologically $(D^{2}\times S^{1})\setminus
B^{3},$ so that $\partial\Sigma = S^{2}$ with a single end of the form
$T^{2}\times {\mathbb R}^{+}.$ Metrically, the end is asymptotic to one
of the (static) Kasner metrics. This is of course not a counterexample
to the static black hole uniqueness theorem, since the end is not AF.
Note that since $\pi_{1}(\Sigma) = {\mathbb Z}$ here, one may take
non-trivial covering spaces of the Myers metrics. This leads to static
vacuum solutions with an arbitrary finite number, or even an infinite
number, of black holes in static equilibrium. This situation is of
course not possible in Newtonian gravity, and so is a highly non-linear
effect of general relativity.
\section{Lorentzian Analogues and Open Problems.}
\setcounter{equation}{0}
In this section, we discuss potential analogues of the results of \S 2
and \S 3 for Lorentzian metrics on 4-manifolds. The main interest is in
space-times ({\bf M, g}) for which one has control on the Ricci
curvature of {\bf g}, or via the Einstein equations, control on the
energy-momentum tensor $T$. In particular, the main focus will be on
vacuum space-times, $Ric_{{\bf g}} = 0$.
\medskip
One would like to find conditions under which one can take limits of
vacuum space-times. One natural reason for trying to do this is the
following. There are now a number of situations where global stability
results have been proved, namely: the global stability of Minkowski
space-time [19], and of deSitter space-time [24], the global future
stability of the Milne space-time [10], and the future $U(1)$ stability
of $U(1)$ Bianchi models [18]. These results are {\it openness}
results, which state that the basic features of a given model, e.g.
Minkowski, are preserved under suitably small perturbations of the
initial data. It is then natural to consider what occurs when one tries
to pass to limits of such perturbations.
The issue of being able to take limits is also closely related with
the existence problem and singularity formation for the vacuum Einstein
evolution equations. From this perspective, suppose one has an
increasing sequence of domains $(\Omega_{i}, {\bf g_{i}}), \Omega_{i}
\subset \Omega_{i+1}$ with ${\bf g_{i+1}}
|_{\Omega_{i}} = {\bf g_{i}},$ which are evolutions of smooth Cauchy
data on some fixed initial data set. If ${\bf M} = \cup\Omega_{i}$ is
the maximal Cauchy development, then understanding {\bf (M, g)} amounts
to understanding the limiting behavior of $(\Omega_{i}, {\bf g_{i}}).$
\medskip
There are two obvious but essential reasons why it is much more
difficult to develop a Lorentzian analogue of the Cheeger-Gromov
theory, in particular with bounds only on the Ricci curvature. The
first is that the elliptic nature of the P.D.E. for Ricci curvature
becomes hyperbolic for Lorentz metrics, and hyperbolic P.D.E. are much
more difficult than elliptic P.D.E. The second is that the compact
group of Euclidean rotations $O(4)$ is compact, while the group of
proper Lorentz transformations $O(3,1)$ is non-compact.
\medskip
\noindent
{\bf A: $1^{\rm st}$ Level Problem.}
Consider first the problem of controlling the space-time metric {\bf
g} in terms of bounds, say $L^{\infty},$ on the space-time curvature
{\bf R},
\begin{equation} \label{e5.1}
|{\bf R}|_{L^{\infty}} \leq K < \infty,
\end{equation}
since already here there are significant issues.
First, the norm of curvature tensor
$|{\bf R}|^{2} = {\bf R}_{ijkl}{\bf R}^{ijkl}$ is no longer
non-negative for
Lorentz metrics, and so a bound on $|{\bf R}|^{2}$ does not imply a
bound on
all the components ${\bf R}_{ijkl}.$ In fact, for a Ricci-flat
4-metric,
there are exactly two scalar invariants of the curvature tensor:
\begin{equation} \label{e5.2}
< {\bf R}, {\bf R}> = |{\bf R}|^{2} = {\bf R}_{ijkl}{\bf R}^{ijkl} \
{\rm and} \
<{\bf R}, *{\bf R}> = {\bf R}_{ijkl}(*{\bf R}^{ijkl}).
\end{equation}
Both of these invariants can vanish identically on classes of
Ricci-flat non-flat space-times; for instance this is the case for the
class of plane-fronted gravitational waves, given by
$${\bf g} = -dudv + (dx^{2} + dy^{2}) - 2h(u,x,y)du^{2},$$
$$\Delta_{(x,y)}h = 0,$$
cf. [13,\S 8] and references therein. Here, $h$ is only required to
harmonic in the variables $(x,y)$, and is arbitrary in $u$. The class
of such space-times is highly non-compact, and so one has no local
control of the metric in any coordinate system under bounds on the
quantities in (5.2).
\medskip
Thus, one must turn to bounds on the components of {\bf R} in some
fixed coordinate system or framing. The most efficient way to do this
is to choose a unit time-like vector $T = e_{0},$ say future directed,
and extend it to an orthornormal frame $e_{\alpha},$ 0 $\leq \alpha
\leq $ 3. Since the space $T^{\perp}$ orthogonal to $T$ is space-like
and $O(3)$ is compact, the particular framing of $T^{\perp}$ is
unimportant. One may then define the norm w.r.t. $T$ by
\begin{equation} \label{e5.3}
|{\bf R}|_{T}^{2} = \sum ({\bf R}_{ijkl})^{2},
\end{equation}
where the components are w.r.t. the frame $e_{\alpha}.$ This is
equivalent to taking the norm of {\bf R} w.r.t. the Riemannian metric
$$g_{E} = {\bf g} + 2T\otimes T.$$
If, at a given point $p$, $T$ lies within a compact subset $W$ of the
future interior null cone $T_{p}^{+}$, then the norms (5.3) are all
equivalent, with constant depending only on $W$. Of course if $D$ is a
compact set in the space-time ({\bf M, g}) and the vector field $T$ is
continuous in $D$, then $T$ lies within a compact subset of $T^{+}D,$
where $T^{+}D$ is the bundle of future interior null cones in the
tangent bundle $TD$.
\medskip
It is quite straightforward to prove that if $(M, g)$ is a smooth
Riemannian manifold with an $L^{\infty}$ bound on the full curvature,
$|R| \leq K$ then there are local coordinate systems in which the
metric is $C^{1,\alpha}$ or $L^{2,p},$ with bounds depending only on
$K$ and a lower volume bound, cf. Remark 2.4(i).
However, this has been an open problem for Lorentzian metrics,
apparently for some time, cf. [20],[40] for instance. The following
result gives a solution to this problem.
\medskip
To state the result, let $\Omega$ be a domain in a smoooth Lorentz
manifold ({\bf M, g}), of arbitrary dimension $n+1$. Then $\Omega$
satisfies the {\sf size conditions} if the following holds. There is a
smooth time function $t$, with $T = \nabla t / |\nabla t|$ the
associated unit time-like vector field on $\Omega$, such that, for $S =
S_{0} = t^{-1}(0)$, the 1-cylinder
\begin{equation} \label{e5.4}
C_{1} = B_{p}(1) \times [-1, 1] \subset \subset \Omega,
\end{equation}
i.e. $C_{1}$ has compact closure in $\Omega$. Here $B_{p}(1)$ is the
geodesic ball of radius 1 about $p$, w.r.t. the metric $g$ induced on
$S$ and the product is identified with a subset of $\Omega$ by the flow
of $T$.
It is essentially obvious that any point $q$ in a Lorentz manifold
has a neighborhood satisfying the size conditions, when the metric {\bf
g} is scaled up suitably.
Let $D = Im T|_{C_{1}} \subset \subset T^{+}\Omega$.
\begin{theorem} \label{t 5.1}
Let $\Omega$ be a domain in a vacuum $(n+1)$-dimensional space-time
$({\bf M, g})$. Suppose $\Omega$ satisfies the size conditions, and
that there exist constants $K < \infty $ and $v_{o} > $ 0 such that
\begin{equation} \label{e5.5}
|{\bf R}|_{T} \leq K, \ \ vol_{g}B_{p}(\textstyle{\frac{1}{2}}) \geq
v_{o}.
\end{equation}
Then there exists $r_{o} > 0$, depending only on $K, v_{o}$ and $D$,
and coordinate charts on the $r_{o}$-cylinder
$$C_{r_{o}} = B_{p}(r_{o}) \times [-r_{o}, r_{o}],$$
such that the components of the metric ${\bf g}_{\alpha\beta}$ are in
$C^{1,\alpha}\cap L^{2,p}$, for any $\alpha < 1$, $p < \infty$.
Further, there exists $R_{o}$, depending only on $K$, $v_{o}$, $D$
and $p$, such that, on $C_{r_{o}}$,
\begin{equation} \label{e5.6}
||{\bf g}_{\alpha\beta}||_{L^{2,p}} \leq R_{o}.
\end{equation}
\end{theorem}
Here, the components ${\bf g}_{\alpha\beta}$ are the full space-time
components of {\bf g}, and the estimate (5.6) gives bounds on both
spatial and time derivatives of {\bf g}, up to order 2, in $L^{p}$,
where $L^{p}$ is measured on spatial slices of $C_{r_{o}}$.
This result is formulated in such a way that it is easy to pass to
limits. Thus, if one has a sequence of smooth space-times $({\bf
M}_{i}, {\bf g}_{i})$ satisfying the hypotheses of the Theorem, (with
fixed constants $K$, $v_{o}$ and uniformly compact domains $D$), then
it follows that, in a subsequence, there is a limit $C^{1,\alpha}\cap
L^{2,p}$ space-time $({\bf M_{\infty}, g_{\infty}})$, defined at least
on the $r_{o}$-cylinder $C_{r_{o}}$. Further, the convergence to the
limit is $C^{1,\alpha}$ and weak $L^{2,p}$, and the estimate (5.6)
holds on the limit.
\medskip
We sketch some of the ideas of the proof; full details appear in [9].
First, one constructs a new local time function $\tau$ on small
cylinders $C_{r_{1}}$, with $|\nabla \tau|^{2} = -1$, so the flow of
$\nabla \tau$ is by time-like geodesics. On the level sets
$\Sigma_{\tau}$ of $\tau$, one constructs spatially harmonic
coordinates $\{x_{i}\}$, (w.r.t. the induced Riemannian metric). This
gives a local coordinate system $(\tau, x_{1}, ... , x_{n})$ on small
cylinders about $p$. One then uses the transport or Raychaudhuri
equation, together with the Bochner-Weitzenbock formula, (Simons'
equation), and elliptic estimates to control ${\bf g}_{\alpha \beta}$.
The vacuum Einstein equations are needed in Theorem 5.1 only to prove
the $2^{\rm nd}$ time derivatives $\partial_{\tau}\partial_{\tau}
g_{0\alpha}$ are in $L^{p}$, via use of the Bianchi identity. In place
of vacuum space-times, it suffices to have a rather weak bound on the
stress-energy tensor in the Einstein equations. All other bounds on
$g_{\alpha\beta}$ do not require the Einstein equations.
\medskip
It seems as if this result should be of use in understanding the
structure of the boundary of space-times.
\medskip
If the volume bound on space-like hypersurfaces in (5.5) is dropped,
then it is possible that space-like hypersurfaces may collapse with
bounded curvature, as described in \S 3. Examples of this behavior
occur on approach to Cauchy horizons, (as noted in \S 3 in connection
with the Berger collapse and the Taub-NUT metric). More generally,
Rendall [38] has proved the following interesting general result: if
$\Sigma$ is a {\it compact} Cauchy horizon in a smooth vacuum
space-time in 3+1 dimensions, then nearby space-like hypersurfaces
collapse with bounded curvature on approach to $\Sigma$.
\bigskip
\noindent
{\bf B: $2^{\rm nd}$ Level Problem.}
While Theorem 5.1 represents a first step, one would like to do much
better by replacing the bound on $|{\bf R}|_{T}$ by a bound on the
Ricci curvature of ({\bf M, g}), or assuming for instance the vacuum
Einstein equations. Thus, one may ask if analogues of Theorems 2.2 or
2.3 hold in the Lorentzian setting.
The main ingredients in the proofs of these results are the splitting
theorem - a geometric part - and the strong convergence to limits - an
analytic part obtained from elliptic estimates for the Ricci curvature.
Now one does have a direct analogue of the splitting theorem for vacuum
space-times, (or more generally space-times satisfying the time-like
convergence condition). Thus, by work of Eschenburg, Galloway and
Newman, if ({\bf M, g}) is a time-like geodesically complete, (or a
globally hyperbolic), vacuum space-time which contains a time-like
line, i.e. a complete time-like maximal geodesic, then ({\bf M, g}) is
flat, cf. [11] and references therein.
In analogy to the Riemannian case, define then the 1-cross
$Cro_{1}(x,T)$ of a Lorentzian 4-manifold ({\bf M, g}) at $x$, in the
direction of a unit time-like vector $T$, to be the length of the
longest maximizing geodesic in the direction $T$, with center point
$x$. For $\Omega$ a domain with compact closure in
{\bf M} and $T$ a smooth unit time-like vector field, define
$$Cro_{1}(\Omega ,T) = \inf_{x\in\Omega}Cro_{1}(x,T).$$
What is lacking is the regularity boost obtained from elliptic
estimates. For space-times, the vacuum equations give a hyperbolic
evolution equation, (in harmonic coordinates), for which one does not
have a gain in derivatives. However, the smoothness of initial data is
preserved under the evolution, until one hits the boundary of the
maximal development.
Let $H^{s} = H^{s}(U)$ denote the Sobolev space of functions with $s$
weak derivatives in $L^{2}(U)$, $U$ a compact domain in ${\mathbb
R}^{3}$. For $s > 2.5$, (so that $H^{s}$ embeds in $C^{0}),$ and a
space-like hypersurface $S \subset $ ({\bf M, g}), define the harmonic
radius $\rho_{s}(x)$ of $x\in S$ in the same way as in Definition 2.1,
where the components ${\bf g}_{\alpha \beta}$ and derivatives are in
both space and time directions. For the following, we need only
consider $s \in {\mathbb N}^{+}$, with $s$ large, for instance, $s =
3$.
Now a well-known result of Choquet-Bruhat [17] states that the
maximal vacuum $H^{s}$ development of smooth ($C^{\infty}$) initial
data on $S$ is the same for all $s$, provided $s > 2.5$. Thus, one does
not have different developments of smooth initial data, depending on
the degree of desired $H^{s}$ regularity. Here, one may assume that $S$
is compact, or work locally, within the domain of dependence of $S$.
This qualitative result can be expressed as follows. Let $S_{t}$ be
space-like hypersurfaces obtained by evolution from initial data on $S
= S_{0}$. If $x_{t} \in S_{t}$, then
\begin{equation} \label{e5.7}
\rho_{s}(x_{t}) \geq c_{1} \Rightarrow \rho_{s+1}(x_{t}) \geq c_{2},
\end{equation}
where $c_{2}$ depends on $c_{1}$ and the $(C^{\infty})$ initial data on
$S_{0}$.
We raise the following problem of whether the qualitative statement
(5.7) can be improved to a {\it quantitative} statement.
{\bf Regularity Problem.}
Can the estimate (5.7) be improved to an estimate
\begin{equation} \label{e5.8}
\inf_{x_{t} \in S_{t}}\rho_{s+1}(x_{t}) \geq c_{0}\inf_{x_{t} \in
S_{t}}\rho_{s}(x_{t}),
\end{equation}
where $c_{0}$ depends only on the initial data on $S$? One may assume,
w.l.o.g, that $t \leq 1$.
The important point of (5.8) over (5.7) is that the estimate (5.8) is
scale-invariant. Here, we recall that $\rho_{s}(x)$ measures the degree
of concentration of derivatives of the metric in $H^{s}$, so that
$\rho_{s} \rightarrow 0$ corresponds to blow-up of the metric in
$H^{s}$ locally.
If (5.8) holds, it serves as an analogue of the regularity boost. In
such circumstances, one can imitate the proof of Theorems 2.2 or 2.3 to
obtain similar results for sequences of space-times $({\bf M, g}_{i})$.
In fact, the validity of (5.8) would have numerous interesting
applications, even if it could be established under some further
restrictions or assumptions.
\bigskip
Suppose next one drops any assumption on the 1-cross of ({\bf M, g})
and maintains only a lower bound on the volumes of geodesic balls, as
in (5.5), on space-like hypersurfaces. This leads directly to issues of
singularity formation and the structure of the boundary of the vacuum
space-time, where comparatively little is known mathematically.
A useful problem, certainly simple to state, is the following: for
simplicity, we work in the context of compact, (i.e. closed, without
boundary), Cauchy surfaces.
\smallskip
\noindent
{\bf Sandwich Problem.}
Let $({\bf M, g}_{i})$ be a sequence of vacuum space-times, and let
$\Sigma_{i}^{1}, \Sigma_{i}^{2}$ be two compact Cauchy surfaces in
${\bf M}$, with $\Sigma_{i}^{2}$ to the future of $\Sigma_{i}^{1}$ and
with
$$1 \leq dist_{{\bf g}}(x, \Sigma_{i}^{1}) \leq 10,$$
for all $x \in \Sigma_{i}^{2}$. Suppose the Cauchy data $(g_{i}^{j},
K_{i}^{j})$, $j = 1,2$ on each Cauchy surface are uniformly bounded in
$H^{s}$ for some fixed $s > 2.5$, possibly large. Hence the data
$(g_{i}^{j}, K_{i}^{j})$ converge, in a subsequence and weakly in
$H^{s}$, to limit $H^{s}$ Cauchy data $g_{\infty}^{j}, K_{\infty}^{j}$
on $\Sigma^{j}$.
Do the vacuum space-times $A_{i}(1,2) \subset (M, g_{i})$ between
$\Sigma^{1}$ and $\Sigma^{2}$ converge, weakly in $H^{s}$, to a limit
space time,
\begin{equation} \label {e5.9}
(A_{i}(1,2), g_{i}) \rightarrow (A_{\infty}, g_{\infty})?
\end{equation}
This question basically asks if a singularity can form between
$\Sigma_{i}^{1}$ and $\Sigma_{i}^{2}$ in the limit. It is unknown even
if there could be only a single singularity at an isolated point
(event) $x_{0} \in (A_{\infty}, g_{\infty})$.
The existence of such a singularity may be related to the Choptuik
solution. However, both the existence and the smoothness properties of
the Choptuik solution have not been established well mathematically;
cf. [30] for an interesting discussion.
Such a limit singularity would be naked in a strange way. It could be
detected on $\Sigma^{2}$, since light rays from it propagate to
$\Sigma^{2}$, but on $\Sigma^{2}$, no remnant of the singularity is
detectable, since the data is smooth on $\Sigma^{2}$. Thus, the
singularity is invisible to the future (or past) in a natural sense.
A resolution of this problem would be useful in understanding, for
instance, limits of the asymptotically simple vacuum perturbations of
deSitter space, given by Friedrich's theorem [24]. The sandwich problem
above asks: suppose one has control on the space-time near past and
future space-like infinity $\mathcal{I}^{\pm}$, does it follow that one
has control in between?
Similar questions can be posed for non-compact Cauchy surfaces, and
relate for instance to limits of the AF perturbations of Minkowski
space given by Christodoulou-Klainerman, [19].
\section{Future Asymptotics and Geometrization of 3-Manifolds.}
\setcounter{equation}{0}
In this section, we give some applications to the future asymptotic
behavior of cosmological spaces times.
Let ({\bf M, g}) be a vacuum cosmological space-time, i.e. ({\bf M,
g}) contains a compact Cauchy surface $\Sigma $ of constant mean
curvature (CMC). We assume throughout this section that $\Sigma $ is of
non-positive Yamabe type, so that $\Sigma $ admits no metric of
positive scalar curvature. It is well-known that $\Sigma$ then embeds
in a foliation $\mathcal{F} $ by CMC Cauchy surfaces $\Sigma_{\tau},$
all diffeomorphic to $\Sigma = \Sigma_{1},$ and parametrized by their
mean curvature $\tau .$ The parameter $\tau $ thus serves as a time
function, taking values in
\begin{equation} \label{e6.1}
\tau \in (-\infty , 0),
\end{equation}
with $\tau $ increasing towards the future in ({\bf M, g}). The sign of
the mean curvature is chosen so that $vol_{g_{\tau}}\Sigma_{\tau}$ is
increasing with increasing $\tau ,$ i.e. expanding towards the future.
The foliated region ${\bf M}_{\mathcal{F}}$ is thus a subset of {\bf
M}, although in general
${\bf M} \neq {\bf M}_{\mathcal{F}}.$
\medskip
Suppose that ({\bf M, g}) is geodesically complete to the future of
$\Sigma ,$ and that the future is foliated by CMC Cauchy surfaces, i.e.
${\bf M} = {\bf M}_{\mathcal{F}}$ to the future of $\Sigma$. These are
of course strong assumptions, but are necessary if one wants to
understand the future asymptotic behavior of ({\bf M, g}) without the
complicating issue of singularities.
The topology of $\Sigma_{\tau}$ is fixed, and so the metrics
$g_{\tau}$ induced on $\Sigma_{\tau}$ by the ambient metric {\bf g}
give rise to a curve of Riemannian metrics on the fixed manifold
$\Sigma .$ It is not hard to see that $vol_{g_{\tau}}\Sigma
\rightarrow \infty $ as $\tau \rightarrow $ 0, and typically, the
metrics $g_{\tau}$ become flat, due to the expansion, (compare with the
discussion in \S 1).
This is of course not very interesting. As in \S 1 and \S 4, to study
the asymptotic behavior, one should rescale by the distance to a fixed
base point or space-like hypersurface. In this case, the distance is
the time-like Lorentzian distance. Thus, for $x$ to the future of
$\Sigma = \Sigma_{-1},$ let $t(x) = dist_{{\bf g}}(x, \Sigma )$ and let
\begin{equation} \label{e6.2}
t_{\tau} = t_{max}(\tau ) = max\{t(x): x\in\Sigma_{\tau}\} = dist_{{\bf
g}}(\Sigma_{\tau}, \Sigma).
\end{equation}
It is natural to study the asymptotic behavior of the metrics
\begin{equation} \label{e6.3}
\bar g_{\tau} = t_{\tau}^{-2}g_{\tau},
\end{equation}
on $\Sigma_{\tau}.$ Observe that in the rescaled space-time (M, ${\bf
\bar g}_{\tau}),$ the distance of $(\Sigma_{\tau}, \bar g_{\tau})$ to
the ``initial'' singularity, (big bang), tends towards 1, as $\tau
\rightarrow $ 0. Any other essentially distinct scaling would have the
property that the distance to the initial singularity tends towards 0
or $\infty$, and so is not particularly natural.
\smallskip
We need the following definition.
\begin{definition} \label{d 6.1}
{\rm Let $\Sigma $ be a closed, oriented, connected 3-manifold, of
non-positive Yamabe type. A {\it weak} geometrization of $\Sigma $ is
a decomposition
\begin{equation} \label{e6.4}
\Sigma = H \cup G,
\end{equation}
where $H$ is a finite collection of complete, connected hyperbolic
manifolds, of finite volume, embedded in $\Sigma ,$ and $G$ is a finite
collection of connected graph manifolds, embedded in $\Sigma .$ The
union is along a finite collection of embedded tori $\mathcal{T} =
\cup T_{i} = \partial H = \partial G.$
A {\it strong} geometrization of $\Sigma $ is a weak geometrization
as above, for which each torus $T_{i}\in \mathcal{T} $ is
incompressible in $\Sigma ,$ i.e. the inclusion of $T_{i}$ into $\Sigma
$ induces an injection of fundamental groups.}
\end{definition}
Of course it is possible that the collection $\mathcal{T} $ of tori
dividing $H$ and $G$ is empty, in which case weak and strong
geometrizations coincide. In such a situation, $\Sigma $ is then either
a closed hyperbolic manifold or a closed graph manifold. For a strong
geometrization, the decomposition (6.4) is unique up to isotopy, but
this is certainly not the case for a weak geometrization, c.f. the end
of \S 3.
\medskip
In general, no fixed metric $g$ on $\Sigma $ will realize the
decomposition (6.4), unless $\mathcal{T} = \emptyset$. This is because
the complete hyperbolic metric on $H$ does not extend to a metric on
$\Sigma$. However, one can find sequences of metrics $g_{i}$ on $\Sigma
$ which limit on a geometrization of $\Sigma $ in the sense of (6.4).
Thus, the metrics $g_{i}$ may be chosen to converge to the hyperbolic
metric on larger and larger compact subsets of $H$, to be more and more
collapsed with bounded curvature on $G$, and such that their behavior
matches far down the collapsing hyperbolic cusps.
\medskip
Next, to proceed further, we need to impose a rather strong curvature
assumption on the ambient space-time curvature. Thus, suppose there is
a constant $C < \infty $ such that, for $x$ to the future of $\Sigma ,$
\begin{equation} \label{e6.5}
|{\bf R}|(x) + t(x)|\nabla{\bf R}|(x) \leq C\cdot t^{-2}(x).
\end{equation}
Here, the curvature norm $|{\bf R}|$ may be given by $|{\bf R}|_{T}$ as
in (5.3), where $T$ is the unit normal to the foliation
$\Sigma_{\tau}$. Since ({\bf M, g}) is vacuum, this is equivalent to
$|{\bf R}|^{2} = |E|^{2} + |B|^{2}$, where $E$, $B$ is the
electric/magnetic decomposition of {\bf R},
$E(X,Y) = <{\bf R}(X,T)T, Y>$, $B(X,Y) = <(*{\bf R})(X,T)T, Y>$ with
$X$, $Y$ tangent to the leaves. Similarly, $|\nabla{\bf R}|^{2} =
|\nabla E|^{2} + |\nabla B|^{2}.$
The bound (6.5) is scale invariant, and analogous to the bound (4.9)
or (4.10) for stationary space-times, (where it of course holds in
general). The bound on $|\nabla {\bf R}|$ in (6.5) is needed only for
technical reasons, (related to Cauchy stability), and may be removed in
certain natural situations.
\medskip
It is essentially obvious, and in any case easily verified, that the
curvature assumption (6.5) holds for the class of expanding Bianchi
cosmological models, (where the leaves $\Sigma_{\tau}$ are locally
homogeneous). It is natural to conjecture that it also holds at least
for perturbations of the Bianchi models. Similarly, we conjecture it
holds for all vacuum Gowdy space-times.
In fact, there are no known cosmological space-times ({\bf M, g}),
geodesically complete to the future, for which (6.5) is known to fail.
\smallskip
The discussion above leads to the following result from [8], to which
we refer for further discussion and details.
\begin{theorem} \label{t 6.2}
Let ({\bf M, g}) be a cosmological space-time of non-positive Yamabe
type. Suppose that the curvature assumption (6.5) holds, and that
$M_{\mathcal{F}} = {\bf M}$ to the future of $\Sigma$.
Then ({\bf M, g}) is future geodesically complete and, for any
sequence $\tau_{i} \rightarrow $ 0, the slices $(\Sigma_{\tau_{i}},
\bar g_{\tau_{i}})$, cf. (6.3), have a subsequence converging to a weak
geometrization of $\Sigma ,$ in the sense following Definition 6.1.
\end{theorem}
We indicate some of the basic ideas in the proof. The first step is to
show that the bound (6.5) on the ambient curvature {\bf R}, in this
rescaling, gives uniform bounds on the intrinsic and extrinsic
curvature of the leaves $\Sigma_{\tau}.$ The proof of this is similar
to the proof of Theorem 5.1.
Given this, one can then apply the Cheeger-Gromov theory, as described
in \S 2-3. Given any sequence $\tau_{i} \rightarrow 0$, there exist
subsequences which either converge, collapse or form cusps. From the
work in \S 3, one knows that the regions of $(\Sigma_{\tau_{i}}, \bar
g_{\tau_{i}})$ which (fully) collapse, or which are sufficiently
collapsed, are graph manifolds. This gives rise to the region $G$ in
(6.4). It remains to show that, for any fixed $\varepsilon > 0$, the
$\varepsilon$-thick region $\Sigma^{\varepsilon}$ of
$(\Sigma_{\tau_{i}}, \bar g_{\tau_{i}})$ converges to a hyperbolic
metric.
The main ingredient in this is the following volume monotonicity
result:
\begin{equation} \label{e6.6}
\frac{vol_{g_{\tau}} \Sigma_{\tau}}{t_{\tau}^{3}} \downarrow ,
\end{equation}
i.e. the ratio is monotone non-increasing in the distance $t_{\tau}.$
This result is analogous to the Fischer-Moncrief monotonicity of the
reduced Hamiltonian along the CMC Einstein flow, cf. [23]. The
monotonicity (6.6) is easy to prove, and is an analogue of the
Bishop-Gromov volume monotonicity (2.12). It follows from an analysis
of the Raychaudhuri equation, much as in the Penrose-Hawking
singularity theorems, together with a standard maximum principle.
Moreover, the ratio in (6.6) is constant on some interval $[\tau_{1},
\tau_{2}]$ if and only if the annular region $\tau^{-1}(\tau_{1},
\tau_{2})$ is a time annulus in a flat Lorentzian cone
\begin{equation} \label{e6.7}
{\bf g_{o}} = - dt^{2} + t^{2}g_{-1},
\end{equation}
where $g_{-1}$ is a hyperbolic metric. Again, the ratio in (6.6) is
scale invariant, and so
\begin{equation} \label{e6.8}
\frac{vol_{g_{\tau}} \Sigma_{\tau}}{t_{\tau}^{3}} = vol_{\bar
g_{\tau}}\Sigma_{\tau}.
\end{equation}
In the non-collapse situation, $vol_{\bar g_{\tau}}\Sigma_{\tau}$ is
uniformly bounded away from 0 as $\tau \rightarrow 0$, (i.e. $t_{\tau}
\rightarrow \infty$), and hence converges to a non-zero limit. On
approach to the $\tau = 0$ limit, the ratio (6.6) tends to a constant,
and hence the corresponding limit manifolds are of the form (6.7). This
implies that $\varepsilon$-thick regions converge to hyperbolic
metrics, giving rise to the $H$ factor in (6.4).
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\bigskip
\begin{center}
August, 2002
\end{center}
\medskip
\noindent
\address{Department of Mathematics\\
S.U.N.Y. at Stony Brook\\
Stony Brook, N.Y. 11794-3651}
\noindent
\email{anderson@math.sunysb.edu}
\end{document}