# Asymptotical flatness and cone structure at infinity

Anton Petrunin111”★” This work was done while we enjoyed our stay at the MPI
Leipzig and finished during a joint week at the IHES. and Wilderich Tuschmann★

We investigate asymptotically flat manifolds with cone structure at infinity.
We show that any such manifold $M$ has a finite number of ends, and we
classify (except for the case dim $M=4$, where it remains open if one of the
theoretically possible cones can actually arise) for simply connected ends all
possible cones at infinity. This result yields in particular a complete
classification of asymptotically flat manifolds with nonnegative curvature:
The universal covering of an asymptotically flat $m$-manifold with nonnegative
sectional curvature is isometric to ${\mathbb{R}}^{m-2}\times S$, where $S$ is
an asymptotically flat surface.

## 0\. Introduction

Let $(M,g)$ be a complete noncompact Riemannian manifold. Choose a point $p\in
M$ and set

$A(M)=\limsup_{|px|\to\infty}\ \\{|K_{x}|{\cdot}|px|^{2}\\},$

where $|K_{x}|$ denotes the maximal absolute value of the sectional curvatures
at the point $x\in M$.

One easily checks that $A(M)$ does not depend on the choice of the reference
point $p$, so that the quantity $A(M)$ yields a nice geometric invariant of
$M$ which is, in particular, invariant under rescalings of the metric.

Definition.  A noncompact complete Riemannian manifold $(M,g)$ is called
asymptotically flat if $A(M)=0$.

Note that the mere condition of being asymptotically flat places in general no
restrictions whatsoever on the topology of a manifold. For instance, by a
result of Abresch (see [Ab]) any noncompact surface carries a complete and
asymptotically flat Riemannian metric.

Definition.  A noncompact complete Riemannian manifold $(M,g)$ is said to have
cone structure at infinity if there is a metric cone $C$ with vertex $o$ such
that the pointed Gromov–Hausdorff limit of $(M,\varepsilon{\cdot}g,p)$ exists
for any sequence of numbers $\varepsilon>0$ converging to zero and such that
this limit is isometric to $(C,o)$.

As a Gromov–Hausdorff limit of proper metric spaces, that is, metric spaces
such that any closed ball of finite radius is compact, the cone $C$ which
arises in the above definition is in particular proper and locally compact.

Note that large classes of Riemannian manifolds have cone structure at
infinity. In fact, Kasue (see [K1]) showed that under certain lower curvature
limitations (e.g., if for some $C<\infty$ and $\delta>0$ it holds that
$K_{x}\geq-C/|px|^{2+\delta}$) a noncompact (complete) Riemannian manifold
always has this property. Thus in particular any noncompact Riemannian
manifold with faster-than-quadratic curvature decay (that is, any noncompact
Riemannian manifold for which there exists some $C$ and $\delta>0$ such that
$|K_{x}|\leq C/|px|^{2+\delta}$), and, especially, any noncompact Riemannian
manifold with nonnegative curvature has cone structure at infinity.

On the other hand by Abresch’s result one can easily construct asymptotically
flat surfaces $(S,g)$ such that the Gromov–Hausdorff limit of
$(S,\varepsilon_{n}{\cdot}g,p)$ indeed depends on the choice of the sequence
$\varepsilon_{n}\to 0$ and such that for some sequences this limit is not even
a metric cone. In particular, by considering products
${\mathbb{T}}^{m-2}\times(S,g)$ one thus obtains examples of asymptotically
flat manifolds without cone structure at infinity in any dimension $m\geq 2$.
(Actually any such example we know of looks on a big scale always two-
dimensional; for more on this see section $3$).

Theorem A.  Let $M$ be an asymptotically flat $m$-manifold. Assume that $M$
has cone structure at infinity. Then

(i) There exists an open ball $B_{R}(p)\i M$ such that $M\setminus B_{R}(p)$
is a disjoint union $\bigcup_{i}N_{i}$ of a finite number of ends; that is,
$N_{i}$ is a connected topological manifold with closed boundary $\partial
N_{i}$ which is homeomorphic to $\partial N_{i}\times[0,\infty)$. For each
$N_{i}$ the limit $C_{i}=$ GH-$\lim_{\varepsilon\to 0}\varepsilon{\cdot}N_{i}$
exists.

(ii) If the end $N_{i}$ is simply connected, then $N_{i}$ is homeomorphic to
${\mathbb{S}}^{m-1}\times[0,\infty)$.

In this case moreover the following holds:

(a) if $m\not=4$, then $C_{i}$ is isometric to ${\mathbb{R}}^{m}$;

(b) if $m=4$, then $C_{i}$ is isometric to one of the following spaces:
${\mathbb{R}}^{4}$, ${\mathbb{R}}^{3}$, or ${\mathbb{R}}\times[0,\infty)$.

A finiteness of ends statement as in part (i) of Theorem $A$ was proved by
Abresch (see [Ab]) in a related setting.

Part (ii)(a) of Theorem $A$ is new even in the special case of faster-than-
quadratic curvature decay (recall that, as noted above, faster-than-quadratic
curvature decay implies cone structure at infinity).

Theorem $A$ generalizes here work of Greene, Petersen, and Zhu (see Theorem
$1$ in [GPZ], where the same conclusion as in (ii)(a) was proved under the
additional assumptions of faster-than-quadratic curvature decay and
nontriviality of the tangent bundle of $\partial N_{i}$.

When combined with the fact that volume growth of exactly Euclidean order
implies flatness, Theorem $A$ also yields the following result, which
generalizes for manifolds of dimension $m\neq 4$ Theorem $2$ in [GPZ] from
faster-than-quadratic curvature decay to asymptotical flatness with cone
structure at infinity:

Corollary.  Let $M$ be an asymptotically flat $m$-manifold of dimension $m\neq
4$ which has cone structure at infinity. If $M$ has nonnegative Ricci
curvature and one simply connected end, then $M$ is isometric to
${\mathbb{R}}^{m}$.

Part (ii)(b) of Theorem $A$ shows that four-dimensional manifolds play in
Theorem $A$ a peculiar role and indicates that in this dimension special
phenomena can arise. Indeed, Unnebrink (see [U]) showed that there are
examples of asymptotically flat $4$-manifolds which have (cone structure at
infinity and) a simply connected end $N_{1}$ such that, in the notation of
Theorem $A$, $C_{1}={\mathbb{R}}^{3}$. It is not clear if there exists an
asymptotically flat $4$-manifold with cone structure at infinity with a simply
connected end $N_{1}$ so that $C_{1}={\mathbb{R}}\times[0,\infty)$. (We
actually conjecture that there is no such example; see also section $3$.)

Note also that in dimension $2$ all ends are homeomorphic to
${\mathbb{S}}^{1}\times[0,\infty)$, so that the ends of an asymptotically flat
surface are never simply connected.

Combining Theorem $A$ and a result from [GP] we obtain proofs of statements of
Gromov (see [BGS], p.59) which till now have been treated in the literature
(compare [D] and the references therein) as conjectures, and which completely
classify the asymptotically flat manifolds with nonnegative sectional
curvature:

Theorem B.  Let $M$ be an asymptotically flat $m$-manifold with nonnegative
sectional curvature. Then the universal covering of $M$ is isometric to
${\mathbb{R}}^{m-2}\times S$, where $S$ is an asymptotically flat surface. If,
in particular, $M$ is simply connected and $m\geq 3$, then $M$ is isometric to
${\mathbb{R}}^{m}$.

Assuming faster-than-quadratic curvature decay and assuming that the unit
normal bundle of the soul of $M$ has nontrivial tangent bundle, the second
part of Theorem $B$ was proved by Drees ([D]).

As a direct consequence of Theorem $B$ one also obtains an affirmative answer
to a question of Hamilton (see [H], §19; this paper also contains some nice
relations between asymptotical curvatures and the Ricci flow), which is
equivalent to the following one:

Let $M$ be a complete noncompact Riemannian manifold of dimension $m\geq 3$
with positive sectional curvature. Is it true that $A(M)>0$ ?

That in odd dimensions the answer to this question is “yes” was already known
and proved by Eschenburg, Schröder, and Strake ([ESS]).

Our results obviously also have a relation to the positive mass conjecture; in
[GPZ] the reader will find explained the precise connections.

The main idea of the proof of Theorem $A$ can be described as follows:

Inside an end $N_{i}$ of an asymptotically flat manifold with cone structure
at infinity we construct a continuous family of “spheres”, which after
rescaling have uniform curvature bounds and which Gromov–Hausdorff converge to
the “unit sphere” in the cone $C_{i}$.

To this continuous family of spheres we now apply two results from [PRT]: The
first says that any continuously collapsing family with bounded curvature
contains an infinite “stable” subsequence. To this sequence then a corollary
of the Limit of Covering Geometry Theorem from [PRT] applies.

(This corollary actually also holds without using a stability assumption, and
the proof of Theorem $A$ can be given without relying on [PRT], but instead on
results from [PT], see section $1$).

This in turn enables us to prove some inequalities for the ranks of certain
homotopy groups. These imply that in fact collapse is not possible except for
the case where the dimension of the manifold is equal to $4$. Therefore in all
nice cases $C_{i}$ is nothing but ${\mathbb{R}}^{m}$.

There is a vast amount of literature on noncompact complete Riemannian
manifolds whose sectional curvature at infinity is zero (and on many different
specific ways in which the curvature is allowed to go to zero). For a detailed
account of what is known and wanted to be known about such spaces, the reader
is recommended to look at the survey article [Gre] and the paper [GPZ]. Here
we just mention (besides the references already given) some papers in the
field which are most closely related to the results of this note: [ES], [GW1],
[GW2], [KS] and [LS].

The remaining parts of the paper are organized into a preliminaries, a proof,
and a problem section which contains further remarks and several open
questions.

We would like to thank Patrick Ghanaat for pointing out to us a simplified
proof of the sublemma in section $2$ as well as Luis Guijarro for useful
comments.

## 1\. Preliminaries

In this section we review some results about manifolds which collapse with
bounded curvature and diameter. More on this can be found in the references
given in [PRT] and [PT].

Definition.  A sequence of metric spaces $M_{i}$ is called stable if there is
a topological space $M$ and a sequence of metrics $d_{i}$ on $M$ such that
$(M,d_{i})$ is isometric to $M_{i}$ and such that the metrics $d_{i}$ converge
as functions on $M\times M$ to a continuous pseudometric.

Proposition (Continous Collapse implies Stability) ([PRT]).  Suppose that a
simply connected manifold $M$ admits a continuous one-parameter family of
metrics $(g_{t})_{0<t\leq 1}$ with $\lambda\leq K_{g_{t}}\leq\Lambda$ such
that, as $t\to 0$, the family of metric spaces $M_{t}=(M,g_{t})$ Hausdorff
converges to a compact metric space $X$ of lower dimension. Then the family
$M_{t}$ contains a stable subsequence.

The version of the Limit of Covering Geometry Theorem from [PRT] we need in
this paper (it is straightforward to check that the proof given in [PRT] also
proves the result below) can be stated as follows:

Theorem (Limit of Covering Geometry Theorem ([PRT]).  Let $M_{n}$ be a stable
sequence of Riemannian $m$-manifolds with curvature bounds $|K(M_{n})|\leq 1$
such that for $n\to\infty$ the sequence of metric spaces $M_{n}$ Hausdorff
converges to a compact metric space $X$ of lower dimension. Consider any
sequence of points $p_{n}\in M_{n}$ and balls $B_{n}=B_{\pi/2}\i T_{p_{n}}$
which are equipped with the pull back metrics of the exponential maps

$\exp_{p_{n}}\colon T_{p_{n}}\to M_{n}.$

Assume that for any such converging subsequence $B_{n}\to B$, the limit $B$
has curvature $\geq 0$ in the sense of Alexandrov.

Then for any converging subsequence $B_{n}\to B$, the limit $B$ has the same
dimension as the manifolds $M_{n}$, and in a neighbourhood of its center, the
metric on $B$ coincides with that of a metric product ${\mathbb{R}}\times N$,
where $N$ is a manifold with two-sided bounded curvature $0\leq K(N)\leq 1$ in
the sense of Alexandrov.

Our proof of Theorem $A$ will in fact only use the following corollary of this
theorem. At first sight this corollary looks almost obvious, but it doesn’t
seem easy to adopt any of the known proofs of injectivity radius estimates to
this case.

Corollary.  Let $M_{n}$ be a (stable) sequence of closed simply connected
Riemannian manifolds of dimension $m\geq 2$ with curvature $|K(M_{n})|\leq C$
and uniformly bounded diameters. Consider any sequence of points $p_{n}\in
M_{n}$ and balls $B_{n}=B_{\pi/2\sqrt{C}}\i T_{p_{n}}$ which are equipped with
the pull back metrics of the exponential maps

$\exp_{p_{n}}\colon T_{p_{n}}\to M_{n}.$

Assume that for any such converging subsequence $B_{n}\to B$, the limit $B$
has at all interior points curvature $=1$. Then the manifolds $M_{n}$ converge
to a standard sphere.

The stability condition is actually not necessary for the above result to
hold. This can be seen from the following independent proof of the Corollary,
which does not use stability at all. The proof itself is very short, but since
it uses the notion of Grothendieck–Lipschitz convergence and Riemannian
megafolds from [PT], we decided to also incorporate the above [PRT] approach,
which might be easier to understand.

Proof of the Corollary without stability assumption.

The only nontrivial part is to establish a lower positive bound for the
injectivity radii of all manifolds $M_{n}$.

Since because of the Gauss–Bonnet theorem the case $m=2$ is trivial, we may
assume that $m\geq 3$.

If the manifolds $M_{n}$ would collapse, then, after passing to a subsequence
if necessary, one may assume that the manifolds $M_{n}$ Grothendieck–Lipschitz
converge to a Riemannian megafold ${\mathfrak{M}}$ which is not a manifold.

The assumptions of the Corollary imply that the limit ${M}$ has constant
curvature $=1$, so that ${\mathfrak{M}}=({\mathbb{S}}^{m}:G)$, where $G$ is a
commutative group of isometries of ${\mathbb{S}}^{m}$. However, by ([PT],
Theorem A.7) we have that
$0\not=H^{2}_{dR}({\mathfrak{M}})=H^{2}_{dR}({\mathbb{S}}^{m})$, which is a
contradiction. ∎

## 2\. Proofs

Proof of part (i) of Theorem A.

The first statement of the theorem will follow from the fact that the distance
function to $p$, distp, for sufficiently large values does not have any
critical points.

By assumption, for any sequence of numbers $\varepsilon>0$ converging to zero,
the pointed Gromov–Hausdorff limit of $(M,\varepsilon{\cdot}g,p)$ exists and
is isometric to a locally compact metric cone $C$ with vertex $o$. The cone
$C$ obviously has curvature $\geq 0$ (in the sense of Alexandrov) everywhere
except the origin $o\in C$.

Let us assume that there exists a sequence of points $x_{n}$ such that
$|px_{n}|\to\infty$ as $n\to\infty$, and such that each $x_{n}$ is a critical
point for distp.

Consider the sequence of rescaled manifolds $(M,g/|px_{n}|,p)$. By the
assumption of the theorem, this sequence converges to $(C,o)$, and the points
$x_{n}\in(M,g/|px_{n}|)$ (after passing to a subsequence) converge to a point
$x\in C$ which has distance $1$ to the origin $o$.

Since $C$ is a cone, we can consider $y:=2x\in C$. Choose a sequence of points
$y_{n}\in(M,g/|px_{n}|)$ which converge to $y$, and consider minimal geodesics
$x_{n}y_{n}$ from $x_{n}$ to $y_{n}$. Since $x_{n}$ is a critical point of
distp, there is for each $n$ a minimal geodesic $px_{n}$ which makes an angle
less than $\pi/2$ with the minimal geodesic $x_{n}y_{n}$. Therefore
Toponogov’s comparison theorem implies that
$\lim_{n\to\infty}|py_{n}|/|px_{n}|\leq\sqrt{2}$. But obviously
$\lim_{n\to\infty}|py_{n}|/|px_{n}|=|oy|/|ox|=2$, a contradiction.

Thus for some $R>0$ the function distp does not have any critical points
outside the open ball $B_{R}(p)$. In particular, as follows from Morse theory
for distance functions, see ([Grov], Cor. 1.9.), $M$ has finite topological
type; that is, $M$ is homeomorphic to the interior of a compact manifold with
boundary (which in our case is simply the closed ball $\bar{B}_{R}(p)$).

This also implies that the manifold $M$ has only finitely many ends.

Note that the cone $C_{i}$ is nothing but the closure of the connected
component of $C\backslash o$ that corresponds to $N_{i}$, in particular for
each $N_{i}$ the limit $C_{i}=$ GH-$\lim_{\varepsilon\to
0}\varepsilon{\cdot}N_{i}$ exists.

Thus part (i) of Theorem $A$ is proved. ∎

Proof of part (ii) of Theorem A.

The fact that $N$ is homeomorphic to ${\mathbb{S}}^{m-1}\times[0,\infty)$ will
follow directly from the proofs of (ii)(a) and (ii)(b). Therefore we only need
to prove these two statements.

Note that if dim $C_{i}=m=$ dim$M$, then parts (ii)(a) and (ii)(b) of the
theorem are trivially true:

Indeed, if so we have that the curvature of $C_{i}$ is zero everywhere except
the origin. We can assume that $m\not=2$ (otherwise all ends would be
homeomorphic to ${\mathbb{S}}^{1}\times[0,\infty)$, and therefore in
particular they would not be simply connected). It follows that
$C_{i}={\mathbb{R}}^{m}/F$, where $F$ is a finite group of rotations which
acts freely on ${\mathbb{R}}^{m}\backslash 0$. Since $C_{i}\setminus B_{1}(o)$
is homeomorphic to $N_{i}$, it follows that $F=\pi_{1}(\partial N_{i})$. Since
by assumption $\partial N_{i}$ is simply connected, $F$ must be trivial, and
thus for dim $C_{i}=m=$ dim$M$ our claims are proved, since the above also
implies that in this case $N_{i}$ is homeomorphic to
${\mathbb{S}}^{m-1}\times[0,\infty)$.

From now on we will assume that dim $C_{i}<m$.

We can view $C_{i}$ as a cone over its space of directions,
$C_{i}=C(\Sigma_{i})$, where $\Sigma_{i}$ is an Alexandrov space of curvature
$\geq 1$ or dim$\Sigma_{i}=1$. $\Sigma_{i}$ can be viewed as a “unit sphere”
in $C_{i}$.

We will first construct a continuous family of hypersurfaces
$S_{2-1,\varepsilon}$ in $(M,\varepsilon{\cdot}g)$ which collapse to
$\Sigma_{i}$ such that the sectional curvatures of $S_{2-1,\varepsilon}$ stay
uniformly bounded. The following construction is very close to one used by
Kasue in [K2]. We will therefore only explain it here; all of its details can
be found in ([K2, $\S 2$]).

For each rescaling $(M,\varepsilon{\cdot}g,p)$, consider the sphere of radius
$2$, $S_{2,\varepsilon}(p)\i(M,\varepsilon{\cdot}g)$. Its principal curvatures
for outcoming normal directions lie in the range $[-C(\varepsilon),\infty]$,
where $C(\varepsilon)\to 1/4$ as $\varepsilon\to 0$.

Next consider, for an inward direction (to $p$) the equidistant hypersurface
$S_{2-1,\varepsilon}$ at distance $1$ to $S_{2,\varepsilon}(p)$. Then
$S_{2-1,\varepsilon}$ has uniformly bounded principal curvatures which in fact
lie in the range $[-C^{\prime}(\varepsilon),C^{\prime}(\varepsilon)]$, where
$C^{\prime}(\varepsilon)\to 1$ as $\varepsilon\to 0$.

Therefore, since $M$ is asymptotically flat, $S_{2-1,\varepsilon}$ has
uniformly bounded sectional curvature as $\varepsilon\to 0$. For sufficiently
small $\varepsilon$ it follows that $S_{2-1,\varepsilon}$ (equipped with the
induced intrinsic metric) is a continuous family which, as $\varepsilon\to 0$,
collapses to $\Sigma_{i}$.

Key Lemma.  Take any sequence of points $p_{\varepsilon_{n}}\in
S_{2-1,\varepsilon_{n}},\ \varepsilon_{n}\to 0$. Consider the balls
$B_{n}=B_{1}(0)\i T_{p_{\varepsilon_{n}}}(S_{2-1,\varepsilon_{n}})$, equipped
with the pull back metrics.

Then as $n\to\infty$, the $B_{n}$ Lipschitz converge to the ball of radius $1$
in ${\mathbb{S}}^{l-1}\times{\mathbb{R}}^{m-l}$, for some fixed $l$ depending
on $M$.

Moreover $\Sigma_{i}=\widetilde{\mathbb{S}}^{l-1}/A$, where $A$ is an Abelian
group of isometries of $\widetilde{\mathbb{S}}^{l-1}$ (here by
$\widetilde{\mathbb{S}}^{l-1}$ we understand the standard $l-1$-sphere if
$l\geq 3$, ${\mathbb{R}}$ if $l=2$, and a point if $l=1$).

The proof of the Key Lemma will be given below. Let us now continue with the
proof of part (ii) of Theorem $A$:

Obviously all $S_{2-1,\varepsilon}$ are homeomorphic to $\partial N_{i}$ and
therefore simply connected. Now applying the Corollary in section $1$ we see
that $l<m$.

Using that $\partial N_{i}$ is simply connected we can moreover show that the
group $A$ in the Key Lemma is connected: Let $A^{o}$ be the identity component
of $A$. Then $\widetilde{\Sigma}_{i}={\mathbb{S}}^{l-1}/A^{o}$ is a branched
covering of $\Sigma_{i}$, and it is easy to see that one can find a covering
$\widetilde{\partial N_{i}}\to\partial N_{i}$ which is a lifting of
$\widetilde{\Sigma}_{i}\to\Sigma_{i}$. But since $\partial N_{i}$ is simply
connected we have that $\widetilde{\Sigma}_{i}=\Sigma_{i}$. Therefore
$A^{o}=A$; that is, $A$ is connected.

Now if $l=1$, then $\Sigma_{i}$ is a point, so that $\partial N_{i}$ must be
an infranil manifold. But any infranil manifold has infinite fundamental
group, which contradicts the fact that $\partial N_{i}$ is simply connected.

If $l=2$ it follows that $\Sigma_{i}$ is homeomorphic to a point or
${\mathbb{R}}$, since $A$ is connected. The first case cannot occur by the
above reasoning, and the second contradicts that $C$ is locally compact.

Therefore the only serious case to deal with is the case $l\geq 3$.

From the above we have that in this case $\Sigma_{i}$ is isometric to
${\mathbb{S}}^{l-1}/{\mathbb{T}}^{k^{\prime}}$.

Since for $\varepsilon\to 0$ the hypersurfaces $S_{2-1,\varepsilon}$ collapse
to $\Sigma_{i}$ and since $S_{2-1,\varepsilon}$ is homeomorphic to $\partial
N_{i}$, we know that $\Sigma_{i}$ is homeomorphic to $\partial
N_{i}/{\mathbb{T}}^{k}$ and that this homeomorphism can be chosen to preserve
the natural stratifications of these spaces.

Let us now do some topological calculations:

Let $O_{{\mathbb{T}}^{k}}$ be a regular orbit of the ${\mathbb{T}}^{k}$ action
on $\partial N_{i}$. Consider the relative homotopy sequence of pairs

$\pi_{2}(\partial
N_{i},O_{{\mathbb{T}}^{k}})\to\pi_{1}({\mathbb{T}}^{k})={\mathbb{Z}}^{k}\to\pi_{1}(\partial
N_{i})=0.$

Therefore rk${}_{\mathbb{Q}}\,\pi_{2}(N_{i},O_{{\mathbb{T}}^{k}})\geq k$.

Next consider the corresponding homotopy sequence for ${\mathbb{S}}^{l-1}$:

$0=\pi_{2}({\mathbb{T}}^{k^{\prime}})\to\pi_{2}(S^{l-1})\to\pi_{2}({\mathbb{S}}^{l-1},O_{{\mathbb{T}}^{k^{\prime}}})\to\pi_{1}({\mathbb{T}}^{k^{\prime}})={\mathbb{Z}}^{k^{\prime}}\to\pi_{1}({\mathbb{S}}^{l-1})$

Therefore
rk${}_{\mathbb{Q}}\,\pi_{2}({\mathbb{S}}^{l-1},O_{{\mathbb{T}}^{k^{\prime}}})=k^{\prime}+$
rk${}_{\mathbb{Q}}\,\pi_{2}({\mathbb{S}}^{l-1})$.

On the other hand one has that $k=$ dim$\partial N_{i}-$ dim$\Sigma_{i}$ and
$k^{\prime}=l-1-$ dim$\Sigma_{i}$.

Let $\Sigma_{i}^{\\#}$ denote $\Sigma_{i}$ with the singular sets removed.
Consider now the following three cases:

1\. $\Sigma_{i}$ has no boundary. Then obviously

$\hbox{rk}_{\mathbb{Q}}\,\pi_{2}({\mathbb{S}}^{l-1},O_{{\mathbb{T}}^{k^{\prime}}})=\hbox{rk}_{\mathbb{Q}}\,\pi_{2}(\Sigma_{i}^{\\#})=\hbox{rk}_{\mathbb{Q}}\,\pi_{2}(\partial
N_{i},O_{{\mathbb{T}}^{k}}).$

2\. $\partial\Sigma_{i}$ has one component. Then

$\hbox{rk}_{\mathbb{Q}}\pi_{2}({\mathbb{S}}^{l-1},O_{{\mathbb{T}}^{k^{\prime}}})=\hbox{rk}_{\mathbb{Q}}\pi_{2}(\Sigma_{i}^{\\#})+1=\hbox{rk}_{\mathbb{Q}}\pi_{2}(\partial
N_{i},O_{{\mathbb{T}}^{k}}).$

3\. $\partial\Sigma_{i}$ has more than one component.

In both case $1$ and case $2$ it follows that $k^{\prime}\geq k-$
rk${}_{\mathbb{Q}}\pi_{2}({\mathbb{S}}^{l-1})$, and hence

$m\leq l+\text{rk}_{\mathbb{Q}}\pi_{2}({\mathbb{S}}^{l-1}).$

However, this contradicts the fact that $l<m$, except if $m=4,l=3$. In this
particular case it follows that $\Sigma_{i}={\mathbb{S}}^{2}/A^{rot}$.
Therefore, since $\Sigma_{i}$ has not more than one boundary component, we
have that $A^{rot}$ is trivial and $\Sigma_{i}={\mathbb{S}}^{2}$. Thus
$C_{i}=C(\Sigma_{i})={\mathbb{R}}^{3}$ (and that this indeed can happen was
shown in [U]).

Case $3$ can only occur if $\Sigma_{i}$ is homeomorphic to $[0,1]$. Then,
since $N_{i}$ is simply connected, it must hold that $k$, $k^{\prime}\leq 2$.
Since the ${\mathbb{T}}^{k}$ action on $\partial N_{i}$ has empty fixed point
set, we have that $k=2$, and since $l<m$, we have that $k^{\prime}=1$.
Therefore $m=4$, $l=3$ and $\Sigma_{i}$ is isometric to
${\mathbb{S}}^{2}/{\mathbb{S}}^{1}=[0,\pi]$ , so that
$C_{i}=C(\Sigma_{i})={\mathbb{R}}\times[0,\infty)$.

The proof of Theorem $A$ is complete. ∎

Proof of Theorem B.

Let $M$ be an asymptotically flat $m$-manifold with nonnegative sectional
curvature. Then $M$ has cone structure at infinity, and by [GP] the soul $S$
of $M$ is flat. This forces the universal cover $\tilde{M}$ of $M$ to split
isometrically as a Euclidean part, coming from the soul $S$, and a
nonnegatively curved complete manifold $F$ homeomorphic to ${\mathbb{R}}^{k}$.

Now $F$ is also asymptotically flat and has one end
${\mathbb{S}}^{k-1}\times[0,\infty)$. Therefore by Theorem $A$, if $k\neq
2,4$, then the cone at infinity of $F$ is isometric to ${\mathbb{R}}^{k}$.
Since by Toponogov’s Comparison Theorem any line in the cone corresponds to a
line in $F$, it follows from the Toponogov Splitting Theorem that $F$ itself
is isometric to ${\mathbb{R}}^{k}$.

Thus to finish the proof we must only exclude the case $k=4$. By Theorem $A$,
if $k=4$ we have that $C=$ GH-$\lim_{\varepsilon\to 0}\varepsilon{\cdot}F$ is
isometric to one of the following: ${\mathbb{R}}^{4}$, ${\mathbb{R}}^{3}$ or
${\mathbb{R}}\times[0,\infty)$. In all of these cases we have that $C$
contains a line, and therefore $F$ splits isometrically as ${\mathbb{R}}\times
F^{\prime}$. But since $F$ is asymptotically flat it follows that $F^{\prime}$
is flat, and therefore $F$ is isometric to ${\mathbb{R}}^{4}$. ∎

Proof of the Key Lemma.

Consider a $\nu$-neighbourhood $U\supset\Sigma_{i}\i C_{i}$. From the results
of [CFG] (see section $1$ of [PRT], where also further references can be
found) we have an $N$-structure $\pi\colon E_{\varepsilon}\to U$, where
$E_{\varepsilon}$ is a subset of $(M,\varepsilon{\cdot}g)$ containing the
hypersurface $S_{2-1,\varepsilon}$. Since $E_{\varepsilon}$ is homotopically
equivalent to $\partial N_{i}$, it follows that $E_{\varepsilon}$ is simply
connected. Therefore the $N$-structure is given by an almost isometric smooth
${\mathbb{T}}^{k}$-action without fixed points (see again section $1$ in
[PRT]).

Now take a point $x\in\Sigma_{i}\i C_{i}$ (so $|ox|=1$) and consider a
spherical neighbourhood of $U_{x}\ni x$. Consider the preimage
$V_{\varepsilon}=\pi^{-1}(U_{x})\i E_{\varepsilon}$ and let
$\widetilde{V}_{\varepsilon}$ be its universal Riemannian covering. Then the
${\mathbb{T}}^{k}$-action induces an almost isometric ${\mathbb{R}}^{k}\times
F$ action on $\widetilde{V}_{\varepsilon}$, where $F$ is a finite Abelian
group.

From [CFG] one has a uniform bound for the injectivity radius of
$\widetilde{V}_{\varepsilon}$, so that, as $\varepsilon\to 0$,
$\widetilde{V}_{\varepsilon}$ converges to a flat manifold $\widetilde{V}_{0}$
with boundary and isometric ${\mathbb{R}}^{k}\times F$ action (for the
convergence claim see the first part of Lemma 2.1.4 in [PRT]).

Since the interior of $\widetilde{V}_{0}$ is flat, there exists a map
$\widetilde{V}_{0}\to{\mathbb{R}}^{m}$ which is for all interior points a
local isometry. Therefore the ${\mathbb{R}}^{k}\times F$ action on
$\widetilde{V}_{0}$ can be extended to an action of whole ${\mathbb{R}}^{m}$,
and the local factors $U\i{\mathbb{R}}^{m}/{\mathbb{R}}^{k}$ are isometric to
local branched coverings of subsets of $C_{i}$. (Here by local factors we
understand factorizing $U$ by the connected components of the
${\mathbb{R}}^{k}$-orbits in $U$, as is illustrated in the following picture).

Now the above group ${\mathbb{R}}^{k}$ can be regarded as an Abelian group of
isometries of Euclidean space ${\mathbb{R}}^{m}$. We will show that in our
case ${\mathbb{R}}^{k}$ actually splits into a direct sum of translations and
rotations.

To this means first note the following:

Sublemma.  Let a connected Abelian group $H$ act on Euclidean space
${\mathbb{R}}^{m}$ by isometries. Then one can represent ${\mathbb{R}}^{m}$ as
an orthonormal sum $V\oplus W$ such that $H$ is contained in a direct sum of
translations and rotations,

$H<A^{tr}\oplus A^{rot},$

so that the following holds: The group $A^{tr}=V$ is the group consisting of
all parallel translations along $V$, and $A^{rot}\i O(W)$ is an Abelian
subgroup of rotations of $W$.

Proof of the Sublemma.

By [Al] one orbit of $H$ is an affine subspace $V$ (in fact, such an orbit
corresponds to the origin $o$ of $C_{i}$). Choose the origin of affine space
${\mathbb{R}}^{m}$ so that it is contained in this subspace. Each element
$\alpha\in H$ can be viewed as $(r_{\alpha},\phi_{\alpha})\in V\times O(m)$,
such that $\alpha(x)=r_{\alpha}+\phi_{\alpha}(x)$ for any
$x\in{\mathbb{R}}^{m}$.

Then $V$ can be viewed as the set of all pure translations of $H$,

$A^{tr}=V=\\{r:(r,\phi)\in H\ \hbox{for some}\ \phi\in O(m)\\}.$

Let $A^{rot}:=\\{\phi:(r,\phi)\in H$ for some $r\\}$ be the group of pure
rotations of $H$. If each $\phi\in A^{rot}$ acts trivially on $V$, then
obviously $H<A^{tr}\oplus A^{rot}$, which is exactly what we want.

Therefore we only have to prove that for any $\phi\in A^{rot}$ and any $v\in
V$ we have that $\phi(v)=v$.

Take any $(r,\phi)\in H$ and $v\in V$. For all $n\in{\mathbb{N}}$ there exists
$\phi_{n}\in O(m)$ such that $(nv,\phi_{n})\in H$. Since $H$ is Abelian, it
follows that $(r,\phi)(nv,\phi_{n})=(nv,\phi_{n})(r,\phi)$ and therefore
$nv+\phi_{n}r=r+\phi nv$. Dividing by $n$ and letting $n\to\infty$ thus
implies $\phi v=v$. ∎

Thus our group ${\mathbb{R}}^{k}$ is contained in a direct sum
$A^{tr}\oplus\tilde{A}^{rot}$, where $\tilde{A}^{rot}$ is universal covering
Lie group of $A^{rot}$. Now note that since the local factors by
${\mathbb{R}}^{k}$ have a cone structure, ${\mathbb{R}}^{k}$ moreover itself
splits as ${\mathbb{R}}^{k}=A^{tr}\oplus\tilde{A}^{rot}$:

Indeed, since the local factors $U/{\mathbb{R}}^{k}$ admit a cone structure,
in radial directions their sectional curvatures must be zero. But this is
impossible unless ${\mathbb{R}}^{k}$ is itself a direct product
${\mathbb{R}}^{k}=\tilde{H}=A^{tr}\oplus\tilde{A}^{rot}$:

To prove this, we only have to show that (in the notation of the Sublemma) it
holds that $A^{rot}\i H$. Assume that this is wrong. Then we can find a ray
$c\colon[0,\infty)\to{\mathbb{R}}^{m}$ which is orthogonal to $V$, and there
will be an element $\alpha\in a^{rot}$ in the Lie algebra of $A^{rot}$ which
is not contained in the Lie algebra of $H$, so that $\alpha$ defines a linear
Jacobi field on the ray $c$ which can assumed to be non-zero.

Consider now the projection $\bar{c}$ of $c$ along some local factor. Then
$\bar{c}$ is a piece of a ray in the cone $C$ and the projection $\bar{J}$ of
the field $J$ is also a Jacobi field. But since $C$ is a cone, any Jacoby
field along $\bar{c}$ must be linear. On the other hand it is straightforward
to show that $|\bar{J}(t)|$ is a strictly concave function, and this is a
contradiction.

Therefore the local factors $W/A^{rot}$ are isometric to local branched
coverings of $C_{i}$ (everywhere except the origin). Thus $C_{i}\setminus o$
is isometric to a factor of its universal covering, $\widetilde{(W\setminus
0)}/A$, by an Abelian Lie group $A$. Restricting this last isometry to the
unit spheres of both cones it follows that
$\Sigma_{i}=\tilde{\mathbb{S}}^{l-1}/A$, and the second part of the Key Lemma
is proved.

Let $\rho\colon\widetilde{V}_{\varepsilon}\to V_{\varepsilon}$ be the covering
map and $\widetilde{S}_{2-1,\varepsilon}=\rho^{-1}(S_{2-1,\varepsilon})$. It
converges to the preimage of $\Sigma_{i}$ under the map $V_{0}\to
V_{0}/A=U_{x}\i C_{i}$, so that it locally coincides with the cylinder
$V\times{\mathbb{S}}^{l-1}$, where ${\mathbb{S}}^{l-1}$ is the unit sphere in
$W$. Therefore, since $x\in\Sigma_{i}$ can be chosen to be arbitrary, the
covering geometry of $S_{2-1,\varepsilon}$ converges to the one of
$V\times{\mathbb{S}}^{l-1}$, and this finishes the proof of the first part of
the Key Lemma. ∎

## 3\. Remarks and Open Questions

Question 1.  Let $M$ be an asymptotically flat manifold, and let the sequence
$(M,\varepsilon_{n}{\cdot}g,p)$ converge to $(G,o)$ as $\varepsilon_{n}\to 0$.
Assume that dim$G\geq 3$ and that $G\backslash o$ has only one connected
component.

Is it true that $G$ is a metric cone with origin $o$?

A positive answer to this question could possibly lead to a general
classification of asymptotically flat manifold of higher dimension. To obtain
such a classification is particularly interesting because of the fact that
Gromov (see [Grom], p.96 and also [LS]) showed that any (smooth paracompact)
noncompact manifold $M$ admits a complete Riemannian metric whose asymptotic
curvature satisfies $A(M)\leq C$, where $C$ depends only on the dimension of
$M$.

Question 2.  Does there exist in each dimension $m$ a positive constant $C(m)$
such that any noncompact complete Riemannian $m$-manifold $M$ with $A(M)\leq
C(m)$ admits an asymptotically flat metric?

Note that the answer (positive or negative) to the following question would
give a complete classification of the cone structures at infinity of simply
connected ends of asymptotically flat manifolds:

Question 3.  Can the cone ${\mathbb{R}}\times[0,\infty)$ be a cone at infinity
of a simply connected end of an asymptotically flat $4$-manifold?

It seems not possible to obtain such an example by a direct generalization of
Unnebrink’s example. Namely, one can exchange the Berger spheres
${\mathbb{S}}^{3}_{f(t),h(t)}$ (in the notation of [U]) in Unnebrink’s example
by ${\mathbb{S}}^{3}_{a(t),f(t),h(t)}$, where the number $a(t)$ describes
along which one-dimensional subgroup of the ${\mathbb{T}}^{2}$-action on the
standard ${\mathbb{S}}^{3}$ we shrink the distance (so
${\mathbb{S}}^{3}_{a,f,h}$ is a Berger sphere if $a=\pm 1$). But direct
calculation then shows that there is no triple of functions $a,f,h$ which
would give an asymptotically flat 4-manifold with
${\mathbb{R}}\times[0,\infty)$ as a cone at infinity.

However, on the other hand, if one would take as $a$ a constant which is close
to $1$, then as a result one obtains an end $N$ whose asymptotic curvature
$A(N)$ is arbitrarily small and which has ${\mathbb{R}}\times[0,\infty)$ as
cone at infinity.

Remark.  The same arguments as the one which we used in the proof of Theorem
$A$ actually also make it possible to characterize the cones at infinity of
complete noncompact manifolds whose asymptotic curvature is small:

Namely, if for some given sequence of simply connected $m$-dimensional ends
$N_{n}$ with $A(N_{n})\to 0$ as $n\to\infty$ their cones at infinity $C_{n}$
Gromov–Hausdorff converge to some metric space $C$ (which then must be a
cone), then for sufficently large $n$ it holds that $N_{n}$ is homeomorphic to
${\mathbb{S}}^{m-1}\times[0,\infty)$ and moreover the following is true:

(a) if $m\not=4$, then $C$ is isometric to ${\mathbb{R}}^{m}$;

(b) if $m=4$, then $C$ is isometric to one of the following spaces:
${\mathbb{R}}^{4}$, ${\mathbb{R}}^{3}$, or ${\mathbb{R}}\times[0,\infty)$.

The above modification of the Unnebrink construction for constant $a$ shows
that for manifolds with small asymptotic curvature all cones which are
mentioned in part (b) actually do arise.

As a last point we would like to mention that the methods we used in this
paper do not distinguish between spaces which are asymtotically flat and
sequences of spaces whose asymptotic curvature goes to zero.

Therefore, no matter how special our question whether
${\mathbb{R}}\times[0,\infty)$ can be a cone at infinity of a simply connected
end of an asymptotically flat $4$-manifold might at first sight look like, any
negative answer to it will require more sensitive collapsing techniques.

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