# On the genericity of singularities in spacetimes with weakly trapped
submanifolds

Ivan P. Costa e Silva  Department of Mathematics, Universidade Federal de
Santa Catarina, 88.040-900
Florianópolis-SC, Brazil<EMAIL_ADDRESS>and  Victor L. Espinoza
Department of Mathematics, Universidade Federal de Santa Catarina, 88.040-900
Florianópolis-SC, Brazil<EMAIL_ADDRESS>
###### Abstract.

We investigate suitable, physically motivated conditions on spacetimes
containing certain submanifolds - the so-called weakly trapped submanifolds \-
that ensure, in a set of neighboring metrics with respect to a convenient
topology, that the phenomenon of nonspacelike geodesic incompleteness (i.e.,
the existence of singularities) is generic in a precise technical sense. We
obtain two sets of results. First, we use strong Whitney topologies on spaces
of Lorentzian metrics on a manifold $M$, in the spirit of [34], and obtain
that while the set of singular Lorentzian metrics around a fiducial one
possessing a weakly trapped submanifold $\Sigma$ is not really generic, it is
nevertheless prevalent in a sense we define, and thus still quite “large” in
this sense. We prove versions of that result both for the case when $\Sigma$
has codimension 2, and for the case of higher codimension. The second set of
results explore a similar question, but now for initial data sets containing
MOTS. For this case, we use certain well-known infinite dimensional, Hilbert
manifold structures on the space of initial data and use abstract functional-
analytic methods based on the work of [12] to obtain a true genericity of null
geodesic incompleteness around suitable initial data sets containing MOTS.

###### Key words and phrases:

Global Lorentzian geometry; singularity theorems; generic properties

###### 2020 Mathematics Subject Classification:

53C50, 53C80

##  1\. Introduction

Let $(M^{n},g)$ be spacetime, i.e., a connected time-oriented Lorentz manifold
of dimension $n\geq 2$. An embedded spacelike submanifold (without boundary)
$\Sigma\subset M$ is said to be (future) trapped (in $(M,g)$) if its mean
curvature vector field $H^{\Sigma}$ is past-directed timelike everywhere on
$\Sigma$. (Such submanifolds are also known as a future-converging, comp. [38,
Def. 14.57], and the time-dual definitions of past trapped/past-converging
submanifolds are understood throughout even if they will be hardly
mentioned.). In particular, compact, codimension-two trapped submanifolds
(often called closed trapped surfaces in the physics literature) were
introduced by Roger Penrose in 1965 in the context of his celebrated first
singularity theorem proven in [40]. Penrose’s insightful use of trapped
submanifolds to model gravitational collapse set up the template for all the
other singularity theorems in the literature (those described in [9, 22, 23]
being only the most famous ones). The importance of theorems of this kind in
geometric theories of gravity - those in which gravitational fields are
described via the geometry of some spacetime irrespective of detailed field
equations - cannot be overstated, and indeed the 2020 Nobel prize in physics
was shared by Penrose in a direct recognition of this fact.

It is well known that all singularity theorems establish the existence of
incomplete, inextendible causal geodesics in spacetimes under certain
physically motivated geometric assumptions, the presence of compact trapped
submanifolds being a primary one. Now, their role in these theorems is not as
independent physical structures, but rather to provide an abstract,
differential-geometric description of the onset of gravitation collapse. As
such, trapped submanifolds are widely believed to appear in any physically
realistic model where “large enough concentrations of mass gather in small
enough regions”. However, due to their abstract character, a precise
mathematical formulation of the informal statement between quotations marks
and an actual proof that compact trapped surfaces should indeed develop under
suitable conditions are notoriously difficult. Even in the simplest and best-
known geometric theory of gravity, general relativity, these proofs require
very specific, highly technical hypotheses in order to control the analytic
and geometric features of a long-term development of initial data for the
Einstein field equations, see e.g., [14, 29, 28]. (Geometric theories other
than general relativity, with different field equations, would have to be
analyzed on a case-by-case basis.) Therefore, the existence of suitable
trapped submanifolds remains in most cases an added, extra assumption.

Yet, since the appearance of incomplete causal geodesics are the primary
concern in singularity theorems, one might try to find a way to deduce their
presence indirectly, without imposing the existence of a trapped submanifold
at the outset. For instance, insofar as the interior of black holes are
expected to be prime places to find singularities in physics, one might hope
to link their presence to the detection of the black horizons themselves.
(Indeed, insofar as one cannot “peek” behind event horizons, this is arguably
the only way one can actually do it in physics!). Mathematically, however,
black hole horizons have a global definition that seems ill-suited to make
them good substitutes for “quasi-local” objects like closed trapped surfaces
(see, e.g., [24, pp. 315-323] and [25, pp. 27-74]) in singularity theorems.
But an early, general quasi-local description of black hole horizons have been
given via marginally outer trapped surfaces (MOTS), which are closely related
to (and sometimes conflated with) the so-called apparent horizons [24, 44].
Spatial slices of totally geodesic null hypersurfaces - such as Killing
horizons, a class that includes the event horizons of stationary black holes
in general relativity - give key examples of MOTS. Although in general,
dynamic, non-stationary black holes MOTS do not coincide with slices of black
hole horizons, they often appear together. Indeed, they are much used in
numerical relativity in “horizon-finder“ algorithms even in a highly dynamic
regime far away from stationarity [35, 43].

We shall say that an embedded spacelike submanifold (without boundary)
$\Sigma\subset M$ is weakly (future) trapped (in $(M,g)$) if at each
$p\in\Sigma$ its mean curvature vector $H^{\Sigma}_{p}$ is either zero or
past-directed causal. Any trapped submanifold is thus weakly trapped, but so
are MOTS. The question now is: can we prove singularity theorems in the
presence of weakly trapped submanifolds? Put in this naive way, the answer is
of course generally negative: It is easy (via, say, simple isometric
identifications of Minkowski spacetime) to find geodesically complete Ricci-
flat spacetimes containing weakly trapped submanifolds of any codimension
$\geq 1$. However, one intuitively expects that in the presence of weakly
trapped submanifolds, “arbitrarily small perturbations” of the spacetime
metric might produce trapped submanifolds and hence singularities. But it
might still occur that such perturbations would have to be very ”special” to
produce trapped manifolds. Since on physical grounds any field can only have
its values ascertained within a certain empirical precision, one may change
the question and ask whether the occurrence of singularities is a “generic”
feature in a suitable set of metrics “near” some metric containing MOTS or
other weakly trapped submanifolds. If so, we “almost always” should expect
singularities to occur, given the large ensemble of black hole horizons
believed to occur throughout the visible universe. We only need to make these
terms between quotation marks mathematically more precise.

A first hint as to how this can be done is given by a theorem by [15, Prop.
1.1], in which the density of outer trapped surfaces near MOTS on a suitable
space of initial data sets in general relativity is established, and since the
presence of outer trapped surfaces are enough to establish a singularity
theorem in that context, this can be regarded as definite evidence of the
genericity mentioned above. In a different vein, there are certain “generic
singularity theorems” such as those proven by [41] which rely on a variant of
the so-called generic condition on the curvature tensor (cf. [9, Section
2.5]). This is an assumption already used in the Penrose-Hawking classic
singularity theorem ([23]), and although the generic condition at first seems
to be a somewhat contrived curvature constraint, its “truly generic” character
has been analyzed at tangent spaces by [10], as well as globally in the
master’s thesis of [30]. These results suggest a natural, broader conceptual
perspective on the singularity theorems in [15, 41]: one can try and obtain
density/genericity of a whole class of singular spacetimes near (with respect
to a suitable topology on the space of Lorentzian metrics) a spacetime
containing a closed weakly trapped submanifold such as a MOTS.

In order to turn these general musings into definite theorems, we are inspired
by two separate approaches. One is the seminal work by [34], who first
introduced a natural framework to discuss such stability and genericity issues
in mathematically rigorous terms and in full nonlinear generality, namely the
strong Whitney $C^{s}$ topologies on the space of Lorentzian metrics on a
given manifold. Lerner presented a cogent case for the special suitability of
these topologies in mathematical relativity, analyzing and establishing the
stability of a number of causal and curvature properties used in the original
singularity theorems (see also [9, Ch. 7] for a detailed discussion and
further results and references on the subject).

Although they are geometrically flexible and conceptually satisfying, Lerner’s
methods apply to spacetimes as a whole. As opposed to that, an initial data
set analysis is arguably of greater physical importance. In fact, the latter
approach seems to be the only viable one in numerical relativity, for example
[17], which in turn is the only practical means of producing theoretical
templates to be confronted with astrophysical observations. In this context,
one wishes to investigate how often the existence of singularities - i.e.,
incomplete causal geodesics inextendible in the maximal Cauchy development of
a given initial data - can be inferred. The natural means here is to use a
suitable Banach manifold structure of the class of initial data sets on a
given manifold [2, 7, 16] and analytic methods quite different from those of
Lerner’s. The work of [15] (see in particular Thm. 1.2 therein) cited above
provides a particularly relevant example of this approach.

The goal of this paper is to combine the spirit of both [34] and [15] \-
though with different technical details - to obtain novel conditions under
which causal geodesic incompleteness of spacetimes can be regarded as a
generic phenomenon.

Reflecting these two motivating philosophies, we divide our work into two main
parts. Our first set of results use the strong Whitney topologies on the set
of Lorentzian metrics on a fixed manifold, in order to analyze how frequently
causal geodesic incompleteness occurs in spacetimes containing weakly trapped
submanifolds. (Recall, these comprise a class that includes MOTS.) Using a
concrete example, we show, in particular, that if we consider only those
spacetimes with certain physically motivated but strictly enforced geometric
conditions, then singularities are not generic with respect to Whitney
topologies. But upon allowing slight perturbations of these conditions - which
seems to be a physically sensible thing to do anyway - then they are shown to
be still prevalent in a suitable technical sense we explain.

On the other hand, in the second part we do obtain a strict genericity result
directly influenced by the arguments in [15]. We emphasize that whereas these
authors obtain density, our goal in this section is to obtain genericity of
incompleteness near MOTS. (To see the difference, consider the standard
example of the set of irrational real numbers versus rational ones in
$\mathbb{R}$: both subsets are dense therein, but only the former is generic;
thus “almost every number is irrational”.) We also use abstract infinite
dimensional (Banach/Hilbert) manifold structures on the set of initial data
(under some reasonable restrictions, cf. [7] and [2]), and apply the most
basic technical tool to obtain genericity results in this contexts, namely the
Sard-Smale theorem ([42]).

We view these two sets of results as complementary perspectives on the same
problem. As one might expect, there are advantages and disadvantages in each
approach. Among the perks of the Whitney topology approach we might count: (i)
the proofs are less technical, and rely on already fairly well-known
topological techniques, (ii) our curvature assumption in the codimension 2
case (the so-called strong energy condition) is strictly weaker than the
dominant energy condition often imposed on initial data sets, (iii) our
results require only the presence of weakly trapped submanifolds, a class far
larger than just MOTS, (iv) we include a result for higher codimension for
little extra cost, and last but not least (v) the causality requirements on
spacetime - basically stable causality - are much weaker than those in the
second part. The latter point is relevant especially in physical applications,
because unless strong cosmic censorship applies, any incomplete causal
geodesic one predicts in the - necessarily globally hyperbolic - maximal
Cauchy development of a given initial data set might still be complete on an
isometric extension of lower causality. This can be neatly illustrated in the
case of initial data induced on a suitable smooth partial Cauchy hypersurfaces
in anti-de Sitter spacetime, which is geodesically complete, stably causal but
not globally hyperbolic. The causal geodesics are all complete in the whole
spacetime, but all incomplete in the Cauchy development of the partial Cauchy
hypersurface viewed as a spacetime on its own right.

The initial data approach, on the other hand, is technically more involved,
but it is much more convenient to treat the specific case of MOTS, which are
after all natural models for (slices of stationary) black hole horizons.
Initial data sets have broader applicability in PDE analysis of the Einstein
fields equations of general relativity, and also in numerical methods [8, 17].
From a physical perspective, again, one might argue that one can hardly expect
to glean actual, direct information of spacetimes as a whole. Rather, all one
can expect is to make predictions from current data, and initial data sets are
just the model for such a situation. While we can only guarantee genericity of
causal incompleteness for the Cauchy development of initial data sets, and in
particular only for globally hyperbolic spacetimes, these already cover a vast
amount of interesting and relevant cases.

The rest of the paper is organized as follows. In section 2 we recall some
basic notions and fix notation and conventions, and in subsection 2.3 we
briefly review some of the main results in [34] we shall use. Section 3
discusses the codimension 2 case, and the respective theorem 3.7 is proven in
subsection 3.1. Also, we are able to answer a problem raised by Lerner
(example 3.4) that as far we know was left unanswered, and expand on such
point (proposition 3.6). Section 4 discusses analogous results in larger
codimension of the weakly trapped submanifold, while theorem 4.6 itself is
proven in subsection 4.1. Finally section 5 is devoted to the initial
data/MOTS case. We start with an abstract, Banach manifold genericity method
especially adapted from [12]. This abstract approach has the enormous
advantage of flexibility: we can choose among a number of variants of
Banach/Hilbert manifold structures on initial data sets and on the set of
embeddings extant in the literature, subject only to relatively mild technical
restrictions. It also bypasses most of the tremendously involved
technicalities arising from looking too closely uponthese structures. We then
obtain definite genericity results in (theorem 5.10 and theorem 5.13) as a
relatively straightforward consequence of the abstract machinery.

##  2\. Preliminary Notions

In this initial section we briefly introduce terminology and notation that
will recur throughout this paper, and also review some topological and
geometrical results that will be referenced in our main discussion.

We shall assume the reader is familiar with the basic concepts of Lorentzian
geometry, and refer to the textbooks by [9, 38] for details.

Throughout this work, $M$ denotes a fixed connected, non-compact smooth (i.e.
$C^{\infty}$) real manifold without boundary of dimension $m+k\geq 3.$

###  2.1. Genericity and Prevalence

As discussed in the Introduction, the goal of this paper is to establish,
under suitable conditions, that the appearance of singularities in a spacetime
is a generic phenomenon. This is made precise via the following standard
definition.

######  Definition 2.1.

Let $X$ be a topological space. A subset $A\subset X$ is said to be _residual_
(in $X$) if it contains a countable union of open dense sets. A property $P$
of elements of $X$ is said to be _generic_ if the subset of all those elements
of $X$ that possess the property $P$ is residual.

The complementary notion of a residual set is called a _meager set_ , that is,
a set $S\subset X$ contained in a countable union of nowhere dense subsets
(sets such that the interior of their closure is empty) of the topological
space $X$. If every residual set is dense, we say that the topological space
$X$ is a _Baire space_. The well-known Baire category theorem then states that
completely metrizable spaces or locally compact Hausdorff spaces are Baire
spaces ([33], Thm. 4.68).

While genericity of course implies density on Baire spaces, the converse is
certainly false: the set of rational numbers is dense in the real line, but as
a countable union of singleton sets it is also a meager set.

The idea of searching for generic properties is that its complementary meager
set is _topologically negligible_ : a nowhere dense set is so “topologically
small” that the interior of its closure is empty, and a meager set is
contained in a countable union of such “small” sets, which is also viewed as
small. While such notion of smallness seems hardly as good as a the measure-
theoretical idea of a null set applicable on $\mathbb{R}^{n}$, and more
generally on finite-dimensional manifolds, it is widely adopted in the context
we are interested in here. Part of the reason is because there are no
infinite-dimensional analogues of the Lebesgue measure. (See [27] for a
discussion on this topic and for an alternative notion of smallness in
infinite dimensional linear spaces. While closer to measure theory, it is
unclear how to adapt their definitions and results to infinite dimensional
manifolds, and thus they will not be pursued here. See also [39] for a survey
of the advantages and shortcomings of Baire category “smallness” as opposed to
a measure-theoretic one). In any case, the notions of residual sets and
genericity, as well as its dual notion of topological smallness are very
standard throughout the literature, and have proven to be useful in many
geometrical situations.

We introduce here the following alternative notion of topological smallness
that will be relevant for our first set of results.

######  Definition 2.2.

Let $X$ be a topological space and let $C\subseteq X$ be a closed set. We
shall say that a set $A\subseteq C$ is _prevalent_ 111This nomenclature is
inspired by the work of [27], but it is unrelated to the measure-theoretic
notion of prevalence introduced by these authors. in C if $C\setminus A$ is a
meager set in $X.$

It is clear from this definition that if $A\subset A^{\prime}\subset C$ and
$A$ is prevalent in the closed set $C\subset X$, then $A^{\prime}$ is also
prevalent in $C$. In particular, the (topological) closure in $X$ (or
equivalently in $C$) of a prevalent set in $C$ is also prevalent in $C$.
Arbitrary unions and countable intersections of prevalent subsets of $C$ are
also prevalent in $C$. Finally, it should be obvious that the notion of
prevalence is nontrivial only when $C$ has itself nonempty interior in $X$,
something which will always occur in all our intended applications, and thus
will be implicitly assumed from now on.

The following elementary lemma will be of use later on.

######  Lemma 2.3.

Let $C$ be a closed subset of a topological space $X,$ and let $U\subseteq C$
be an open subset of $X$ such that $int(C)\subseteq\overline{U}.$ Then
$C\setminus U$ is nowhere dense in $X$ (and in particular, $U$ is prevalent in
$C$).

###### Proof.

Since $C\setminus U$ is closed in $X$, $int\left({\overline{C\setminus
U}}\right)=int(C\setminus U).$ Now, $int(C\setminus U)\subseteq
int(C)\setminus\overline{U}=\varnothing$ by assumption, implying that
$int\left(\overline{C\setminus{U}}\right)=\varnothing,$ and therefore
$C\setminus U$ is nowhere dense in $X.$ ∎

As an already mentioned simple concrete example, if we take $C=\mathbb{R}$
with its standard topology, then the set of irrational numbers is prevalent in
the real line, whereas the set of rationals is not. More generally, it is
evident that when $C=X$ then every residual set is prevalent in $C$. This can
be extended to proper closed subsets of the topological space $X$ as follows.

######  Proposition 2.4.

Let $C\subseteq X$ be a closed subset. Then every residual set of $C$ (with
respect to the subspace topology on $C$) is prevalent in $C$.

###### Proof.

Let $A\subseteq C$ be residual in $C.$ Then $A$ contains the intersection
$\bigcap_{n}\mathcal{O}_{n}$ of a countable collection
$\\{\mathcal{O}_{n}\\}_{n\in\mathbb{N}}$ of open dense subsets of $C.$ We have
$C\setminus A\subseteq\bigcup_{n}C\setminus\mathcal{O}_{n},$ with each
$C\setminus\mathcal{O}_{n}$ being closed in $C,$ therefore closed in $X.$ Now,
$int_{X}(C\setminus\mathcal{O}_{n})$ is an open subset of $C$ contained in
$C\setminus\mathcal{O}_{n}$ that does not intersect $\mathcal{O}_{n}$. Since
$\mathcal{O}_{n}$ is dense in $C,$ we conclude that
$int_{X}(C\setminus\mathcal{O}_{n})$ must be empty, and $C\setminus A$ is thus
a meager set of $X.$ ∎

As the next example shows, the converse to proposition 2.4 is false even if
both $X$ and $C$ are Baire spaces.

######  Example 2.5.

Let $X={\bm{\mathbb{R}}}^{2}$ with the usual topology, and $C=\\{y\leq
0\\}\cup\\{x=0,y\geq 0\\}.$ Let $A=\\{y\leq 0\\}.$ Since $C\setminus
A=\\{x=0,y>0\\},$ and the latter line is nowhere dense in $\mathbb{R}^{2}$, we
conclude that $A$ is a closed prevalent subset in $C.$

However, $C\setminus A=\\{x=0,y>0\\}$ has nonempty interior with respect to
the subspace topology (indeed, this whole half line is itself open in $C$).
But then $C\setminus A$ cannot be contained in a meager set in $C$, since $C$
is a Baire space and any meager set therein must have empty interior in $C$.
We conclude that $A$ cannot be residual in $C$ with respect to the subspace
topology.

In spite of not being in general residual in the closed set $C\subset X$, a
prevalent subset $A\subset C$ is still “large” in $C$ relative to the broader
space $X$, and our position here is that this notion of “largeness” is still
relevant in the context explored in this paper.

###  2.2.  Lorentzian metrics and Whitney topologies

Let $Sym^{2}(M)$ be the vector bundle of $(0,2)$-type symmetric tensors on
$M$. We denote by $L\subset Sym^{2}(M)$ the smooth subbundle whose sections
are Lorentzian metric tensor fields (or Lorentzian metrics for short) on $M$,
which is open in $Sym^{2}(M)$ when the latter is endowed with its standard
manifold topology. In other words, a section of $L$ is a map $g$ which
associates with each $p\in M$ a symmetric nondegenerate bilinear form
$g_{p}:T_{p}M\times T_{p}M\rightarrow\mathbb{R}$ of index 1. Let
$\Gamma^{r}(L)$ be the set of $r$-differentiable sections with $0\leq
r\leq\infty$, that is, the set of Lorentzian metrics $g$ on $M$ whose
components $g_{ij}$ in local coordinates have continuous partial derivatives
up to order $r$ for $r\geq 1$, or that are simply continuous when $r=0$222We
avoid to refer to these as “(of class) $C^{r}$” to avoid notational confusion
with the Whitney topologies discussed ahead.. By a metric we always mean here
a Lorentzian metric unless explicitly stated otherwise, but we specify its
degree of differentiability as needed.

Let us briefly describe the relevant topologies we shall adopt on
$\Gamma^{r}(L)$. Again, since this is fairly standard material we shall only
do so in outline here. In general, given any other smooth manifold $N$, we
denote by $C^{r}(M,N)$ the set of all $r$-differentiable functions $f:M\to N.$
Over this set we shall always adopt the strong $C^{s}$ Whitney topologies,
$0\leq s\leq r$ (see, e.g., [26] or [37] for a comprehensive introduction to
Whitney topologies, and [36] for some of the more technical results used
here). A basis for the (strong) $C^{s}$ Whitney topology on $C^{r}(M,N)$
($s<\infty$) can be most readily defined via jet bundles as follows. Given the
bundle $J^{s}(M,N)$ of $s$-jets of $r$-differentiable maps, for each open set
$\mathcal{O}\subseteq J^{s}(M,N)$ put

$B_{s}(\mathcal{O})=\\{f\in C^{r}(M,N):j^{s}f(M)\subseteq\mathcal{O}\\},$

so the collection of all sets of this form is the desired basis. A basis for
the so-called Whitney $C^{\infty}$ topology on $C^{\infty}(M,N)$ is defined by
taking as a basis the collection of all $C^{s}$-open sets, for all $0\leq
s<\infty.$

In particular, consider a smooth fiber bundle $E$ over $M.$ Then the set
$\Gamma^{r}(E)$ of its $r$-differentiable sections will also said to be given
the $C^{s}$ topology when we take $\Gamma^{r}(E)\subseteq C^{r}(M,E)$, adopt
on the latter set the $C^{s}$ Whitney topology, and endow $\Gamma^{r}(E)$ with
the corresponding induced topology.

An alternative, perhaps more concrete way of defining Whitney topologies is by
using local charts. However, this discussion is somewhat lengthy and we refer
to the standard literature for the details (conf., e.g., [26, 35]) or [37, pp.
237-239]).

$\Gamma^{r}(L)$ will always be assumed to be endowed with the (induced)
$C^{s}$ topology as described. In most of our main results we shall take
$s=2\leq r\leq\infty$. In case we fix some $s\in\mathbb{Z}_{+}$ in connection
with the Whitney topology $C^{s}$ in a statement, that statement is meant to
hold separately on $\Gamma^{r}(L)$ for each $s\leq r\leq\infty$.

In the seminal work [34] Lerner obtained a number of key results using the
strong Whitney topologies on $\Gamma^{r}(L)$ \- some of which are briefly
reviewed in the next section - and these have ever since been widely accepted
as the natural topologies to adopt in the particular geometric setting of
interest here. In any case, we work exclusively with the strong Whitney
topologies on $\Gamma^{r}(L)$ in this paper, and from now on by _$C^{s}$
topology_ we always mean the strong $C^{s}$ Whitney topology.

Recall that such topologies are not even first countable if the domain
manifold is not compact, so we will rely on net convergence arguments when
needed. Also, recall that by a $C^{s}$ _stable property_ in $C^{r}(M,N),$
[resp. $\Gamma^{r}(E)$] we mean a property that is valid for all functions in
a $C^{s}$ open subset of $C^{r}(M,N)$ [resp. $\Gamma^{r}(E)$]. For each $t>s$
the $C^{t}$ topology is finer than the $C^{s}$ topology, so a stable propriety
in $C^{s}$ is also stable in $C^{t}.$

###  2.3. Previous stability results

To keep the presentation reasonably self-contained as well as to establish
further notation, we reproduce here, for later reference, some classic results
established in [34] that will be relevant for us later on.

Fix some $0\leq r$ which is either an integer or else $r=\infty$. Denote by
$\mathscr{ST}^{r}\subset\Gamma^{r}(L)$ the set of $r$-differentiable metrics
$g$ such that $(M,g)$ is time-orientable. The first relevant result is the
$C^{0}$-stability of time-orientability.

######  Proposition 2.6 ([34], prop. 4.7).

Let $X\in\mathfrak{X}(M)$ be an everywhere nonzero vector field. Then the set

$\mathscr{ST}^{r}(X):=\\{g\in\Gamma^{r}(L)\;:\;\text{$X$ is $g$-timelike}\\}$

is $C^{0}$-open in $\Gamma^{r}(L)$. In particular, $\mathscr{ST}^{r}$ is
$C^{0}$\- open (and therefore $C^{s}$-open for each $0\leq s\leq r$) in
$\Gamma^{r}(L)$. $\Box$

A convenient aspect of working with $\mathscr{ST}^{r}(X)$ is that we can
simultaneously choose the time-orientation in all of its elements so that $X$
is future-directed, and we shall implicitly assume this choice from now on.
Fix a codimension $k$ submanifold $\Sigma\subseteq M$, and denote by
$\mathscr{S}_{\Sigma}^{r}$ the set of metrics in $\Gamma^{r}(L)$ for which
$\Sigma$ is spacelike.

######  Proposition 2.7 ([34], prop. 4.2).

$\mathscr{S}_{\Sigma}^{r}$ is $C^{0}$-open in $\Gamma^{r}(L)$. $\Box$

Consider also the set $\mathcal{SC}^{r}\subset\mathscr{ST}^{r}$ of (time-
orientable) metrics that are _stably causal_ , meaning that
$g\in\mathcal{SC}^{r}$ if there is a $C^{0}$-neighborhood $\mathcal{U}\ni g$
in $\Gamma^{r}(L)$ such that every $g^{\prime}\in\mathcal{U}$ is a time-
orientable causal metric. The set $\mathcal{SC}^{r}$ is nonempty because $M$
is noncompact [34, p. 27, item (a)]. It is $C^{0}$-open by definition, and if
we denote as $\mathcal{CH}^{r}\subset\mathscr{ST}^{r}$ the set of
chronological, time-orientable metrics, then this set is $C^{0}$-closed in
$\mathscr{ST}^{r}$, with $\overline{\mathcal{SC}}=\mathcal{CH}$ (closure in
the $C^{0}$ topology)[34, p. 27, item (b)].

Thus, stably causal metrics are _$C^{0}$ -generic_ in the set of chronological
metrics, in the sense that they form an open dense set of the latter, or
equivalently, that chronological but not stably causal metrics form a nowhere
dense subset of the set of all chronological metrics.

We emphasize, however, that unlike stability, this does not imply
$C^{s}$-genericity for $s>0$. This is so, of course, because while
$\mathcal{SC}^{r}$ would still be $C^{s}$-open in $\mathcal{CH}^{r}$, it might
no longer be dense in this finer topology. A consequence for us here is that
in order to obtain our higher-order genericity results we shall need to work
with stably causal metrics even if the needed singularity theorems only
require chronology.

Assume now $r\geq 2$ and denote by $\mathcal{SE}^{r}$ the set of
$r$-differentiable metrics $g\in\Gamma^{r}(L)$ for which the respective Ricci
tensor, denoted by $Ric(g),$ satisfies

$Ric(g)(v,v)>0,\quad v\in TM\text{ $g$-causal}.$

This set is $C^{2}$-open in $\Gamma^{r}(L)$ [34, Prop. 4.3]. Importantly for
us here, for each $g\in\mathcal{SE}^{r}$ all $g$-causal vectors are generic in
$(M,g)$ in the sense of [9, ch. 2].

Similarly, consider $\mathcal{E}^{r}$ the set of $r$-differentiable metrics
$g\in\Gamma^{r}(L)$ satisfying

$Ric(g)(v,v)\geq 0,\quad v\in TM\text{ $g$-causal}.$

This set is $C^{2}$-closed, with
$\overline{\mathcal{SE}^{r}}\subseteq\mathcal{E}^{r}$, where now the overbar
indicates $C^{2}$-closure [34, p. 28, item 4.4]. The main result [34, Prop.
4.5] we will need in our later arguments is the following relation between
$\mathcal{E}^{r}$ and $\mathcal{SE}^{r}$:

######  Theorem 2.8.

In the $C^{2}$ strong Whitney topology,
$int(\mathcal{E}^{r})=\mathcal{SE}^{r}$, for all $2\leq r\leq\infty$. In
particular, $\mathcal{SE}^{r}$ is prevalent in $\mathcal{E}^{r}.$ $\Box$

(The first sentence in theorem 2.8 is actually the statement as proved by
Lerner, but the second sense follows immediately by lemma 2.3.)

######  Remark 2.9.

The set $\mathcal{E}^{r}$ consists precisely of those metrics $g$ satisfying
the so-called timelike convergence condition: $Ric(g)(v,v)\geq 0,\forall v\in
TM$ timelike. (The inequality also applies to null vectors via limits). This
is often referred to as the strong energy condition in the physics literature,
because it arises via the Einstein field equation in the context of general
relativity, by coupling the spacetime metric with physically relevant
classical matter fields, most of which satisfy it. Hence, it is a very common
assumption in singularity theorems. Theorem 2.8 has thus a simple but very
suggestive meaning: only those metrics in a “negligibly small” subset of the
metrics satisfying the timelike convergence condition do not admit of an
arbitrarily close approximation by a metric in $\mathcal{SE}^{r}$, which is
often causal geodesic incomplete.

###  2.4. Initial Data and MOTS

Since we shall focus on initial data sets that have a MOTS on the last part of
the paper (section 5), we now review these notions and some main results
related to it.

######  Definition 2.10 (Initial Data Set).

An _initial data set_ is a triple $(\mathcal{S}^{n},h,\mathcal{K})$ where
$(\mathcal{S}^{n},h)$ is a Riemannian manifold and $\mathcal{K}$ is a
symmetric $(0,2)$-tensor field on $\mathcal{S}$. Given an initial data
$(\mathcal{S},h,\mathcal{K})$, we define a function $\rho\in
C^{\infty}(\mathcal{S})$ and a one-form $J\in\Omega^{1}(\mathcal{S})$, called
respectively the _energy density_ and _energy-momentum current_ associated
with the data by

(1) $\displaystyle\rho$
$\displaystyle=\frac{1}{2}\left[\mathrm{Scal}_{h}-|\mathcal{K}|_{h}^{2}+(\operatorname{tr}_{h}\mathcal{K})^{2}\right],$
$\displaystyle J$
$\displaystyle=\operatorname{div}_{h}\mathcal{K}-d(\operatorname{tr}_{h}\mathcal{K}).$

The seminal work of [13] shows that if $(\mathcal{S},h,\mathcal{K})$ a vacuum
initial data set, i.e., one for which $\rho=0$ and $J=0$, then the manifold
$\mathcal{S}$ can be viewed as a suitably embedded Cauchy spacelike
huypersurfaces in a uniquely defined maximal globally hyperbolic Ricci-flat
spacetime. We summarize this result in the following theorem:

######  Theorem 2.11.

Let $(\mathcal{S}^{n},h)$ be a vacuum data set. Then there exists an
$(n+1)$-dimensional Ricci-flat spacetime $(M^{n+1},g)$ such that
$(\mathcal{S}^{n},h)$ isometrically embeds into $(M,g)$ as a Cauchy
hypersurface with second fundamental form $\mathcal{K}$. Furthermore, there is
a unique (up to isometry) _maximal_ such spacetime, in the sense that any
other spacetime satisfying these conditions can be isometrically embedded
therein. $\Box$

For our applications we are interested in MOTS and its null expansion scalar
from an initial data set point of view. Let
$\Sigma^{n-1}\subseteq\mathcal{S}^{n}$ be an embedded submanifold, and recall
that $\Sigma$ is _two-sided_ if it has a trivial normal bundle in
$\mathcal{S}.$

######  Definition 2.12 (Null expansion and MOTS - Initial data version).

Let $(\mathcal{S},h,\mathcal{K})$ be an initial data set and let
$\Sigma\subseteq\mathcal{S}$ be a two-sided embedded codimension one
submanifold, with one of the two normal vectors
$\bm{\upnu}\in\mathfrak{X}^{\perp}(\Sigma)$ fixed, and referred to as the
_outward-pointing unit normal_ vector field on $\Sigma.$ The outward _null
expansion_ $\theta_{+}$ [resp. inward _null expansion_ $\theta_{-}$] of
$\Sigma$ is defined as

(2) $\theta_{\pm}=\operatorname{tr}_{\Sigma}\mathcal{K}\pm H_{\bm{\upnu}},$

where $H_{\bm{\upnu}}$ is the mean curvature scalar of $\Sigma$ with respect
to the normal ${\bm{\upnu}}$ and $\mathcal{K}$ is the second fundamental form
of $\Sigma$ with respect to the normal ${\bm{\upnu}}$ and the (partial) trace
on $\Sigma$ is taken with respect to the induced metric. Then, $\Sigma$ is
said to be

  * •

outer trapped if $\theta_{+}<0,$

  * •

weakly outer trapped if $\theta_{+}\leq 0,$

  * •

marginally outer trapped (MOTS) if $\theta_{+}=0.$

As mentioned in the Introduction, MOTS are important in general relativity
because they provide a “quasi-local” analogue of the fully global notion of
(black hole) event horizon, and as such can be adapted to numerical studies,
for example [17]. In purely mathematical terms, however, MOTS are important
because they provide spacetime/initial data analogues of minimal surfaces (cf.
[31], section 7.5). Indeed, any Riemannian manifold $(\mathcal{S},h)$ can be
viewed as an initial data set with $\mathcal{K}\equiv 0$, and MOTS therein are
precisely the minimal surfaces. However, while minimal surfaces can be
described via a variational formulation, there is no known such formulation
for general MOTS.

####  2.4.1. MOTS stability operator

A powerful tool for studying minimal surfaces is the notion of stability,
which comes from the sign of the second variation of the volume measure; this
concept of stability can be generalized to the setting of MOTS through the
linearization of the null expansion $\theta_{+}.$ Such notion was introduced
by [4] (see also [21] for an in- depth review of the subject). For now, the
relevant definition is the following.

######  Definition 2.13 (MOTS stability operator - Initial data).

Let $\Sigma^{n-1}$ be a closed (that is, compact without boundary) MOTS within
an initial data set $(\mathcal{S}^{n},h,\mathcal{K})$. We define its
_stability operator_ $L:C^{\infty}(\Sigma)\rightarrow C^{\infty}(\Sigma)$ as
the elliptic differential operator given by

(3) $L(\psi)=-\Delta\psi+2\langle
X,\operatorname{grad}\psi\rangle+(Q+\operatorname{div}X-\|X\|^{2})\psi,\quad\forall\psi\in
C^{\infty}(\Sigma),$

where

(4)
$Q=\frac{1}{2}\mathrm{Scal}_{\Sigma}-[J(\bm{\upnu})+\rho]-\frac{1}{2}|\mathcal{K}_{\bm{\upnu}}+\mathcal{K}|^{2}.$

Here, all the geometric objects are defined on $\Sigma$, $\bm{\upnu}$ is the
outward pointing unit normal vector field on $\Sigma,$
$\mathcal{K}_{\bm{\upnu}}$ is the scalar second fundamental form of $\Sigma$
with respect to the induced metric from $(\mathcal{S},h)$ on the direction
$\bm{\upnu},$ $X$ is a vector field, the metric dual to the one-form
$\mathcal{K}(\bm{\upnu},\cdot)$ on $\Sigma$. Finally, $\rho$ and $J$ are
defined as in definition 2.10.

In the case of time-symmetric initial data ($\mathcal{K}=0$), the operator $L$
reduces to the self-adjoint, classic stability (or Jacobi) operator of minimal
surface theory, which basically gives the second variation of the volume.
Although the MOTS stability operator $L$ is not self-adjoint in general, it
still possesses certain key spectral properties summarized in the next
proposition.

######  Proposition 2.14 ([4], [19]).

Let $\Sigma$ be a closed MOTS within an initial data set
$(\mathcal{S}^{n},h,\mathcal{K})$. The following statements hold for the MOTS
stability operator $L$.

  1. (1)

There is a real eigenvalue $\lambda_{1}=\lambda_{1}(L)$, called the _principal
eigenvalue_ of $L$, such that for any other eigenvalue $\mu$,
$Re(\mu)\geq\lambda_{1}$. The associated eigenfunction $\phi\in
C^{\infty}(\Sigma)$, $L\phi=\lambda_{1}\phi$, is unique up to a multiplicative
constant, and can be chosen to be strictly positive.

  2. (2)

$\lambda_{1}\geq 0$ (resp., $\lambda_{1}>0$) if only and if there exist some
$\psi\in C^{\infty}(\Sigma),\psi>0$, such that $L(\psi)\geq 0$ (resp.,
$L(\psi)>0$). $\Box$

###  2.5. A singularity theorem in the presence of a MOTS

Let $\Sigma^{n-1}\subseteq\mathcal{S}^{n}$ be connected manifolds. We say that
$\Sigma$ _separates_ $\mathcal{S}$ if $\mathcal{S}\setminus\Sigma$ is not
connected. In this case, we write
$\mathcal{S}\setminus\Sigma=\mathcal{S}_{+}\sqcup\mathcal{S}_{-},$ where
$\mathcal{S}_{\pm}$ are open submanifolds of $\mathcal{S}.$ We now give a
singularity theorem for spacetimes containing MOTS that will be of crucial
importance for our discussion in the second and final part of this paper.

######  Theorem 2.15.

Let $(M^{n+1},g)$ be a globally hyperbolic spacetime. Assume the following
conditions.

  * 1)

The _lightlike convergence condition_ holds on $(M,g)$, i.e., $Ric(v,v)\geq
0,\forall v\in TM$ lightlike.

  * 2)

There exists a spacelike Cauchy hypersurface $\mathcal{S}^{n}\subseteq M$ and
a connected, compact MOTS $\Sigma^{n-1}\subseteq\mathcal{S}$ without boundary
that separates $\mathcal{S},$ with $\mathcal{S}\setminus\Sigma=S_{+}\sqcup
S_{-},$ where $S_{\pm}$ are open disjoint sets in $S$ and $\overline{S_{+}}$
is not compact.

  * 3)

The principal eigenvalue for the stability operator of $\Sigma$ is not zero.

Then, $(M,g)$ is null geodesically incomplete.

###### Proof.

(Sketch) Our hypotheses are a restriction of a more general study for
singularities in the presence of MOTS done by [41], so we just give here an
outline of the argument.

Following [41], propositions 2.1 and 2.4, one shows using $(2)$, and even when
$\Sigma$ is not a MOTS, the existence of a future-inextendible null
$\Sigma$-ray $\eta:[0,a)\to M$ with $\eta^{\prime}(0)$ parallel at the point
$\eta(0)$ to $\ell_{+}\in\mathfrak{X}^{\perp}(\Sigma)$, which is the unique
future-pointing normal lightlike vector field that projects on $\Sigma$ onto
the outward unit vector field defining $\theta_{+}$.

We assume in $(3)$ that the stability operator of $\Sigma$ has a non-zero
principal eigenvalue $\lambda_{1}\neq 0$, and since by proposition 2.14 its
eigenfunction $\phi$ can be chosen to be strictly positive. Consider the
normal variation $t\mapsto\Sigma_{t}$ of $\Sigma$ in $\mathcal{S}$ defined by
with vector field $V=\phi\upnu,$ where $\bm{\upnu}$ is the outward pointing
unit normal vector field of $\Sigma$ in $\mathcal{S}.$ One can show that

$\left.\frac{\partial\theta_{+}}{\partial
t}\right|_{t=0}=L(\phi)=\lambda_{1}\phi;$

therefore, under this variation we can deform $\Sigma$ to be outer-trapped
(i.e. $\theta_{+}<0$) by moving $\Sigma$ along the variation either outwards
or inwards according to the sign of $\lambda.$ Still denoting this displaced
manifold as $\Sigma,$ we then have an outward-pointing null $\Sigma$-ray
$\eta:[0,a)\to M$ normal to this outer-trapped $\Sigma$ as argued above, and
this null geodesic if future-incomplete because if it were complete, then by
using $(1)$ and prop. 10.43 in [38], there would exist a focal point of
$\Sigma$ along $\eta$ which is a contradiction since $\eta$ is a $\Sigma$-ray
and hence globally maximal. ∎

######  Remark 2.16.

The assumption that $\mathcal{S}$ is separated by $\Sigma$ is not essential in
the previous theorem, because even if that is not the case one can show (the
proof is analogous to that of prop. 14.48 of [38]) that there exists a smooth
covering manifold of $M$ for which there exists a spacelike Cauchy
hypersurface $\tilde{\mathcal{S}}$ covering $\mathcal{S}$ and containing a
isometric copy of $\Sigma$ separating $\tilde{\mathcal{S}}$. Now, insofar as
we are interested in geodesic incompleteness, one may as well work in covering
manifolds. However, the non-compactness of the outer region $\overline{S_{+}}$
must still be enforced in that covering. Otherwise, one may consider the
following counterexample. Let $(\mathbb{S}^{n},\omega_{n})$ be the round
$n$-sphere with its standard round metric $\omega_{n}$, and let
$(M:=\mathbb{R}\times\mathbb{S}^{n},g:=-dt^{2}+\omega_{n})$ be the
$(n+1)$-dimensional Einstein cylinder. One directly checks that $(M,g)$ is
geodesically complete and satisfies the lightlike convergence condition.
Moreover, $\mathcal{S}:=\\{0\\}\times\mathbb{S}^{n}$ is a totally geodesically
spacelike Cauchy hypersurface, so any minimal sphere
$\mathbb{S}^{n-1}\subset\mathbb{S}^{n}$ defines a two-sided separating MOTS
$\Sigma\subset\mathcal{S}$ in the obvious manner. However, for any such the
stability operator is shown by direct computation to reduce on $\Sigma$ to

(5) $L(\phi)=-\Delta\phi+c\cdot\phi$

with $c>0$ is some positive constant. Taking $\phi$ to be a strictly positive
principal eingenfunction, integration of (5) over $\Sigma$ allows one to
conclude that $\lambda_{1}=c$. Therefore, this example satisfies all
conditions in theorem 2.15 except the non-compactness of the Cauchy
hypersurface, and the conclusion fails.

##  3\. Weakly trapped submanifolds and prevalence of singularities I:
codimension two

We fix both both this section and the next one a smooth embedded codimension
$k\geq 2$ submanifold $\Sigma^{m}\subseteq M^{m+k}$. In some of our main
results we shall need that $\Sigma$ be compact and without boundary, and then
we will simply say that $\Sigma$ is _closed_ , and let the context distinguish
this usage from the tandard topological closure. In order to simplify the
notation, unless stated otherwise we work in $\Gamma^{2}(L)$ endowed with the
$C^{2}$ Whitney topology, with the understanding the everything remains valid
for $r$-differentiable metrics with $r\geq 2$, and thus omit any $r$
superscripts in what follows.

Let $g\in\Gamma^{2}(L)$, and let $\nabla=\nabla^{g}$ denote its Levi-Civita
connection. Recall that if $\Sigma$ is spacelike in $(M,g)$, then we can
define the second fundamental form tensor (or shape tensor for short) of
$\Sigma$ by

$II^{\Sigma}(g)(V,W):=(\nabla_{V}W)^{\perp},\quad\forall
V,W\in\mathfrak{X}(\Sigma).$

where $\perp$ denotes the normal part with respect to $g$. Given any local
$g$-orthonormal frame $\\{E_{1},\ldots,E_{m}\\}\subset\mathfrak{X}(\Sigma)$ on
$\Sigma$, the associated mean curvature vector field of $\Sigma$ is

$H^{\Sigma}(g):=tr_{\Sigma}\,II^{\Sigma}(g)=\sum_{i=1}^{m}II^{\Sigma}(g)(E_{i},E_{i}).$

When the underlying metric is unambiguously understood, we shall denote the
associated mean curvature vector field simply by $H^{\Sigma}.$ Straightforward
coordinate computations and net convergence arguments similar to the ones
given in the proof of [34, Prop. 4.7(b)] show that the mapping

(6) $g\in\Gamma^{r}(L)\mapsto H^{\Sigma}(g)\in\Gamma^{0}(TM|_{\Sigma})$

is continuous in the $C^{r}$ topology on $\Gamma^{r}(L)$ for all $1\leq
r\leq\infty$, where $TM|_{\Sigma}$ is the restriction of the tangent bundle
$TM$ to $\Sigma$.

Fix a nowhere-vanishing vector field $X\in\mathfrak{X}(M)$. We now proceed to
define the topological space of metrics we shall consider in this section.

Consider the set
$\mathscr{ST}(X)\cap\mathscr{S}_{\Sigma}\cap\mathcal{SC}\subset\Gamma^{r}(L)$
of $C^{2}$ stably causal, time-oriented metrics for which $X$ is future-
directed timelike, and for which $\Sigma$ is spacelike. Recall this set
$C^{2}$-open in $\Gamma(L)$ conf. props. 2.6 and 2.7). In that set, consider
the subset $\mathcal{A}$ of metrics for which $\Sigma$ is also a future-
trapped submanifold. Recall that $\Sigma$ is future-trapped if and only if
$H^{\Sigma}_{p}$ is past-directed timelike for each $p\in\Sigma$.

More precisely,

$\mathcal{A}=\\{g\in\mathscr{ST}(X)\cap\mathscr{S}_{\Sigma}\cap\mathcal{SC}:g(H^{\Sigma},H^{\Sigma})<0,g(H^{\Sigma},X)>0\\}.$

Since $H^{\Sigma}$ only depends on the metric coefficients and their first
derivatives in suitable coordinates (conf. e.g. [38, Ex. 1, p. 123]) the
definition of $\mathcal{A}$ readily implies (again by arguments entirely
analogous to those in the proof of [34, Prop. 4.7(b)]) that this set is
$C^{1}$-open in $\Gamma^{2}(L)$, and hence $C^{2}$-open therein as well.
Consider also the set

$\mathcal{FA}=\\{g\in\mathscr{ST}(X)\cap\mathscr{S}_{\Sigma}\cap\mathcal{SC}:g(H^{\Sigma},H^{\Sigma})\leq
0,g(H^{\Sigma},X)\geq 0\\}.$

Observe that for $g\in\mathcal{FA}$ and a point $p\in\Sigma,$ either
$H_{p}^{\Sigma}(g)$ is past-directed and causal, or else it is zero.
Therefore, $\Sigma$ is weakly future-trapped for a metric
$g\in\mathscr{ST}(X)\cap\mathscr{S}_{\Sigma}\cap\mathcal{SC}$ if and only if
$g\in\mathcal{FA}.$ In the context of a fixed $\Sigma$ as we have here, we
informally refer to the metrics in $\mathcal{A}$ themselves as “future-trapped
metrics,” and analogously we say the metrics in $\mathcal{FA}$ are “weakly
future-trapped”.

In particular, $\mathcal{FA}$ includes metrics $g$ for which $\Sigma$ is an
extremal submanifold, i.e., $H^{\Sigma}(g)\equiv 0$ identically, which occurs
for example if $\Sigma$ is totally geodesic with respect to $g$.

######  Remark 3.1.

For another key class of examples of elements of $\mathcal{FA}$, assume for
the moment that $\Sigma$ is closed and with codimension $k=2$, and that the
normal bundle of $\Sigma$ is trivial. (Of course, this can be ensured
independently of the choice of the ambient metric provided suitable
orientability assumptions on $\Sigma$ are made.) Then, given
$g\in\mathscr{ST}(X)\cap\mathscr{S}_{\Sigma}\cap\mathcal{SC}$ we may choose
two normal future-directed null vector fields
$\ell_{\pm}\in\mathfrak{X}^{\perp}(\Sigma)$ globally defined on $\Sigma$ and
spanning its normal bundle. The null expansion scalars $\theta_{\pm}\in
C^{1}(\Sigma)$ associated with this choice are defined by

$\theta_{\pm}:=-g(H^{\Sigma}(g),K_{\pm}).$

Observe that in this case $g\in\mathcal{A}$, i.e., it is future-trapped, if
and only if $\theta_{\pm}<0$. If by convention we say that $\ell_{+}$ is
outward-pointing, then $\Sigma$ is a marginally outer trapped surface
(MOTS333MOTS are being redefined here just because we had previously
considered them only on initial data sets. The relationship between the two
definitions is elementary but the details are not important for us here, so we
omit them.) if $\theta_{+}\equiv 0$ on $\Sigma$. Then, for each $p\in\Sigma$,
since since $H^{\Sigma}_{p},K_{+}(p)\in(T_{p}\Sigma)^{\perp}$ and the latter
vector space is a two-dimensional Lorentz space, they can be orthogonal if and
only if either $H^{\Sigma}_{p}$ is zero or null (parallel to $K_{+}$), and
hence $g\in\mathcal{FA}$. In other words, $C^{2}$ metrics for which $\Sigma$
is a MOTS are all in $\mathcal{FA}$, i.e., MOTS are weakly future-trapped.

Returning now to our main discussion, we evidently have
$\mathcal{A}\subseteq\mathcal{FA}.$ We will be able to prove much more
regarding $\mathcal{A}$ and $\mathcal{FA},$ the main point being that
$\mathcal{A}$ is prevalent in $\mathcal{FA}.$ Informally, weakly future-
trapped spacetime metrics can “almost always” be arbitrarily
$C^{2}$-approximated by future-trapped ones.

A key tool for these proofs will be conformal perturbations of the metric, so
let us briefly recall how the mean curvature vector of $\Sigma$ transforms
under a conformal change of the metric. Let $g\in\Gamma^{2}(L)$ and consider
the usual conformal transformation $\widehat{g}=e^{2f}g,$ where
$f:M\to{\bm{\mathbb{R}}}$ is any smooth function. The Levi-Civita connection
transforms as (conf. [32], p. 217)

$\widehat{\nabla}_{X}Y=\nabla_{X}Y+(Xf)Y+(Yf)X-g(X,Y)\,grad_{g}f,$

for $X,Y$ smooth vector fields over $M.$ Then, the shape tensor of $\Sigma$
associated with the metric $\widehat{g}$ is

$\widehat{II}(V,W)=(\widehat{\nabla}_{V}W)^{\perp}=II(V,W)-g(V,W)\,(grad_{g}f)^{\perp},$

for $V,W$ smooth vector fields tangent to $\Sigma.$ It now readily follows
that

(7) $\widehat{H}^{\Sigma}=e^{-2f}H^{\Sigma}-me^{-2f}(grad_{g}f)^{\perp},$

and the scalar product of $H^{\Sigma}$ transforms as

(8) $\displaystyle\widehat{g}(\widehat{H}^{\Sigma},\widehat{H}^{\Sigma})={}$
$\displaystyle
e^{-2f}g(H^{\Sigma},H^{\Sigma})-2e^{-2f}mg(H^{\Sigma},grad_{g}f)$
$\displaystyle+e^{-2f}m^{2}g((grad_{g}f)^{\perp},(grad_{g}f)^{\perp}).$

We are now ready to prove the announced result.

######  Theorem 3.2.

$\mathcal{FA}$ is $C^{2}$-closed in
$\mathscr{ST}(X)\cap\mathscr{S}_{\Sigma}\cap\mathcal{SC},$ with
$\mathcal{FA}=\overline{\mathcal{A}},$ and we also have
$int(\mathcal{FA})=\mathcal{A}.$ In particular, by lemma 2.3 ${\mathcal{A}}$
is prevalent in $\mathcal{FA}.$

###### Proof.

First, we show that $\mathcal{FA}$ is $C^{2}$-closed, thus concluding that
$\overline{\mathcal{A}}\subseteq\mathcal{FA}.$ Consider a net of metrics
$\\{g_{\lambda}\\}_{\lambda\in\Lambda}$ in $\mathcal{FA}$ converging to a
metric $g\in\mathscr{ST}(X)\cap\mathscr{S}_{\Sigma}\cap\mathcal{SC}$ in the
$C^{2}$ topology. Denoting by $H_{\lambda}^{\Sigma}$ the mean curvature vector
of $\Sigma$ associated with the metric $g_{\lambda},$ by the continuity in (6)
we have $H^{\Sigma}_{\lambda}\to H^{\Sigma}$ in the $C^{0}$ topology on
$\Gamma(TM|_{\Sigma})$, and since $C^{2}$ convergence implies $C^{0}$
convergence, we have a pointwise convergence
$g_{\lambda}(H^{\Sigma}_{\lambda},H^{\Sigma}_{\lambda})\to
g(H^{\Sigma},H^{\Sigma}),$ which implies $g(H^{\Sigma},H^{\Sigma})\leq 0.$
Similarly, we have a pointwise convergence
$g_{\lambda}(X,H^{\Sigma}_{\lambda})\to g(X,H^{\Sigma}),$ thus showing that
$g(X,H^{\Sigma})\geq 0,$ i.e., $g\in\mathcal{FA},$ which is therefore closed
as claimed.

In order to show that $\mathcal{FA}\subseteq\overline{\mathcal{A}},$ let
$g\in\mathcal{FA}$; if we can find a sequence $g_{n}\in\mathcal{A}$ of metrics
converging to $g$ in the $C^{2}$ topology to $g$ such that

(9) $g_{n}(H_{n},H_{n})<0\quad\text{and}\quad g_{n}(H_{n},X)>0,$

then we may conclude that $g\in\overline{\mathcal{A}}.$

We find such a sequence as follows. Since $(M,g)$ is stably causal, there
exists a $C^{2}$ $g$-temporal function $t:M\to{\bm{\mathbb{R}}}$ [11] with
$grad_{g}\,t$ future-directed timelike with respect to $g$. Since $\Sigma$ is
compact, we can find precompact open neighborhoods $V$ and $U$ of $\Sigma$ in
$M$ such that $\overline{V}\subseteq U.$ Consider a smooth bump function
$\phi$ that is constant to 1 in $\overline{V}$ and with support in $U.$ By
defining $f=\phi t,$ then $f$ is zero outside of a compact set, and has the
same $g$-gradient as $t$ for points of $\Sigma,$ so we define $f_{n}=f/n$ and
$g_{n}=e^{2f_{n}}g.$ Since $f_{n}\to 0$ in the $C^{2}$ topology (see e.g. [20,
p.43-44]), we see that $e^{2f_{n}}$ converges to 1 in the $C^{2}$ topology,
and then also $g_{n}\to g$ in the $C^{2}$ topology.

To see that $g_{n}$ indeed satisfies (9), we make a few elementary
computations. By (8),

$\displaystyle g_{n}(H_{n},H_{n})={}$
$\displaystyle{}e^{-2f_{n}}g(H^{\Sigma}(g),H^{\Sigma}(g))-\frac{2me^{-2f_{n}}}{n}g(H^{\Sigma}(g),grad_{g}\,t)$
$\displaystyle+\frac{m^{2}e^{-2f_{n}}}{n^{2}}g((grad_{g}\,t)^{\perp},(grad_{g}\,t)^{\perp}).$

Since $grad_{g}\,t$ future-directed and $H^{\Sigma}(g)$ is either zero or
past-directed (with respect to $g$), one has $g(H^{\Sigma}(g),grad_{g}\,t)\geq
0,$ and $g(H^{\Sigma}(g),H^{\Sigma}(g))\leq 0,$ but also the strict inequality
$g((grad_{g}\,t)^{\perp},(grad_{g}\,t)^{\perp})<0$, and thus
$g_{n}(H_{n},H_{n})<0.$ Similarly, by eq. 7,

$g_{n}(H_{n},X)=e^{-2f_{n}}g(H^{\Sigma}(g),X)-\frac{me^{-2f_{n}}}{n}g((grad_{g}\,t)^{\perp},X),$

and since $g(H^{\Sigma}(g),X)\geq 0$ and $g((grad_{g}\,t)^{\perp},X)<0$
because $grad_{g}\,t$ is also future-directed, we have obtained the desired
sequence.

$\mathcal{A}\subseteq\mathcal{FA}$ is $C^{2}$-open in $\Gamma^{2}(L)$, so we
immediately have $\mathcal{A}\subseteq int({\mathcal{FA}}).$ Consider now
$g\in\mathcal{FA}\setminus\mathcal{A}.$ We shall exhibit a sequence
$g_{n}\notin\mathcal{FA}$ with $g_{n}\to g$ in the $C^{2}$ topology, so that
every $C^{2}$-open neighborhood of $g$ will have a metric not in
$\mathcal{FA},$ meaning $g\notin int({\mathcal{FA}}),$ whence we conclude that
$\mathcal{FA}\subseteq\mathcal{A}$.

Since $g$ is not a metric in $\mathcal{A},$ there exists a point $p\in M$ for
which either $g(H^{\Sigma}_{p},H^{\Sigma}_{p})\geq 0,$ or else
$g(H^{\Sigma}_{p},X_{p})\leq 0.$ But $g\in\mathcal{FA},$ so either
$H_{p}^{\Sigma}(g)$ is lightlike, or otherwise $H^{\Sigma}_{p}(g)=0.$

For the lightlike case, choose a past-directed lightlike vector $v\in
T_{p}\Sigma^{\perp}$ not collinear to $H^{\Sigma}_{p}$ (which exists because
$k=codim\,\Sigma\geq 2$). We easily find some relatively compact neighborhood
$U$ of $p$ in $M$ and a smooth function $\phi\in C^{\infty}(U)$ such that
$grad_{g}\phi|_{p}=v.$ We then extend $\phi$ globally to a smooth real-valued
function $M$ with support in $U$ by a standard bump function argument, and
define the sequence $\phi_{n}=\phi/n$. This sequence converges to the zero
function in the $C^{2}$ topology, so that $e^{2\phi_{n}}$ converges to the
constant function 1 in the $C^{2}$ topology, and thus $g_{n}=e^{2\phi_{n}}g$
converges to $g$ in the $C^{2}$ topology on $\Gamma^{2}(L)$. Denoting by
$H^{n}$ the mean curvature vector of $\Sigma$ associated with the metric
$g_{n},$ we can now use (8). Since $H^{\Sigma}_{p}(g)$ is (past-directed)
$g$-lightlike, and $grad_{g}\phi|_{p}=v$ is also past-directed lightlike and
$g$-normal to $\Sigma,$ we obtain

$g_{n}(H^{n}_{p},H^{n}_{p})=-\frac{2m}{n}e^{2\phi_{n}(p)}g(H^{\Sigma}_{p}(g),v)>0.$

The case $H_{p}^{\Sigma}=0$ is very similar: we just choose $v\neq 0$
$g$-spacelike in $T_{p}\Sigma^{\perp}$, and the rest of the argument proceeds
analogously, but now applying (8) we get

$g_{n}(H^{n}_{p},H^{n}_{p})=\frac{m^{2}}{n}e^{-2\phi_{n}(p)}g(v,v)>0.$

In any case, we obtain $g_{n}\notin\mathcal{FA},$ as desired. ∎

######  Remark 3.3.

Some important remarks are appropriate here.

  1. (1)

By time-duality, we obviously have analogous results for past-trapped
submanifolds, with an analogous notion of weakly past-trapped submanifolds.

  2. (2)

Consider now metrics in $\Gamma^{\infty}(L)$, that is, smooth Lorentz metrics
on $M$. Then $\mathcal{A}$ is still $C^{\infty}$-open, $\mathcal{FA}$ is
$C^{\infty}$-closed, and $\overline{\mathcal{A}}=\mathcal{FA}$ ($C^{\infty}$
closure). To see this, note that the proof of theorem 3.2, because $\phi$ had
compact support actually the sequence $\phi_{n}$ of smooth functions converges
in any $C^{r}$ topology ($r$ finite) to $0,$ and thus $g_{n}$ converges to $g$
in any $C^{r}$ topology. Therefore $g_{n}\to g$ in the $C^{\infty}$ topology,
and theorem 3.2 can be restated for the $C^{\infty}$ topology. The exact same
kind of proof inspection in [34] gives an analogous “$C^{\infty}$ statement”
for theorem 2.8 provided one considers only smooth metrics.

In [34] the question was raised that perhaps
$\overline{\mathcal{SE}}=\mathcal{E}$, but the issue was left open in that
reference. To the best of our knowledge, this question does not seem to have
been addressed in the literature ever since. However, using later results on
$C^{1}$-stability of causal geodesic completeness in globally hyperbolic
spacetimes ([9, cor. 7.37]), we are now able to give a very simple
counterexample showing one may indeed have
$\mathcal{E}\setminus\overline{\mathcal{SE}}\neq\emptyset$.

######  Example 3.4.

In the usual Minkowski spacetime ${\bm{\mathbb{R}}}^{m+1}_{1}$ consider the
quotient manifold $M={\bm{\mathbb{R}}}^{m+1}/{\bm{\mathbb{Z}}}^{m}$ with
induced metric $g$, where the ${\bm{\mathbb{Z}}}^{m}$ isometric action defined
by $(t,x^{1},\dotsc,x^{m})\sim(t,x^{1}+n_{1},\dotsc,x^{m}+n_{m}).$ Thus,
$(M,g)$ is a flat ($g\in\mathcal{E}$), globally hyperbolic geodesically
complete spacetime. The spacelike Cauchy hypersurface $\Pi=\\{t=0\\}$ in
${\bm{\mathbb{R}}}^{m+1}$ is an $m$-torus and thus compact. Since global
hyperbolicity, causal geodesic completeness and the spacelike character of
$\Pi$ are all $C^{1}$-stable via the cited result, there is a open $C^{2}$
neighborhood $\mathcal{U}$ of $g$ such that all metrics in $\mathcal{U}$
satisfy these properties. Now, if we had $g\in\overline{\mathcal{SE}},$ then
there would exist some $h\in\mathcal{SE}\cap\mathcal{U}.$ However, in that
case the spacetime $(M,h)$ would satisfy all the conditions in the well-known
Hawking-Penrose singularity theorem (conf. [9, Thm. 12.47]), while being
causally geodesically complete, a contradiction. Therefore
$g\in\mathcal{E}\setminus\overline{\mathcal{SE}}.$

###  3.1. Singularities in codimension two

We can now combine these results with the “energy” conditions for the Ricci
tensor (theorem 2.8). As pointed out in remark 3.3(ii), the results in theorem
2.8 and theorem 3.2 can be stated for the $C^{\infty}$ topology. Because the
latter assumption is the most common one in geometry, the following discussion
will be carried out for smooth metrics, while easily restated for $C^{r}$
metrics with the $C^{r}$ topology with $r\geq 2$.

Recall that the set $\mathcal{A}$ is $C^{\infty}$-open in
$\Gamma^{\infty}(L)$, and so is $\mathcal{SE}$. Therefore,
$\mathcal{M}:=\mathcal{A}\cap\mathcal{SE}$ is also $C^{\infty}$-open in
$\Gamma^{\infty}(L)$, and contained in
$\mathscr{ST}(X)\cap\mathscr{S}_{\Sigma}\cap\mathcal{SC}.$

Let $\mathcal{FM}:=\mathcal{FA}\cap\mathcal{E}$ be the set of smooth spacetime
metrics which are both weakly trapped and satisfy the timelike convergence
conditions. This set is $C^{\infty}$-closed in
$\mathscr{ST}(X)\cap\mathscr{S}_{\Sigma}\cap\mathcal{SC}$; thus
$\overline{\mathcal{M}}\subseteq\mathcal{FM},$ and also

$int(\mathcal{FM})=int(\mathcal{FA})\cap int(\mathcal{E})=\mathcal{M},$

therefore $\mathcal{M}$ is prevalent in $\mathcal{FM}.$

######  Remark 3.5.

Just as in the discussion showing that
$\mathcal{E}\setminus\overline{\mathcal{SE}}$ is not necessarily empty given
in example example 3.4, the same spacetime given therein works to show that
$\mathcal{FM}\setminus\overline{\mathcal{M}}$ may also not be empty: just
consider in addition the spacelike surface originating from the quotient of
$S=\\{t=x^{1}=0\\}$. Therefore $g\in\mathcal{FM},$ but the possibility that
$g\in\overline{\mathcal{M}}$ leads to an analogous contradiction.

Our considerations in these examples can be summarized as follows.

######  Proposition 3.6.

Both $\mathcal{E}$ and $\mathcal{FM}$ are Baire spaces in the $C^{2}$
topology. Moreover

  1. (i)

$\mathcal{SE}$ is prevalent in $\mathcal{E},$ but in general it is not
residual in $\mathcal{E}$ (in the subspace topology);

  2. (ii)

Analogously, $\mathcal{M}$ is prevalent in $\mathcal{FM},$ but in general it
is not residual in $\mathcal{FM}.$

###### Proof.

The fact that $\mathcal{E}$ and $\mathcal{FM}$ are Baire spaces follows from
their functional forms as pointwise closed inequalities: these are preserved
by pointwise convergence of 2-jets of metrics, and standard properties of
weakly closed sets etablish they are indeed Baire in the strong topology
(conf. [26], Thms. 4.2-4.4).

For item (i) (the case (ii) is entirely analogous), we saw above that
$\mathcal{SE}$ is prevalent in $\mathcal{E}$ (theorem 2.8) but by example 3.4,
we can indeed have $\mathcal{E}\neq\overline{\mathcal{SE}}$ in general. But in
this case, since $\mathcal{E}$ is a Baire space, $\mathcal{SE}$ cannot be
residual, for otherwise it would also be dense; however,
$\overline{\mathcal{SE}}=\overline{\mathcal{SE}}\vphantom{E}^{\mathcal{E}}\neq\mathcal{E}.$

∎

Now, consider the following key observation. Assume that $(i)$
$g\in\mathcal{M}$ and $(ii)$ $\Sigma$ is a closed submanifold of codimension
$k=2$ (following the physics usage we simply say that $\Sigma$ is a _closed
surface_ in this case). Then, since in particular we will have that $(M,g)$ is
chronological and $Ric(g)(v,v)>0$ for any $g$-causal vector, the generic
condition for causal vectors will be trivially satisfied (conf. [9, Prop.
2.12, p. 39]). But then all the hypotheses in the Hawking-Penrose singularity
theorem [23] hold and therefore $(M,g)$ must possess at least one incomplete
causal geodesic444For this particular result, if $(M,g)$ is chronological it
is enough that $g\in\mathcal{FA}\cap\mathcal{SE}$ ([41])..

Now, since all the metrics we consider are restricted to the set
$\mathscr{ST}(X)\cap\mathscr{S}_{\Sigma}\cap\mathcal{SC}$ (so they are in
particular chronological), for every $g\in\mathcal{M},$ the spacetime $(M,g)$
is causally geodesically incomplete. Since this set is prevalent in
$\mathcal{FM}$, we conclude that “nearly all” metrics in $\mathcal{FM}$ are
arbitrarily $C^{\infty}$-close to causally incomplete metrics. More precisely,
we summarize the discussion of this subsection in the following theorem.

######  Theorem 3.7.

Let $(M,g)$ be a spacetime of dimension $\geq 3$ and smooth metric containing
a spacelike closed surface $\Sigma$. Assume that the following conditions
hold.

  1. (1)

$\Sigma$ is weakly future-trapped;

  2. (2)

$Ric(g)(v,v)\geq 0$ for all $g$-timelike $v\in TM$;

  3. (3)

$(M,g)$ is stably causal.

There exists a prevalent set $\chi$ of $\mathcal{FM}$ such that if $g\in\chi,$
then there exists a net $(g_{\lambda})_{\lambda\in\Lambda}$ of smooth metrics
on $M$ such that $g_{\lambda}\rightarrow g$ in the $C^{\infty}$ topology and
such that for each $\lambda\in\Lambda$, the spacetime $(M,g_{\lambda})$
satisfies the following conditions:

  1. (i)

$\Sigma$ is future-trapped;

  2. (ii)

$Ric(g_{\lambda})(v,v)>0$ for all $g_{\lambda}$-causal $v\in TM$;

  3. (iii)

$(M,g_{\lambda})$ is stably causal.

In particular, $(M,g_{\lambda})$ has at least one incomplete causal geodesic.
$\Box$

##  4\. Weakly trapped submanifolds and prevalence of singularities II: higher
codimensions

Our goal now is to establish a version of Theorem 3.7 that is valid for all
higher codimensions $k\geq 2$ of $\Sigma^{m}\subset M.$ Since singularity
theorems in the presence of higher co-dimensional submanifolds are more
delicate, we must introduce some further notation and results as to have a
better control of the Riemannian curvature as well.

For a Lorentz metric $g\in\Gamma^{2}(L)$ on $M$ denote by $Riem(g)$ the
covariant Riemann curvature $(0,4)$-tensor associated with $g.$ Applying a jet
bundle argument completely analogous to the one used to prove continuity of
the Ricci tensor $Ric$ as a function of $g$ (conf. [34, 23-24]), one readily
sees that the function

$Riem:g\in\Gamma^{r}(L)\mapsto Riem(g)\in\Gamma^{0}(T^{(0,4)}M)$

is continuous with respect to the $C^{r}$-topology on $\Gamma^{r}(L)$ for each
$2\leq r\leq\infty$, where now $T^{(0,4)}M$ denotes the smooth vector bundle
of $(0,4)$-tensors over $M$. Consider in $\Gamma^{2}(L)$ the subset

$\mathcal{P}=\\{g\in\Gamma^{2}(L):Riem(g)(w,v,v,w)>0\text{ $\forall v\in TM$
$g$-causal and all $w$ non-collinear to $v$}\\}.$

######  Proposition 4.1.

The set $\mathcal{P}$ is $C^{2}$-open in $\Gamma^{2}(L)$.

###### Proof.

Our arguments adapt some ideas in the proof of [34, Prop. 4.3], but with a
number of modifications of detail. Denote by $Curv(M)$ the vector subbundle of
$T^{(0,4)}M$ of all curvature-like tensors over $M$, i.e., if $F\in Curv(M)$,
then for all $p\in M$ and all vectors $w,x,y,z\in T_{p}M$ we have

  * CL1)

$F(w,z,x,y)=-F(w,z,y,x)$;

  * CL2)

$F(w,z,x,y)=-F(z,w,x,y)$;

  * CL3)

$F(w,z,x,y)+F(w,x,y,z)+F(w,y,z,x)=0$;

  * CL4)

$F(x,y,w,z)=F(w,z,x,y)$.

Fix $g\in\mathcal{P}$. Denote by $\widehat{TM}$ the tangent bundle minus all
zero vectors, and consider, for $C\in\Gamma^{0}(Curv(M)),$ the set

$V_{C}=\\{v\in\widehat{TM}:C(w,v,v,w)>0\text{ for all $w$ non-collinear to
$v$}\\}.$

We argue that $V_{C}$ is open in $\widehat{TM}$. Indeed, suppose not. Then
there is a sequence $v_{n}\in\widehat{TM}\setminus V_{C}$ converging to some
vector $v\in V_{C}$. For each $v_{n},$ not being an element of $V_{C}$ implies
that there exists $w_{n}$ not collinear to $v_{n}$ and based at the same
points respectively such that $C(w_{n},v_{n},v_{n},w_{n})\leq 0.$ Now, fix
some background Riemannian metric $h$ for $M$. By taking into account the
symmetries for $C$ we can assume without loss of generality that all vectors
are $h$-unitary, and each $w_{n}$ is $h$-normal to $v_{n}.$ By the compactness
of the $h$-sphere bundle over compact subsets of $M$ (since $v_{n}$ is
convergent), passing to a subsequence if necessary we can assume that $w_{n}$
converges to a nonzero $w$ with same base point in $M$ as $v,$ and such that
$h(v,w)=0,$ implying non-collinearity. Thus, on the one hand, $v\in V_{C},$
and hence $C(w,v,v,w)>0,$ and on the other hand the convergence implies
$C(w,v,v,w)\leq 0,$ a contradiction.

Denote by $\mathcal{C}_{g}\subseteq\widehat{TM}$ the set of all $g$-causal
vectors. Now, since $g\in\mathcal{P},$ we have $\mathcal{C}_{g}\subseteq
V_{Riem(g)}$. However, $\mathcal{C}_{g}$ is evidently closed in
$\widehat{TM}$, therefore, since the latter is a manifold and hence a normal
topological space, there exists an open set $U\subseteq\widehat{TM}$
satisfying $\mathcal{C}_{g}\subseteq U\subseteq\overline{U}\subseteq
V_{Riem(g)}$. In addition, we can assume without loss of generality that $U$
can be chosen to satisfy $v\in U\implies\alpha v\in U,$ for nonzero $\alpha$
since both sets $\mathcal{C}_{g}$ and $V_{Riem(g)}$ have this property.

Denoting as $\pi:Curv(M)\to M$ the bundle standard projection, we now prove
the following statement for this chosen $U$: for each point $p\in M,$ there
exists an open set $W\subseteq Curv(M)$ containing $Riem(g)(p)$ with the
property that, if for $C\in\Gamma^{0}(Curv)$ and all $q\in\pi(W)$ we have
$C(q)\in W,$ then $\overline{U}\cap T_{q}M\subseteq V_{C}\cap T_{q}M.$ Suppose
by way of contradiction that this is false. Thus we have a nested sequence of
sets $W_{n}\supseteq W_{n+1},$ all neighborhoods of $Riem(g)(p)$ in $Curv(M)$
with $\bigcap W_{n}=\\{Riem(g)(p)\\},$ a sequence $C_{n}\in\Gamma^{0}(Curv)$
and points $q_{n}\in\pi(W_{n})$ such that there exists some
$v_{n}\in(\overline{U}\setminus V_{C_{n}})\cap T_{q_{n}}M,$ which means that
there is some $w_{n}\in T_{q_{n}}M$ not collinear to $v_{n}$ satisfying
$C_{n}(q_{n})(w_{n},v_{n},v_{n},w_{n})\leq 0.$ Since $C_{n}(q_{n})\to
Riem(g)(p)$ in $Curv(M)$ and $q_{n}\to p$ in $M$, an argument using a
background Riemannian metric similar as the one in the first part of this
proof shows that $v_{n}\to v$ and $w_{n}\to w$ up to a subsequence, for some
nonzero $v\in\overline{U}\cap T_{p}M,$ and $w\in T_{p}M$ not collinear to $v$.
Thus, on the one hand, taking limits we get $Riem(g)(p)(w,v,v,w)\leq 0,$ and
on the other hand $v\in\overline{U}\subset V_{Riem(g)},$ so
$Riem(g)(p)(w,v,v,w)>0,$ which is the desired contradiction.

The rest of the proof is quite similar to the final part that of [34, Prop.
4.3, p. 28]. Namely, using the local triviality of the bundle $Curv(M)$ and
the neighborhoods $W$ above, we can obtain a $C^{0}$ neighborhood
$\mathcal{W}=\mathcal{W}(Riem(g))\subset\Gamma^{0}(Curv(M))$ of $Riem(g)$ with
the property that for any $C\in\mathcal{W}$ we have $\overline{U}\subset
V_{C}$. Now, by the continuity of the map $Riem$ we have that
$\mathcal{Z}:=Riem^{-1}(\mathcal{W})$ is a $C^{2}$-open set containing $g$ and
by construction $\mathcal{Z}\subset\mathcal{P}$ as desired. ∎

Now, let us define

$\mathcal{FP}=\\{g\in\Gamma^{2}(L):Riem(g)(w,v,v,w)\geq 0\text{ $\forall v\in
TM$ $g$-causal and all $w$ non-collinear to $v$}\\}.$

If we have a net $(g_{\lambda})_{\lambda\in\Lambda}$ in $\mathcal{FP}$
converging in $C^{2}$ to $g$, we have that $Riem(g_{\lambda})\rightarrow
Riem(g)$ in the $C^{0}$ topology on $\Gamma^{0}(T^{(0,4)}M)$, whence we
conclude that $g\in\mathcal{FP}$; that is, $\mathcal{FP}$ is $C^{2}$-closed.
We now have:

######  Theorem 4.2.

In the $C^{2}$ topology on $\Gamma^{2}(L)$ we have
$int(\mathcal{FP})=\mathcal{P},$ so $\mathcal{P}$ is prevalent in
$\mathcal{FP}.$

###### Proof.

Clearly, $\mathcal{P}\subset int(\mathcal{F})$. To prove the other inclusion,
we adapt the arguments in the proof of theorem 3.2 above. Specifically, we fix
$g\in\mathcal{FP}\setminus\mathcal{P},$ and build via conformal rescalings a
sequence $g_{n}\notin\mathcal{FP}$ of metrics converging in $C^{2}$ to $g$. We
will use the standard formulas for how global conformal rescalings of the
metric change the Riemann curvature tensor. (See e.g. [32, Eq. 7.44]. Although
presented for Riemannian metric, the formulas do remain valid in this
Lorentzian context.)

Since $g\notin\mathcal{P},$ we can pick a $g$-causal vector $v\in T_{p}M$ for
some $p\in M,$ and some $w\neq 0$ at the same base point not collinear to $v$
with $Riem(g)(w,v,v,w)=0.$ Assume first that $v$ is $g$-timelike, so $w$ can
be assumed to be spacelike and normal to $v,$ and both can also be assumed to
be $g$-unit vectors. Choose a $g$-normal neighborhood system
$(x^{1},\ldots,x^{dim\,M})$ centered at $p,$ which can be chosen such that $v$
has components $(1,0,\dotsc,0),$ and also such that first component of $w$ is
zero. Using such coordinates, consider the function
$\xi(x^{1},\dotsc,x^{dim\,M}):=e^{x^{1}}.$ We globally extend $\xi$ with usual
bump function arguments, (denoting he extension the same way) so that $\xi$
is, in coordinates, $e^{x^{1}}$ around $p,$ and zero outside some compact
subset of $M$ containing $p.$ Define, for each $n\in\mathbb{N}$,
$\xi_{n}=\xi/n,$ $g_{n}=e^{2\xi_{n}}g,$ so that again $g_{n}\to g$ in the
$C^{2}$ topology. With our choice of coordinates, a straightforward
computation gives

$Riem(g_{n})_{p}(w,v,v,w)=-\frac{e^{2/n}}{n}<0,$

so $g_{n}\notin\mathcal{FP}$ as desired. The case when $v$ is $g$-lightlike is
slightly more involved. In this case we pick normal coordinates
$(x^{1},\ldots,x^{dim\,M})$ centered at $p$ such that
$e_{i}:=\partial/\partial x^{i}(p)$ ($i=1,\ldots,dim\,M$) form a
$g$-orthonormal basis with $e_{1}$ timelike and $v=e_{1}+e_{2}$. Let
$\ell:=e_{1}-e_{2}$. $\ell$ is also null, and not collinear with $v$. Now,
write

$w=a\cdot v+b\cdot\ell+\sum_{i=3}^{dim\,M}w^{i}\cdot e_{i}.$

Since $w$ is not collinear with $v$, at least one of the $b,w^{i}$ are
nonzero, and since the part parallel to $v$ gives no contribution to
$Riem(g)(w,v,v,w)$ due to the curvature symmetries we can assume without loss
of generality that $a=0$. Observe that the vector
$\sum_{i=3}^{dim\,M}w^{i}\cdot e_{i}$ is normal to both $v$ and $\ell,$ and we
have the possibility that either $w$ is lightlike (when each $w^{i}\equiv 0$)
or spacelike. Consider first the spacelike case. Define
$\xi(x^{1},x^{2},\dotsc,x^{n})=(x^{1}+x^{2})^{2}$. Proceeding thenceforth just
as in the timelike case, we now obtain

$Riem(g_{n})(w,v,v,w)=-\frac{4}{n}g(w,w)<0.$

Suppose now $w$ is lightlike. Rescaling, we can assume $w=\ell$. Thus, define
$\xi(x^{1},x^{2},\dotsc,x^{n})=(x^{1})^{2}$. Ckmputing as before we obtain

$Riem(g_{n})(w,v,v,w)=-\frac{8}{n}<0.$

Therefore in each case we conclude that $g_{n}\notin\mathcal{FP}$ as desired,
thus completing the proof. ∎

######  Remark 4.3.

Again, by arguments analogous to those in remark 3.3(ii), theorem 4.2 remains
valid for the $C^{\infty}$ topology, provided we work with smooth metrics.

The following set is actually more relevant: let
$\mathcal{O}\subset\Gamma^{2}(L)$ be the the of $2$-differentiable metrics for
which the associated tidal force operators (see [9, pp. 35 and 38] for
definitions and notation) along causal directions are positive semidefinite,
that is, for $g\in\Gamma^{2}(L)$ and $v\in TM$ $g$-causal, the linear
operators

$\begin{cases}R^{g}_{v}:v^{\perp}\to v^{\perp},\quad v\text{ for
$g$-timelike,}\\\
\overline{R^{g}}_{v}:\overline{v^{\perp}}\to\overline{v^{\perp}},\quad v\text{
for $g$-lightlike,}\end{cases}$

are both positive semidefinite. It is straightforward to see that

(10) $\displaystyle\mathcal{P}$ $\displaystyle\subseteq\mathcal{SE};$ (11)
$\displaystyle\mathcal{P}$
$\displaystyle\subseteq\mathcal{O}\subseteq\mathcal{FP}\subset\mathcal{E}$

Therefore,
$\overline{\mathcal{P}}\subseteq\overline{\mathcal{O}}\subseteq\mathcal{FP},$
and since $int(\mathcal{FP})=\mathcal{P},$ then
$int(\mathcal{FP})\subseteq\mathcal{O},$ and the following corollary of
theorem 4.2 is immediate.

######  Corollary 4.4.

In the $C^{2}$ topology, $\mathcal{O}$ is prevalent in $\mathcal{FP}.$ $\Box$

######  Remark 4.5.

It is straightforward to check that $g\in\mathcal{FP}$ if and only if

$Riem(g)(w,v,v,w)\geq 0,\forall v\in TM\text{ $g$-timelike and all $w\in
v^{\perp_{g}}$}.$

###  4.1. Singularities in higher codimensions

We consider a similar analysis as the one carried out in section 3.1 when the
codimension $k$ of the submanifold $\Sigma\subset M$ is higher than 2.
However, we cannot use the Hawking-Penrose singularity theorem in this
context, but rather use some analogous singularity theorem in the presence of
future-trapped submanifolds of codimension higher than 2. Just such a result
has been obtained by [18]. Also, similar to what we did in section 3.1 (cf.
remark 4.3), the following discussion uses the $C^{\infty}$ topology on the
set of smooth metrics, and is easily adapted to $C^{2}$ metrics.

We need only to check that the positive-semidefiniteness of the tidal force
operators along causal directions encoded in the set $\mathcal{O}$ does imply
the required conditions. Specifically, let $g\in\mathcal{A}\cap\mathcal{O}$
and assume that $\Sigma$ is closed. Following [18], let
$\gamma:[0,b)\rightarrow M$ be some future-directed causal geodesic with
$\gamma(0)\in\Sigma$ and $\gamma^{\prime}(0)$ normal to $\Sigma$. Consider
$e_{1},\dotsc,e_{m}$ some coordinate basis of tangent vectors for
$T_{\gamma(0)}\Sigma,$ and let $E_{1},\dotsc,E_{m}$ denote their parallel-
transport vector fields along $\gamma$. Write $g_{ab}=g(E_{a},E_{b})$ (which
is constant along $\gamma$). Along this geodesic, since we have assumed that
$R_{\gamma^{\prime}}$ (or $\overline{R_{\gamma^{\prime}}}$ if $\gamma$ is
null) is positive semidefinite in the subspaces spanned by the vectors
$E_{a},$ the trace

$g^{ab}Riem(\gamma^{\prime},E_{a},E_{b},\gamma^{\prime})\geq 0.$

But this is precisely the condition 3.1 in [18]. Observe also that the
inclusions (11) imply that the positive semidefiniteness of the tidal forces
also entails the timelike convergence condition. Therefore, if in addition we
assume that $(i)$ $(M,g)$ is also chronological and $(ii)$ satisfies the
generic condition for causal vectors, then $(M,g)$ is causally geodesically
incomplete by [18, Thm. 3]. Therefore, the arguments in section 3.1 by taking
proposition 4.1, theorem 4.2, remark 4.5 and the inclusions (10), (11) into
account can be now adapted as follows.

######  Theorem 4.6.

Let $(M,g)$ be a spacetime of dimension $\geq 3$ and smooth metric containing
a spacelike closed submanifold $\Sigma$ of codimension $k\geq 2$. Assume that
the following conditions hold.

  1. (1)

$\Sigma$ is weakly future-trapped;

  2. (2)

$Riem(g)(w,v,v,w)\geq 0$ for all $g$-timelike $v\in TM$ and all $w$ at the
same base point $g$-orthogonal to $v$;

  3. (3)

$(M,g)$ is stably causal.

There exists a prevalent set $\chi$ in $\mathcal{FP}$ such that if $g\in\chi$,
then there exists a net $(g_{\lambda})_{\lambda\in\Lambda}$ of smooth metrics
on $M$ such that $g_{\lambda}\rightarrow g$ in the $C^{\infty}$ topology and
such that for each $\lambda\in\Lambda$, the spacetime $(M,g_{\lambda})$
satisfies the following conditions

  1. (i)

$\Sigma$ is future-trapped;

  2. (ii)

$Ric(g_{\lambda})(v,v)>0$ for all $g_{\lambda}$-causal $v\in TM$, and thus the
causal genericity condition holds in $(M,g_{\lambda})$;

  3. (iii)

the tidal force operators along $g_{\lambda}$-causal directions are all
positive semidefinite;

  4. (iv)

$(M,g_{\lambda})$ is stably causal.

In particular, each $(M,g_{\lambda})$ has at least one incomplete causal
geodesic. $\Box$

##  5\. Genericity of singularities in Cauchy developments of initial data
sets

We now turn to an investigation of genericity of singular spacetimes (under
suitable conditions) representing maximal Cauchy developments of initial data
sets containing a MOTS. We shall do so by considering a separable Hilbert
manifold structure on the set of triples $(h,\mathcal{K},\psi),$ where
$(h,\mathcal{K})$. is an initial data on a fixed connected $n$-manifold
$\mathcal{S}$ and $\psi:\Sigma\to\mathcal{S}$ a MOTS embedding, where again
$\Sigma$ is a fixed compact and connected manifold (without boundary). The
genericity statement we seek will follow from an abstract functional-analytic
approach together with the well-known Sard-Smale theorem.

###  5.1. An abstract Banach manifold genericity result

In order to obtain our main genericity results, we shall adopt a suitable
Banach manifold structure both on the set of initial data on a fixed connected
$n$-manifold $\mathcal{S}$, and on the set of embeddings of a fixed connected
compact $(n-1)$-manifold $\Sigma$ into $\mathcal{S}$. However, we shall first
establish in this section a key but purely abstract result on Banach
manifolds, via a relatively straightforward adaptation of the main analysis in
[12]. Our first lemma is a direct adaptation of lemma 2.2 in that reference.
For later reference, we also list other a few other results from [12] that
will also be needed.

Let $W$ be a closed vector subspace in a Hilbert space $H.$ We denote by
$P_{W}:H\to W$ the associated orthogonal projection onto $W.$

Recall that a bounded linear operator $T:E\to F$ between Banach spaces is
_Fredholm_ if $\ker T$ and $\operatorname{coker}{T}$ have finite
dimension555Some authors explicitly require the range of $T$ to be closed;
however, this condition follows easily from the finite dimensionality of the
cokernel. , where $\operatorname{coker}{T}=E/\operatorname{Im}T$ (the
dimension of the cokernel is also called the _codimension_ of
$\operatorname{Im}T$). The _index_ of a Fredholm operator $T$ is defined as
$\operatorname{ind}(T)=\dim\ker T-\dim\operatorname{coker}{T}.$ For later
reference, recall also that if $M,N$ are (perhaps infinite dimensional) Banach
manifolds and $a$ is an integer number, then a $C^{1}$ map $f:M\rightarrow N$
is said to a Fredholm map of index $a$ if its derivative
$df_{p}:T_{p}M\rightarrow T_{f(p)}N$ is a Fredholm linear operator of fixed
index $a$ for every $p\in M$.

######  Lemma 5.1.

Let $E,F$ be Banach spaces, and let $H$ be a Hilbert space with inner product
$\langle\,\,,\,\rangle.$ Consider $T:E\to H,$ $S:F\to H$ bounded linear
operators, with $\operatorname{Im}S$ closed and such that
$P_{\operatorname{Im}S^{\perp}}(\operatorname{Im}T)$ is closed in
$\operatorname{Im}S^{\perp}$ (this happens for example if $S$ is a Fredholm
operator). Then the sum operator $T\oplus S:E\times F\to H$ (given by $T\oplus
S(x,y)=T(x)+S(y)$) is surjective if and only if
$\operatorname{Im}T^{\perp}\cap\operatorname{Im}S^{\perp}=\\{0\\}.$

###### Proof.

Assuming the sum operator to be surjective, given
$z\in\operatorname{Im}T^{\perp}\cap\operatorname{Im}S^{\perp},$ then there is
a pair $(x,y)$ with $z=T(x)+S(y),$ but also by perpendicularity

$\|z\|^{2}=\langle z,z\rangle=\langle z,T(x)+S(y)\rangle=0.$

Assuming now
$\operatorname{Im}T^{\perp}\cap\operatorname{Im}S^{\perp}=\\{0\\},$ take some
$z\in H\setminus\\{0\\}.$ With $\operatorname{Im}S$ closed,
$H=\operatorname{Im}S\oplus\operatorname{Im}S^{\perp},$ so $z=S(x)+u,$
$u\in\operatorname{Im}S^{\perp}.$ We can also split
$\operatorname{Im}S^{\perp}$ (since it is a Hilbert space under the
restriction of the inner product and
$P_{\operatorname{Im}S^{\perp}}(\operatorname{Im}T)$ is closed in
$\operatorname{Im}S^{\perp}$) as follows:

$\operatorname{Im}S^{\perp}=[P_{\operatorname{Im}S^{\perp}}(\operatorname{Im}T)]\oplus[(P_{\operatorname{Im}S^{\perp}}(\operatorname{Im}T))^{\perp}\cap\operatorname{Im}S^{\perp}].$

Now, $u=P_{\operatorname{Im}S^{\perp}}(T(x))+v,$ where
$v\in(P_{\operatorname{Im}S^{\perp}}(\operatorname{Im}T))^{\perp}\cap\operatorname{Im}S^{\perp}.$
Also, $T(x)=S(\hat{y})+P_{\operatorname{Im}S^{\perp}}(T(x)),$ and we have
$z=T(x)+S(y)-S(\hat{y})+v.$ If $v=0$ we are done. Now,
$v\in\operatorname{Im}S^{\perp},$ and for any $T(e),$ $e\in E,$ again
$T(e)=S(f)+P_{\operatorname{Im}S^{\perp}}(T(e)),$ and we obtain,

$\langle v,T(e)\rangle=\langle v,S(f)\rangle+\langle
v,P_{\operatorname{Im}S^{\perp}}(T(e))\rangle=0,$

meaning $v\in\operatorname{Im}T^{\perp},$ implying $v=0.$ ∎

######  Remark 5.2.

For the Banach adjoint $T^{*}:H^{*}\to E^{*},$ it is a standard textbook
result ([6], p. 285) that we have $\operatorname{Im}T^{\perp}=\ker T^{*},$ so
the trivial intersection of images in lemma 5.1 can be substituted with
trivial intersection of the kernels of adjoint maps.

Recall also that in a Banach space $E,$ a closed subspace $V\subseteq E$ is
said to be _complemented_ if there is another another closed subspace
$W\subseteq E,$ called a _complement of $V$,_ satisfying $E=V\oplus W.$

######  Proposition 5.3 ([12], lemma 2.4).

Let $E,F,G$ be Banach spaces, and $T:E\to G,$ $S:F\to G$ bounded linear
operators, with $\ker S$ complemented in $F$ and $\operatorname{Im}S$ finite
codimensional in $G$ (this happens for example if $S$ is a Fredholm operator).
Then the kernel of $T\oplus S$ is complemented in $E\times F.$

######  Proposition 5.4 ([12], lemma 2.3).

Let $U,V$ be vector spaces, $L:U\to V$ a linear map, and $S\subseteq V$ a
finite codimensional subspace. Then $L^{-1}(S)$ has finite codimension in $U,$
with

(12)
$\operatorname{codim}_{U}{(L^{-1}(S))}=\operatorname{codim}_{V}{(S)}-\operatorname{codim}_{V}{(S+\operatorname{Im}L)}.$

For the abstract Banach manifold result we seek, we fix a separable Banach
manifold $X$, a Hilbert manifold $Y$, and a Hilbert space $V.$ We apply the
general functional-analytic results above to prove the following proposition,
an adaptation of [12], prop. 3.1 with more suitable hypotheses for our
concrete case of interest.

######  Proposition 5.5.

Consider $A\subseteq X\times Y$ an open set, and let $f:A\to V$ be a $C^{k}$
function, $k\geq 1.$ For every $(x_{0},y_{0})\in A$ such that
$f(x_{0},y_{0})=0,$ assume that the partial derivative $\dfrac{\partial
f}{\partial y}(x_{0},y_{0}):T_{y_{0}}Y\to V$ is a Fredholm operator. Then $0$
is a regular value of $f$ if and only if

(13) $\left(\operatorname{Im}\frac{\partial f}{\partial
x}(x_{0},y_{0})\right)^{\perp}\cap\left(\operatorname{Im}\frac{\partial
f}{\partial y}(x_{0},y_{0})\right)^{\perp}=\\{0\\}.$

###### Proof.

0 is a regular value of $f$ if and only if for all $(x_{0},y_{0})\in
f^{-1}(0),$ $df_{(x_{0},y_{0})}:T_{x_{0}}X\times T_{y_{0}}Y\to V$ is
surjective with complemented kernel. Now $df_{(x_{0},y_{0})}$ can be written
as the sum operator

$df_{(x_{0},y_{0})}=\frac{\partial f}{\partial
x}(x_{0},y_{0})\oplus\frac{\partial f}{\partial y}(x_{0},y_{0}).$

Applying lemma 5.1 since $\dfrac{\partial f}{\partial y}(x_{0},y_{0})$ is
Fredholm, it follows that if $df_{(x_{0},y_{0})}$ is surjective then (13)
holds. Conversely if (13) holds, by lemma 5.1 we obtain surjectivity, and also
by proposition 5.3 the kernel of $df_{(x_{0},y_{0})}$ is complemented. ∎

######  Remark 5.6.

For the sake of brevity, we shall name the intersection condition (13) as the
_T condition_. As stated in remark 5.2, we can substitute this condition for

$\ker{\left(\frac{\partial f^{*}}{\partial
x}(x_{0},y_{0})\right)}\cap\ker{\left(\frac{\partial f^{*}}{\partial
y}(x_{0},y_{0})\right)}=\\{0\\}$

at points $(x_{0},y_{0})\in f^{-1}(0).$ This might eventually be helpful in
concrete situations as it directly states the condition in the form of system
of equations for the adjoints.

Under the hypothesis of proposition 5.5, assuming (13) to be valid, then the
level set $M=f^{-1}\\{0\\}\subseteq A$ is an embedded submanifold or $X\times
Y,$ with tangent space at some $(x_{0},y_{0})\in M$ given by

$T_{(x_{0},y_{0})}M=\left\\{\,(v,w)\in T_{x_{0}}X\times
T_{y_{0}}Y:\frac{\partial f}{\partial x}(x_{0},y_{0})(v)+\frac{\partial
f}{\partial y}(x_{0},y_{0})(w)=0\,\right\\}.$

The main abstract genericity result we will need is the following (adapted
from [12, corollary 3.4]).

######  Theorem 5.7.

Under the hypothesis of proposition 5.5, assume also that $\dfrac{\partial
f}{\partial y}$ is a Fredholm operator of index zero for points in $M.$
Denoting by $\Pi:X\times Y\to X$ the canonical projection, then,

  1. (i)

$\Pi|_{M}:M\rightarrow X$ is Fredholm map of index zero.

  2. (ii)

The critical points of $\Pi|_{M}$ are the points $(x_{0},y_{0})\in M$ such
that $y_{0}$ is a critical point of the functional

$y\in A_{x_{0}}\mapsto f(x_{0},y)\in V,$

where $A_{x}=\\{y\in Y:(x,y)\in A\\}.$

  3. (iii)

Assuming $X,Y$ and $V$ to be separable, the set of points $x\in X$ such that
the functional $y\in A_{x}\mapsto f(x,y)\in V$ has no critical points is
generic in $\Pi(A).$

###### Proof.

For (i), given $(x_{0},y_{0})\in M,$ we have $\ker
d\Pi|_{T_{(}x_{0},y_{0})M}=T_{(x_{0},y_{0})}M\cap(\\{0\\}\times T_{y_{0}}Y),$
and this vector space is isomorphic to $\ker\left(\frac{\partial f}{\partial
y}(x_{0},y_{0})\right)$ so it is finite dimensional. Also, one readily sees
that

$\operatorname{Im}d\Pi|_{T_{(x_{0},y_{0})}M}=\left(\frac{\partial f}{\partial
x}(x_{0},y_{0})\right)^{-1}\left(\operatorname{Im}\frac{\partial f}{\partial
y}(x_{0},y_{0})\right),$

so from proposition 5.4, $\operatorname{Im}d\Pi|_{T_{(x_{0},y_{0})}M}$ is
finite codimensional, and since $df_{(x_{0},y_{0})}$ is surjective, by eq. 12

$\operatorname{codim}(\operatorname{Im}d\Pi|_{T_{(x_{0},y_{0})}M})=\operatorname{codim}\left(\frac{\partial
f}{\partial y}(x_{0},y_{0})\right).$

Since we are assuming $\frac{\partial f}{\partial y}(x_{0},y_{0})$ to be
Fredholm of index zero, it follows that $d\Pi|_{T_{(}x_{0},y_{0})M}$ is also
Fredholm of index zero.

For (ii), it is easy to see that $(x_{0},y_{0})\in M$ is a regular point of
$\Pi|_{M}$ if and only if

$\operatorname{Im}\left(\frac{\partial f}{\partial
x}(x_{0},y_{0})\right)\subseteq\operatorname{Im}\left(\frac{\partial
f}{\partial y}(x_{0},y_{0})\right).$

Taking orthogonal complements, this is equivalent to

$\operatorname{Im}\left(\frac{\partial f}{\partial
y}(x_{0},y_{0})\right)^{\perp}\subseteq\operatorname{Im}\left(\frac{\partial
f}{\partial x}(x_{0},y_{0})\right)^{\perp},$

and by condition (13), equivalent to $\operatorname{Im}\left(\frac{\partial
f}{\partial y}(x_{0},y_{0})\right)$ be a zero codimensional subspace, in turn
equivalent to the triviality of $\ker\left(\frac{\partial f}{\partial
y}(x_{0},y_{0})\right).$

To summarize, $(x_{0},y_{0})\in M$ is a regular point of $\Pi|_{M}$ if and
only if $\ker\left(\frac{\partial f}{\partial y}(x_{0},y_{0})\right)=\\{0\\},$
that is, if and only if $y_{0}$ is a regular point of the functional $y\in
A_{x_{0}}\mapsto f(x_{0},y)\in V.$ Item (iii) now follows, because the set of
points $x\in X$ such that $y\in A_{x}\mapsto f(x,y)\in V$ has no critical
points coincides with the set of regular values for $\Pi|_{M},$ and genericity
follows by the Sard-Smale theorem ([42]). ∎

######  Remark 5.8.

We make here an heuristic comment on the plausibility of the hypotheses in
theorem 5.7.

The condition $T$ together with the Fredholm condition is equivalent to $0$
being a regular value of a $C^{k}$ function $f:A\subseteq X\times Y\to V.$
Now, suppose $X,Y,V$ were all finite-dimensional. Then, the simplest version
of elementary transversality theorem (cf. [37], thm. 8.7.7) applied to the
submanifold $W=\\{0\\}\subset V$ would mean the the set of $C^{k}$ functions
$f:A\rightarrow V$ for which $0$ is a regular value is residual with respect
to a Whitney topology. This suggests that the condition $T$ might not be
restrictive.

While we expect to have an analogous situation on a suitable class of infinite
dimensional manifolds, tranversality theory is much harder in this case, and
so we were unable to find a generalizations of the Thom transversality theorem
applicable in our case. Therefore the $T$ condition will remain a technical
condition for us.

###  5.2. Manifold structures on initial data sets

The function we will concretely analyze in order to apply theorem 5.7 is the
expansion scalar $\theta_{+}$ defined on a suitable open set (when manifold
structures are defined) around a triple $(h_{0},\mathcal{K}_{0},\psi_{0}),$
with $(h_{0},\mathcal{K}_{0})$ an initial data set for which
$\psi_{0}:\Sigma\to\mathcal{S}$ is a MOTS embedding. With a domain well
defined, the expansion scalar is a function of the form

$(h,\mathcal{K},\psi)\mapsto\theta_{+}(h,\mathcal{K},\psi)=\operatorname{tr}_{h}\mathcal{K}\circ\psi+H^{\psi}_{h,\bm{\upnu}}.$

First,we describe possible infinite dimensional manifold structures on the set
of vacuum initial data, then discuss a Hilbert manifold structure for the set
of embeddings $\Sigma\to\mathcal{S}.$

These infinite dimensional structures are very technical in nature, and
describing them precisely is out of the scope of this work. Fortunately, we
shall need only very broad features of these here, so we briefly summarize the
ideas behind them. When these structures are established, we can define the
function $\theta_{+}$ more rigorously.

####  5.2.1. Manifold of initial data sets

Denote by $ID(\mathcal{S})$ the set of vacuum initial data $(h,\mathcal{K}).$
We set to establish a reasonable structure for the initial data set
$ID(\mathcal{S}).$

Since we want conditions on $\mathcal{S}$ as to apply theorem 2.15,
$\mathcal{S}$ cannot be compact. For a concrete manifold structure under such
conditions we assume $\mathcal{S}$ to be a non-compact, oriented
_asymptotically flat manifold_ without boundary. Asymptotically flatness here
means in particular that $\mathcal{S}$ has a compact set $K$ such that
$\mathcal{S}\setminus K$ is the union of a finite number of regions
$\mathcal{S}_{1},\dotsc,\mathcal{S}_{\ell}$ called the _ends_ of
$\mathcal{S},$ each diffeomorphic to ${\bm{\mathbb{R}}}^{n}\setminus B,$ where
$B$ denotes the closed ball of radius $1$ centered at zero666The precise
definition of asymptotic flatness also involves detailed falloff conditions
for the metric and its derivatives up to second order that need not concern us
here..

For the sake of simplicity we assume $\mathcal{S}$ has only one end. Under
these conditions we can find a few descriptions of a Banach manifold structure
on the set of initial data on such asymptotically flat manifolds. For our
practical purposes, we choose the smooth Hilbert manifold structure laid out
by [7]. This structure777Originally we wanted to model using a Banach manifold
structure following [16], since their work is not limited to the
asymptotically flat case. However their Banach structure is of the form
$H^{l}\cap C^{r,\alpha}$ (both Sobolev and Hölder) using a Sobolev-Hölder sum
norm, and it is unclear to us if the Banach manifold locally modeled on this
space is separable, as this condition is crucial to apply the Sard-Smale
theorem. is locally modeled on Sobolev spaces $H^{2}\times H^{1},$ and
separable.

####  5.2.2. Manifold of embeddings

Infinite dimensional structures for functions spaces $\Sigma\to\mathcal{S}$
with $\Sigma$ compact is a well-established idea in the literature. Here we
follow [2], section 3.

The idea of the process is to choose a suitable class of maps of regularity
$\mathscr{R}$ lower than $C^{\infty}$ regularity, which densely contains
$C^{\infty}$ and can be continuously embedded into those of $C^{2,\alpha}$
regularity ([2], section 2), then model the set
$\mathscr{R}(\Sigma,\mathcal{S})$ of maps $\Sigma\rightarrow\mathcal{S}$ of
regularity $\mathscr{R}$ as a Banach or Hilbert manifold locally modelled on
$\mathscr{R}(\Sigma,\mathbb{R}^{N})$, and finally argue that the set
$Emb(\Sigma,\mathcal{S})$ of embeddings is open in
$\mathscr{R}(\Sigma,\mathcal{S}).$

Here we choose Sobolev $H^{k+2}$ regularity, with $k,r\in{\bm{\mathbb{N}}}$
and $\alpha\in(0,1)$ (we take $r\geq 2$ since at least two derivatives are
computed in curvature tensors) satisfying $(k+2-r-\alpha)>1/2$ so that the
Sobolev space $H^{k+2}$ is continuously embedded into the Hölder space
$C^{r,\alpha}$ ([5], thms. 2.10 and 2.20). Thus, we do not lose much
regularity by locally modeling the embeddings over Sobolev spaces for large
$k$. More precisely, [2] show that $Emb(\Sigma,\mathcal{S})$ in this
regularity can be viewed as a Hilbert manifold locally modeled888Integrals on
$\Sigma$ are computed with respect to some background Riemannian metric $h$,
but the resulting Sobolev space does not depend on this choice. See [5] for
details. on the real Hilbert space $H^{k+2}(\Sigma).$

###  5.3. The null expansion scalar $\theta_{+}$

To make technical sense of the expansion scalar as a smooth function on the
manifold of initial data sets together with two-sided, framed embeddings
therein, one requires an outward normal direction fixed a priori. As a
consequence we can only work on a neighborhood of a fiducial
$(h_{0},\mathcal{K}_{0},\psi_{0}).$

To be more precise, consider $(h_{0},\mathcal{K}_{0})\in ID(\mathcal{S})$ an
asymptotically flat initial data set, and $\psi_{0}\in
Emb(\Sigma,\mathcal{S})$ a smooth embedding that is a MOTS with respect to
initial data set $(h_{0},\mathcal{K}_{0}),$ with
$\bm{\upnu}_{0}\in\mathfrak{X}^{\perp}(\psi_{0})$ the chosen outward pointing
unit normal vector field of the embedding $\psi_{0}$ under $h_{0}.$ Then

$\theta_{+}(h_{0},\mathcal{K}_{0},\psi_{0})=\operatorname{tr}_{h_{0}}\mathcal{K}_{0}\circ\psi_{0}+H^{\psi_{0}}_{h_{0},\bm{\upnu}_{0}}=0.$

The process is now to “propagate” the normal vector $\bm{\upnu}_{0}$ over a
small neighborhood around $\psi_{0}(\Sigma).$ Since $\psi_{0}(\Sigma)$ is an
smooth compact submanifold of $\mathcal{S},$ there is a smooth vector field
$\mathcal{V}\in\mathfrak{X}(\mathcal{S})$ such that
$\mathcal{V}\circ\psi_{0}=\bm{\upnu}_{0}.$ Restricting to a small enough
neighborhood around $\psi_{0}(\Sigma),$ we can assume
$\mathcal{V}_{p}\notin(d\psi_{0})_{p}(T_{p}\Sigma)$ at all points in this
neighborhood. We can then choose a open neighborhood around our embedding
$\psi_{0}$ in $Emb(\Sigma,\mathcal{S})$ such that all embeddings $\psi$ in
this neighborhood have $\mathcal{V}\circ\psi$ nowhere tangent. (These
embeddings are called _framed embeddings_ , and we see via this argument that
this is a stable property). With initial data $(h,\mathcal{K})$ close enough
to $(h_{0},\mathcal{K}_{0}),$ we consider the $h$-normal part of
$\mathcal{V}\circ\psi:$

$\bm{\upnu}(h,\psi)=\frac{(\mathcal{V}\circ\psi)^{\perp,h}}{\|(\mathcal{V}\circ\psi)^{\perp,h}\|_{h}}.$

Denote by $\mathcal{A}_{0}$ the open set in $ID(\mathcal{S})\times
Emb(\Sigma,\mathcal{S})$ around $(h_{0},\mathcal{K}_{0},\psi_{0})$ as describe
above. Then we can define

(14)
$\displaystyle\theta_{+}:(h,\mathcal{K},\psi)\in\mathcal{A}_{0}\mapsto\operatorname{tr}_{h}\mathcal{K}\circ\psi+H^{\psi}_{h,\bm{\upnu}(h,\psi)},$

###  5.4. Linearization of $\theta_{+}$

With $\theta_{+}$ suitably defined, the formal partial linearization of
$\theta_{+}$ with respect to the embedding is the MOTS stability operator
(definition 2.13): if $\psi:\Sigma\to\mathcal{S}$ is a MOTS embedding for the
initial data $(h,\mathcal{K}),$ we have

$\left.\frac{\partial\theta_{+}}{\partial\psi}\right|_{(h,\mathcal{K},\psi)}(f)=L(f)=-\Delta_{h}f+2\langle
X,\operatorname{grad}_{h}f\rangle_{h}+(Q+\operatorname{div}_{h}X-\|X\|_{h}^{2})f.$

, where

$Q=\frac{1}{2}\mathrm{Scal}_{\Sigma}-[J(\bm{\upnu})+\rho]-\frac{1}{2}|\mathcal{K}_{\bm{\upnu}}+\mathcal{K}|^{2},$

and $X$ the vector field $h$-dual to the one form
$\mathcal{K}(\bm{\upnu},\cdot{})$

An issue that needs addressing in other to apply the analytical machinery
developed is the Fredholm property for the MOTS stability operator. However,
since $L$ is a second order linear eliptic operator with a specific form, we
have the following general result.

######  Proposition 5.9 ([31], cor. A.9).

Let $(\Sigma,h)$ be a compact Riemannian manifold without boundary, and let
$L$ be a second order linear elliptic operator on $(\Sigma,h)$ of the form

$L(f)=-\Delta_{h}f+\langle V,grad_{h}f\rangle_{h}+qf,$

with $V\in\mathfrak{X}(M),q\in C^{\infty}(\Sigma).$ Then

$\displaystyle L:W^{k+2,p}(\Sigma)\to W^{k,p}(\Sigma),$ $\displaystyle
L:C^{k+2,\alpha}(\Sigma)\to C^{k,\alpha}(\Sigma)$

are Fredholm operators of index zero for each $k\in{\bm{\mathbb{N}}}.$ $\Box$

###  5.5. Geometric interpretation

With the abstract analytic machinery in place, a suitable infinite dimensional
manifold structure and $\theta_{+}$ all well established, we now give our main
results.

To simplify our language, let us introduce some terminology: say that a triple
$(h,\mathcal{K},\psi)$ is a _MOTS triple_ if
$\theta_{+}(h,\mathcal{K},\psi)=0$ (i.e. the embedding $\psi$ is a MOTS in the
initial data $(h,\mathcal{K})$), and call a MOTS triple $(h,\mathcal{K},\psi)$
_non-degenerate_ if the associated MOTS stability operator has a non-zero
principal eigenvalue (in such case the stability operator is said to be _non-
degenerate_).

######  Theorem 5.10.

Let $(h_{0},\mathcal{K}_{0})$ be initial data set,
$\psi_{0}:\Sigma\to\mathcal{S}$ an embedding that is a MOTS for the initial
data $(h_{0},\mathcal{K}_{0})$ on $\mathcal{S},$ and consider the neighborhood
$\mathcal{A}_{0}$ where $\theta_{+}$ is defined. Assume $\theta_{+}$ satisfies
the T condition (remark 5.6). Then there exists an open neighborhood
$\mathcal{O}\subseteq ID({\mathcal{S}})$ around $(h_{0},\mathcal{K}_{0})$
where the set

(15) $\mathcal{G}=\\{(h,\mathcal{K})\in\mathcal{O}:\textnormal{every MOTS in
$(h,\mathcal{K})$ are non-degenerate}\\}$

is a residual set in $\mathcal{O}.$

###### Proof.

Since the MOTS stability operator is a Fredholm operator of index 0, we are in
conditions to apply theorem 5.7: here, $M=\theta_{+}^{-1}(0)$ is the
submanifold in $\mathcal{A}_{0}$ of MOTS triples. By item (iii), it is a
generic property in $\Pi(\mathcal{A}_{0})=\mathcal{O}$ the following: initial
data $(h,\mathcal{K})$ such that the functional

$\psi\in\mathcal{(}A_{0})_{(h,\mathcal{K})}\mapsto\theta_{+}(h,\mathcal{K},\psi)$

has no critical points. Having no critical points, then any MOTS embedding
$\psi$ for such $(h,\mathcal{K})$ is such that the associated stabability
operator is non-degenerate. ∎

######  Remark 5.11.

What theorem 5.10 says about the residual set
$\mathcal{G}\subseteq\mathcal{O}$ is that if there is a MOTS in the initial
data $(h,\mathcal{K})\in\mathcal{G}_{\mathcal{O}},$ then this MOTS is non-
degenerate. However our method cannot guarantee that such a MOTS exists for
initial data in $\mathcal{G}.$ In order to obtain this last condition, we have
to make extra assumptions about the fiducial MOTS triple
$(h_{0},\mathcal{K}_{0},\psi_{0}).$ We cite two distinct situations where this
is possible:

  1. (1)

Assume that $(h_{0},\mathcal{K}_{0},\psi_{0})$ is itself non-degenerate. Then
by condition $T$ the partial derivative with respect to the embeddings
variable at $(h_{0},\mathcal{K}_{0},\psi_{0})$ is an isomorphism (this can be
more clearly seen from the abstract case in theorem 5.7). Then by a
straightforward application of the implicit function theorem in Banach spaces
(cf. [1], thm. 2.5.7) We can reduce $\mathcal{A}_{0}$ to a open set of the
form $\mathcal{U}_{0}\times\mathcal{V}_{0},$
$(h_{0},\mathcal{K}_{0})\in\mathcal{U}_{0},$ $\psi_{0}\in\mathcal{V}_{0},$
where now all initial data in $\mathcal{O}=U_{0}$ have a non-degenerate MOTS,
so when the fiducial MOTS triple is non-degenerate we have obtained that
existence of non-degenerate MOTS for initial data in $\mathcal{U}_{0}$ is
stable. (Actually, this argument bypasses the generic step (iii) in theorem
5.7 thanks to condition $T$ and the implicit function theorem.)

  2. (2)

For this second case, we restrict $(h_{0},\mathcal{K}_{0})$ and the dimension
and topology of $\mathcal{S}$ to the following theorem due to [3]:

######  Theorem 5.12 ([3], thm. 3.3).

Let $(h,\mathcal{K})$ be a initial data set for $\mathcal{S},$ where
$3\leq\dim\mathcal{S}\leq 7.$ Assume that there is a connected bounded open
set $\Omega\subseteq\mathcal{S}$ with smooth embedded boundary
$\partial\Omega.$ Assume this boundary consists of two non-empty closed
hypersurfaces $\partial_{+}\Omega$ and $\partial_{-}\Omega,$ possibly
consisting of several components, so that

(16)
$H_{\partial_{+}\Omega}-\operatorname{tr}_{\partial_{+}\Omega}\mathcal{K}>0\quad
and\quad
H_{\partial_{-}\Omega}+\operatorname{tr}_{\partial_{-}\Omega}\mathcal{K}>0,$

where the mean curvature scalar is computed as the tangential divergence of
the unit normal vector field that is pointing out of $\Omega$. Then there
exists a smooth closed embedded MOTS999Other properties of $\Sigma$ are
obtained in [3], but we need not concerned about them here.
$\Sigma\subseteq\Omega$ for the initial data $(h,\mathcal{K})$ that is
homologous to $\partial_{+}\Omega.$ $\Box$

By restricting the topology and dimension of $\mathcal{S}$ as in theorem 5.12,
consider an initial data $(h_{0},\mathcal{K}_{0})$ for $\mathcal{S},$ and then
a MOTS $\psi_{0}$ for this initial data is obtained by the latter theorem.
This MOTS triple $(h_{0},\mathcal{K}_{0},\psi_{0})$ is now our fiducial MOTS
triple, that may or may not be non-degenerated. The point is that conditions
(16) are open, so there is an open set $\mathcal{U}_{0}$ around
$(h_{0},\mathcal{K}_{0})$ where all initial data in $\mathcal{U}_{0}$ have a
MOTS by theorem 5.12. Applying theorem 5.10 under these conditions, the
residual set $\mathcal{G}$ will have the property that all initial data
$(h,\mathcal{K})$ in $\mathcal{G}$ have a non-degenerate MOTS.

Now let us organize theorem 5.10 together with conditions laid out in remark
5.11 as to obtain a generic condition for incompleteness.

Consider a MOTS triple $(h_{0},\mathcal{K}_{0},\psi_{0})$ and its open
neighborhood $\mathcal{A}_{0}$ where $\theta_{+}$ is defined, assume now that
$\psi_{0}(\Sigma)$ separates $\mathcal{S}$ as defined section 2.5. Reducing
$\mathcal{A}_{0}$ to a smaller open set we can assume that all embeddings
separate $\mathcal{S}.$ By theorem 5.10 there is an open set $\mathcal{O}$
around $(h_{0},\mathcal{K}_{0})$ where the property (15) is generic in
$\mathcal{O}.$ If $(h,\mathcal{K})\in\mathcal{G}$ has a MOTS, this MOTS is
non-degenerate, therefore we are in conditions to apply theorem 2.15, and MOTS
for $(h,\mathcal{K})$ can be deformed to be outer-trapped, then the maximal
Cauchy development of $(h,\mathcal{K})$ (cf. theorem 2.11) is null incomplete.

Following remark 5.11, by suitably reducing $\mathcal{A}_{0},$ under condition
(1) null imcompleteness is now stable around $(h_{0},\mathcal{K}_{0}),$ and
under condition (2) it is generic in an open subset around
$(h_{0},\mathcal{K}_{0}).$ We summarize this in the following theorem:

######  Theorem 5.13.

Let $(h_{0},\mathcal{K}_{0})$ be initial data
set,$\psi_{0}:\Sigma\to\mathcal{S}$ an embedding that is a MOTS for the
initial data $(h_{0},\mathcal{K}_{0}),$ where $\psi_{0}(\Sigma)$ separates
$\mathcal{S}.$ Consider the open set $\mathcal{A}_{0}$ around
$(h_{0},\mathcal{K}_{0},\psi_{0})$ where $\theta_{+}$ is defined and all
embeddings separates $\mathcal{S},$ and also assume that $\theta_{+}$
satisfies the $T$ condition. Under these assumptions,

  1. (1)

if $(h_{0},\mathcal{K}_{0},\psi_{0})$ is itself non-degenerate, then there is
an open set $\mathcal{O}$ around $(h_{0},\mathcal{K}_{0})$ where all initial
data $(h,\mathcal{K})\in\mathcal{O}$ are such that their maximal Cauchy
development of $(h,\mathcal{K})$ is null incomplete (null incompleteness is
_stable_);

  2. (2)

if $(h_{0},\mathcal{K}_{0},\psi_{0})$ is a MOTS triple originating from
theorem 5.12 (where the topology and dimensions of $\mathcal{S}$ have to be
further restricted) then there is an open set $\mathcal{O}$ around
$(h_{0},\mathcal{K}_{0})$ and a residual set $\mathcal{G}\subseteq O$ where
all initial data $(h,\mathcal{K})\in\mathcal{G}$ are such that their maximal
Cauchy development of $(h,\mathcal{K})$ is null incomplete (null
incompleteness is _generic_). $\Box$

## Acknowledgements

IPCS is partially supported by the project PID2020-118452GBI00 of the Spanish
government. He warmly thanks Paolo Piccione for discussions and early insights
on the second part of this paper. VLE’s work has been partially supported by
the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil
(CAPES) – Finance Code 001.

Conflict of interest statement: On behalf of all authors, the corresponding
author states that there is no conflict of interest.

Data availability statement: Data sharing not applicable to this article as no
datasets were generated or analysed during the current study.

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