# Nonnegative scalar curvature on manifolds with at least two ends

Simone Cecchini Funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through the Priority Programme “Geometry at Infinity”
(SPP 2026, CE 393/1-1). University of Göttingen
Mathematisches Institut
Germany
<EMAIL_ADDRESS>
simonececchini.org Daniel Räde Funded by a Doctoral Scholarship of the
’Studienstiftung des deutschen Volkes’. Augsburg University
Institut für Mathematik
Germany
<EMAIL_ADDRESS>Rudolf Zeidler Funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID
427320536 – SFB 1442, as well as under Germany’s Excellence Strategy EXC 2044
390685587, Mathematics Münster: Dynamics–Geometry–Structure, and through the
Priority Programme “Geometry at Infinity” (SPP 2026, ZE 1123/2-2). University
of Münster
Mathematisches Institut
Germany
<EMAIL_ADDRESS>
www.rzeidler.eu

###### Abstract

Let $M$ be an orientable connected $n$-dimensional manifold with
$n\in\\{6,7\\}$ and let $Y\subset M$ be a two-sided closed connected
incompressible hypersurface which does not admit a metric of positive scalar
curvature (abbreviated by psc). Moreover, suppose that the universal covers of
$M$ and $Y$ are either both spin or both non-spin. Using Gromov’s
$\mu$-bubbles, we show that $M$ does not admit a complete metric of psc. We
provide an example showing that the spin/non-spin hypothesis cannot be dropped
from the statement of this result. This answers, up to dimension $7$, a
question by Gromov for a large class of cases. Furthermore, we prove a related
result for submanifolds of codimension two. We deduce as special cases that,
if $Y$ does not admit a metric of psc and $\dim(Y)\neq 4$, then
$M:=Y\times\mathbb{R}$ does not carry a complete metric of psc and
$N:=Y\times\mathbb{R}^{2}$ does not carry a complete metric of uniformly psc
provided that $\dim(M)\leq 7$ and $\dim(N)\leq 7$, respectively. This solves,
up to dimension $7$, a conjecture due to Rosenberg and Stolz in the case of
orientable manifolds.

##  1 Introduction

In the 1994 survey article [RoS94, Section 7], Rosenberg and Stolz proposed
the following conjectures concerning the (non-)existence of _positive scalar
curvature_ (abbreviated by _psc_) metrics on certain non-compact manifolds:

######  Conjecture 1.1.

Let $Y$ be a closed manifold of dimension $(n-1)\neq 4$ which does not admit a
metric of positive scalar curvature. Then $Y\times\mathbb{R}$ does not admit a
complete metric of positive scalar curvature.

######  Conjecture 1.2.

Let $Y$ be a closed manifold of dimension $(n-2)\neq 4$ which does not admit a
metric of positive scalar curvature. Then $Y\times\mathbb{R}^{2}$ does not
admit a complete metric with uniformly positive scalar curvature.

Recently Gromov phrased a related conjecture [Gro18, 11.12, Conjecture C]
which is a cornerstone of his program regarding the study of metric
inequalities with scalar curvature:

######  Conjecture 1.3.

Let $Y$ be a closed manifold of dimension $(n-1)\neq 4$ which does not admit a
metric of positive scalar curvature and $X=Y\times[-1,1]$. If $g$ is a
Riemannian metric on $X$ with $\operatorname{scal}(X,g)\geq n(n-1)$, then

$\operatorname{width}(X,g):=\operatorname{dist}_{g}(Y\times\\{-1\\},Y\times\\{1\\})\leq\frac{2\pi}{n}.$

######  Remark 1.4.

The condition $\dim(Y)\neq 4$ (which we have added in our formulation of
Conjectures 1.1 and 1.2) is necessary since in this case there are well known
counterexamples using Seiberg–Witten obstructions to positive scalar
curvature. It is possible to show that there exists a closed simply connected
4-manifold $Y$ which does not admit a metric of positive scalar curvature
while $Y\times\mathrm{S}^{1}$ does (see [29, Counterexample 4.16]). But then
$Y\times\mathbb{R}$ (and consequently $Y\times\mathbb{R}^{2}$) admits a
complete metric with uniformly positive scalar curvature which would violate
Conjectures 1.1, 1.2 and 1.3.

As is common in the study of scalar curvature, there are two broad families of
methods to approach these conjectures: One is based on the spinor Dirac
operator on spin manifolds originating in the work of Lichnerowicz [21].
Indeed, using different variants of index theory on non-compact manifolds, it
is now well established that Conjectures 1.1 and 1.2 can be proved whenever
$Y$ admits an index-theoretic obstruction to positive scalar curvature such as
the Rosenberg index [27], see [1, 17, 35]. The other family of methods is
based on geometric measure theory and originates in the minimal hypersurface
method of [31] [31]. The first established cases of Conjecture 1.3 by Gromov
[14, Section 2] used the classical minimal hypersurface method. Subsequently
also a Dirac operator approach to Conjecture 1.3 was developed by [3] [2, 36,
37, 3].

In [15, Sections 3.6, 5], Gromov proposes a different approach towards
Conjecture 1.3 using a modified version of the minimal hypersurface method
involving so called $\mu$-bubbles. Following this idea, [Rae21] [Rae21] proved
Conjecture 1.3 and generalizations thereof in case that $Y$ is orientable and
$n\leq 7$. Conjectures 1.1 and 1.2 have not been directly approached via
minimal hypersurface techniques so far, in particular due to the non-
compactness inherent to the problem. However, in recent work of various
authors (e.g. [19, 20, 39, 6]), $\mu$-bubbles have turned out to be a useful
tool to deal with non-compact situations. Even though there is no direct
formal implication between Conjecture 1.1 and Conjecture 1.3, it was observed
in [37] that the Dirac operator methods used in Ce20, Zeidler:band-width-
estimates to attack Conjecture 1.3 can be refined to prove a more general
statement (compare [37, Conjecture 7.1]) which imply both Conjectures 1.1 and
1.3 for closed spin manifolds with non-vanishing Rosenberg index.

The main objective of this article is to combine the ideas from [37] with
$\mu$-bubble methods, in particular from [Rae21], to prove a generalization of
Conjecture 1.1 in case $Y$ is orientable and of dimension $\leq 6$. By a
reduction argument to a codimension one situation (compare [Theorem
7.5]GromovLawson:PSCDiracCompleteHankePapeSchick:CodimensionTwoIndex), we also
establish Conjecture 1.2 in case $Y$ is orientable and of dimension $\leq 5$.
We note that ideas related to Conjecture 1.1 have also appeared recently in
the work of [5] [5] in connection with the positive mass theorem.

More generally than Conjecture 1.1, one may ask under which circumstances the
existence of a hypersurface $Y\subset M$ which does not admit psc already is
an obstruction to the existence of a complete psc metric on the ambient
manifold $M$. This question has been discussed by Gromov in [14, §11.6], where
in particular he asked if it would be enough to assume that $Y$ is a two-sided
_incompressible hypersurface_ , that is, the map $\pi_{1}Y\to\pi_{1}M$ induced
by the inclusion is injective. In the case that the ambient manifold is spin
of dimension $n\in\\{6,7\\}$ and under further geometric conditions, a proof
confirming this was sketched in [14, 708] and it was asked if the spin
hypothesis can be dropped. We answer this question in our first theorem below
together with the following Example 1.6. Here we say that a connected manifold
$M$ is _almost spin_ , if its universal covering $\tilde{M}$ is spin, and we
say it is _totally non-spin_ if $\tilde{M}$ is non-spin. Since spin structures
lift to coverings, being almost spin is equivalent to the existence of some
covering which is spin. Unless explicitly stated otherwise, we consider
manifolds without boundary.

######  Theorem 1.5.

Let $M$ be an orientable connected $n$-dimensional manifold with
$n\in\\{6,7\\}$ and let $Y\subset M$ be a two-sided closed connected
incompressible hypersurface which does not admit a metric of positive scalar
curvature. Suppose that one of the following two conditions holds:

  1. (a)

$M$ is almost spin.

  2. (b)

$Y$ is totally non-spin.

Then $M$ does not admit a complete metric of positive scalar curvature. More
precisely, if $g$ is a complete metric of non-negative scalar curvature on
$M$, then $(M,g)$ admits a connected Riemannian covering isometric to
$(N\times\mathbb{R},g_{N}+\mathrm{d}{x}^{2})$, where $(N,g_{N})$ is a closed
Ricci flat manifold.

Note that if $M$ is almost spin, then any two-sided hypersurface $Y\subset M$
is almost spin itself. Conversely, if a two-sided hypersurface is totally non-
spin, then so is the ambient manifold. Thus the alternative hypotheses (a) and
(b) of Theorem 1.5 simply say that either $M$ and $Y$ are both almost spin or
neither is. The following example shows that this restriction cannot be
dropped.

######  Example 1.6.

Fix $n\geq 6$. Let $L$ be the K3 surface, that is, a $4$-dimensional simply-
connected spin manifold such that $\widehat{\mathrm{A}}(L)\neq 0$. Consider
the closed manifold
$M\coloneqq(L\times\mathrm{T}^{n-4})\\#(\mathbb{C}\mathrm{P}^{2}\times\mathrm{S}^{n-4})$.
Then, since $L$ is oriented cobordant to a psc manifold by [12, §3], the
cobordism class represented by $M$ in
$\Omega^{\mathrm{SO}}_{n}(\mathrm{B}\mathbb{Z}^{n-4})$ has a psc
representative. Thus the totally non-spin manifold $M$ itself admits a psc
metric, e.g. by [26, Theorem 2.13] or [8, Theorem 1.5].111Alternatively, the
existence of a psc metric on $M$ can also be deduced from [23, Theorem 5.6].
It contains $Y=L\times\mathrm{T}^{n-5}$ as an incompressible hypersurface
which does not admit psc by [13, Corollary 5.22]. By passing to the covering
$\hat{M}\to M$ with $\pi_{1}\hat{M}=\mathbb{Z}^{n-5}$ corresponding to
$\pi_{1}Y$, we even find an example of a complete manifold which admits a
complete uniform psc metric although it contains a closed incompressible
_separating_ hypersurface which does not.

This shows that the almost spin condition is fundamentally relevant for this
kind of problem even though the proof Theorem 1.5 does not use the Dirac
operator at all. Moreover, even in the spin case, Theorem 1.5 is stronger (in
the dimension range where it applies) than what can be proved using index-
theoretic techniques because there are examples of closed spin manifolds which
do not admit psc even though all known index invariants vanish [30].

The upper dimension bound $n\leq 7$ comes from the usual problem pertaining to
singularities of minimal hypersurfaces and $\mu$-bubbles. Thus, if this issue
was resolved, the upper dimensional bound in Theorem 1.5 (and in all other
results of this paper) should conjecturally be removable. The case $n=8$
likely is more easily accessible via an adaptation of the work of N. Smale
[32].

######  Remark 1.7.

In [14, §11.6], Gromov conjectured that in general, given a complete
Riemannian manifold $M$, the existence of a two-sided incompressible closed
embedded hypersurface which does not carry a psc metric obstructs the
existence of a function $h$ on $M$ such that

$\frac{n}{n-1}h^{2}-2\left|\mathrm{d}h\right|+\operatorname{scal}_{g}\geq 0.$

This condition is motivated by the potential function used to construct
$\mu$-bubbles. From this point of view, Theorem 1.5 gives a complete
description of the case when $h=0$, which includes many geometrically
interesting cases. We point out that our method can be adapted to treat cases
with non-trivial $h$. In the direction of Gromov’s motivating example [14,
708], compare Corollary 3.7. We also point out that, compared to the method
sketched in [14, §11.6], our technique does not require any extra assumption
on the geometry of the manifold.

For $n=5$, Theorem 1.5 fails as mentioned in Remark 1.4. On the other hand,
for $2\leq n\leq 4$, it holds even independently of the hypotheses (a) and
(b), but for a different reason than in high dimensions. To formulate this, we
consider the following notion:

######  Definition 1.8 (compare [Definition
2.20]raede2021scalar[“$\mathcal{C}_{\mathrm{deg}}$”]CPSZ21).

A closed connected oriented manifold $Y$ is called $\mathrm{NPSC^{+}}$ if it
satisfies the following property: No closed oriented manifold $Z$ which admits
a continuous map of non-zero degree $Z\to Y$ admits a metric of positive
scalar curvature.

For instance, it has recently been shown chodosh_li,chodosh2021classifying,
Gromov:5d that closed oriented aspherical manifolds of dimension $\leq 5$ are
$\mathrm{NPSC^{+}}$. Moreover, a closed oriented manifold of dimension $\leq
3$ which does not admit psc necessarily admits a non-zero degree map to an
aspherical $\mathrm{NPSC^{+}}$ manifold.222In dimension $2$ this is a
consequence of the classification of surfaces and the Gauß–Bonnet theorem,
whereas in dimension $3$ it is a consequence of the classification of
$3$-manifolds which admit psc following from Perelman’s work, see e.g. the
discussion in [22]. Thus the low-dimensional counterpart to Theorem 1.5 is
contained in the following theorem which already appeared recently in [5].

######  Theorem 1.9 (compare [5, Theorem 1.1]).

Let $M$ be an orientable connected $n$-dimensional manifold with $n\leq 7$ and
let $\iota\colon Y\hookrightarrow M$ be a two-sided closed hypersurface that
admits a map of non-zero degree $\phi\colon Y\to Y_{0}$ to an aspherical
$\mathrm{NPSC^{+}}$ manifold $Y_{0}$ and such that
$\ker(\pi_{1}Y\xrightarrow{\iota_{\ast}}\pi_{1}M)\subseteq\ker(\pi_{1}Y\xrightarrow{\phi_{\ast}}\pi_{1}Y_{0})$.
Then $M$ does not admit a complete metric of positive scalar curvature. More
precisely, if $g$ is a complete metric of non-negative scalar curvature on
$M$, then $(M,g)$ admits a connected Riemannian covering isometric to
$(N\times\mathbb{R},g_{N}+\mathrm{d}{x}^{2})$, where $(N,g_{N})$ is a closed
Ricci flat manifold.

In particular, applying Theorems 1.5 and 1.9 to $M=Y\times\mathbb{R}$, we
deduce:

######  Corollary 1.10.

Conjecture 1.1 holds for orientable manifolds in dimensions $5\neq n\leq 7$.

However, we note that the low-dimensional cases $n\leq 4$ of Corollary 1.10
were already known because an oriented manifold of dimension $\leq 3$ which
does not admit positive scalar curvature is necessarily spin and has
rationally non-vanishing Rosenberg index, and so the situation is within the
scope of index-theoretic results such as [1, Theorem A].

We now turn to our results corresponding to Conjecture 1.2:

######  Theorem 1.11.

Let $M$ be an orientable connected $7$-dimensional manifold and let $Y\subset
M$ be a closed connected $5$-dimensional submanifold with trivial normal
bundle such that the inclusion induces an injection
$\pi_{1}Y\hookrightarrow\pi_{1}M$ and a surjection
$\pi_{2}Y\twoheadrightarrow\pi_{2}M$. Suppose that $Y$ does not admit a metric
of positive scalar curvature. Then $M$ does not admit a complete metric of
uniformly positive scalar curvature.

Note that, unlike Theorem 1.5, we do not need to impose any conditions
involving spin structures. The intuitive reason for this is that the
hypotheses already imply that the induced map $\tilde{Y}\to\tilde{M}$ between
universal covers is $2$-connected and so $M$ is almost spin if and only if $Y$
is. On the other hand, the surjectivity condition
$\pi_{2}Y\twoheadrightarrow\pi_{2}M$ cannot be omitted as the example
$M=Y\times\mathrm{S}^{2}$ shows.

We also have a codimension $2$ counterpart to Theorem 1.9 exploiting the
$\mathrm{NPSC^{+}}$ property.

######  Theorem 1.12.

Let $M$ be an orientable connected $n$-dimensional manifold with $n\leq 7$ and
let $Y\subset M$ be a closed connected $(n-2)$-dimensional submanifold with
trivial normal bundle such that the inclusion induces an injection
$\pi_{1}Y\hookrightarrow\pi_{1}M$ and a surjection
$\pi_{2}Y\twoheadrightarrow\pi_{2}M$. Suppose that $Y$ admits a map of non-
zero degree $Y\to Y_{0}$ to an aspherical $\mathrm{NPSC^{+}}$ manifold
$Y_{0}$. Then $M$ does not admit a complete metric of uniformly positive
scalar curvature.

Similarly as before, this includes a version of Theorem 1.11 in dimensions
$n\leq 5$:

######  Corollary 1.13.

Let $6\neq n\leq 7$. Let $M$ be an orientable connected $n$-dimensional
manifold and let $Y\subset M$ be a closed connected $(n-2)$-dimensional
submanifold with trivial normal bundle such that the inclusion induces an
injection $\pi_{1}Y\hookrightarrow\pi_{1}M$ and a surjection
$\pi_{2}Y\twoheadrightarrow\pi_{2}M$. Suppose that $Y$ does not admit a metric
of positive scalar curvature. Then $M$ does not admit a complete metric of
uniformly positive scalar curvature.

###### Proof.

For $n=7$, this is a restatement of Theorem 1.11. If $n\leq 5$, then
$\dim(Y)\leq 3$ and so it admits a non-zero degree map $Y\to Y_{0}$, where
$Y_{0}$ is aspherical and $\mathrm{NPSC^{+}}$. Thus Theorem 1.12 is applicable
in this case. ∎

Specializing to $M=Y\times\mathbb{R}^{2}$, we finally obtain:

######  Corollary 1.14.

Conjecture 1.2 holds for orientable manifolds in dimensions $6\neq n\leq 7$.

Again we like to point out that the low-dimensional cases $n\leq 5$ of
Corollary 1.14 were already known due to index-theoretic results
HankePapeSchick:CodimensionTwoIndex[Theorem 1.10]Zeidler:band-width-estimates.

This article is organized as follows: In Section 2, we prepare an abstract
setup for the study of manifolds with at least two ends which underpins our
work. In Section 3, we state quantitative comparison results in the spirit of
Conjecture 1.3 which are then used together with topological arguments in
Section 4 to prove our codimension one results. The codimension two results
are deduced in Section 5. Finally, in Section 6 we provide the analytic proofs
of the comparison statements from Section 3.

### Acknowledgement

The authors thank Georg Frenck for helpful discussions and suggestions.

##  2 Bands

A natural class of manifolds generalizing the situation of Conjecture 1.1 are
connected non-compact manifolds, where we partition the set of ends into two
parts and we consider hypersurfaces separating these parts from each other. To
make this precise, we will make use of the Freudenthal end compactification
[9] of a connected manifold $M$, denoted by $\mathcal{F}M=M\cup\mathcal{E}M$,
where $\mathcal{E}M$ is the space of ends.

######  Definition 2.1.

  1. (i)

An _open band_ is a connected non-compact manifold $M$ without boundary and a
decomposition

$\mathcal{E}M=\mathcal{E}_{-}M\sqcup\mathcal{E}_{+}M,$

where $\mathcal{E}_{\pm}M$ are non-empty closed333While the space of ends is
totally disconnected, it is in general not necessarily discrete. In this case
it is important to assume $\mathcal{E}_{\pm}M$ to be closed (and thus clopen)
subsets of $\mathcal{E}M$. subsets $\mathcal{E}_{\pm}M\subset\mathcal{E}M$; in
particular $M$ has at least two ends.

  2. (ii)

Given an open band $M$, a _separating hypersurface_ $\Sigma\subset M$ is a
compact hypersurface which separates each end in $\mathcal{E}_{-}M$ from every
end in $\mathcal{E}_{+}M$. Moreover, we say a separating hypersurface
$\Sigma\subset M$ is _properly separating_ if every component of $\Sigma$ can
be connected to both $\mathcal{E}_{+}M$ and $\mathcal{E}_{-}M$ inside
$M\setminus\Sigma$.

  3. (iii)

Let $\Sigma_{-},\Sigma_{+}\subset M$ be two properly separating hypersurfaces
in an open band $M$. Then we write $\Sigma_{-}\prec\Sigma_{+}$ if the
hypersurface $\Sigma_{-}$ is contained in the union of those components of
$M\setminus\Sigma_{+}$ that contain the ends in $\mathcal{E}_{-}M$ (or
equivalently, $\Sigma_{+}$ is contained in the union of those components of
$M\setminus\Sigma_{-}$ containing $\mathcal{E}_{+}M$).

The condition of being _properly_ separating simply means that there are no
superfluous components as the observation recorded in the following lemma
illustrates.

######  Lemma 2.2.

Let $\Sigma\subset M$ be a separating hypersurface in an open band $M$. Then
there exists a union of components of $\Sigma$ which is a properly separating
hypersurface in $M$.

###### Proof.

Suppose that $\Sigma$ is a separating hypersurface that contains a component
not connected to both $\mathcal{E}_{-}M$ and $\mathcal{E}_{+}M$ inside
$M\setminus\Sigma$. Then the hypersurface $\Sigma^{\prime}$ obtained from
$\Sigma$ by deleting this component is still a separating hypersurface. This
shows that a minimal collection of components of $\Sigma$ such that its union
is still separating yields the desired properly separating hypersurface. ∎

The next elementary lemma shows that (properly) separating hypersurfaces
always exist and we can find them arbitrarily far out.

######  Lemma 2.3.

Let $M$ be an open band and $K\subset M$ be an arbitrary compact subset. Then
there exists a properly separating hypersurface $\Sigma\subset M$ which also
separates $K$ from $\mathcal{E}_{+}M$ (or $\mathcal{E}_{-}M$, respectively).

###### Proof.

Note that the end compactification $\mathcal{F}M$ is a compact Hausdorff space
which in our case of a connected manifold is also second countable. Thus,
since $\mathcal{E}_{\pm}M$ are two disjoint closed subsets of $\mathcal{F}M$,
Urysohn’s lemma implies the existence of a continuous function
$f\colon\mathcal{F}M\to[-1,1]$ such that $\mathcal{E}_{\pm}M=f^{-1}(\pm 1)$.
Since $K\subseteq M$ is compact, there exists $0<r<1$ such that $K\subseteq
f^{-1}([-r,r])$. Choose $s\in(r,1)$. Then $f^{-1}(s)\subseteq M$ is a compact
subset which separates $\mathcal{E}_{-}M$ from $\mathcal{E}_{+}M$. Now choose
a connected compact $n$-dimensional submanifold $V\subset M$ with boundary,
where $n=\dim(M)$, such that $f^{-1}(s)\subseteq\mathring{V}\subseteq
f^{-1}([s-\varepsilon,s+\varepsilon])$ for some $\varepsilon>0$ with
$r<s-\varepsilon$. Then $\partial V$ is a separating hypersurface and it
contains a properly separating hypersurface $\Sigma\subseteq\partial V$ by
Lemma 2.2. Since by construction $f(x)\leq r<s-\varepsilon\leq f(y)\leq
s+\varepsilon<1$ for each $x\in K$ and $y\in\Sigma$, it follows that $\Sigma$
must separate $K$ from $\mathcal{E}_{+}M$. A completely analogous argument
also provides a properly separating hypersurface that separates $K$ from
$\mathcal{E}_{-}M$. ∎

In the spirit of Conjecture 1.1 we will be interested in open bands with:

######  Property A.

No separating hypersurface admits a metric of positive scalar curvature.

We will also work with compact bands, which may be viewed as a special case of
open bands in the following sense:

######  Definition 2.4.

A _compact band_ , to which we will often simply refer to as a “ _band_ ”, is
a connected compact manifold $X$ together with the structure of an open band
on its interior $\mathring{X}$. In other words, this amounts to a
decomposition $\partial X=\partial_{-}X\sqcup\partial_{+}X$, where
$\partial_{\pm}X$ are (non-empty) unions of boundary components.

The notions of (properly) separating hypersurfaces and Property A make sense
for compact bands by considering them for the interior.

We also note that, if $M$ is an open band and $\Sigma_{\pm}$ are two properly
separating hypersurfaces such that $\Sigma_{-}\prec\Sigma_{+}$, then the
hypersurfaces $\Sigma_{\pm}$ bound a compact band $X\subset M$. In this case,
if $M$ has Property A, then so has $X$.

##  3 The partitioned comparison principle

In this section, we establish the main analytic results on which our main
theorems rely. The central concept we study here is the
$\operatorname{width}(X,g)$ of a compact band $X$ endowed with a Riemannian
metric $g$, that is, the distance between $\partial_{-}X$ and $\partial_{+}X$
with respect to $g$. As is explained in [15, Section 3.6] the width of an
$n$-dimensional Riemannian band $(X,g)$ with $\operatorname{scal}\geq n(n-1)$
is bounded from above by $\frac{2\pi}{n}$ if $X$ has Property A and $n\leq 7$.
We will work in the setting where $(X,g)$ has Property A and is partitioned
into multiple segments with possibly different lower scalar curvature bounds.
It turns out that in this case positivity of the scalar curvature in a single
segment can often have global effects on the geometry of $(X,g)$.

######  Definition 3.1.

Let $X$ be a compact band and let $\Sigma_{i}$, for $i\in\\{1,\dots,k\\}$, be
properly separating hypersurfaces such that $\Sigma_{i-1}\prec\Sigma_{i}$ for
$1<i\leq k$. We call the triple $(X,\Sigma_{i},k)$ a _partitioned band_ and
denote by $V_{j}$, for $j\in\\{1,\ldots,k+1\\}$, the segment of $X$ bounded by
$\Sigma_{j-1}$ and $\Sigma_{j}$, where $\Sigma_{0}=\partial_{-}X$ and
$\Sigma_{k+1}=\partial_{+}X$.

######  Definition 3.2.

A smooth function $\varphi\colon[a,b]\to\mathbb{R}_{+}$ is called
$\log$-_concave_ if

$\frac{\mathrm{d}{}^{2}}{\mathrm{d}{t}^{2}}\log(\varphi(t))=\left(\frac{\varphi^{\prime}(t)}{\varphi(t)}\right)^{\prime}\leq
0$

for all $t\in[a,b]$. If the inequality is strict, then $\varphi$ is called
_strictly $\log$-concave_.

######  Definition 3.3.

Let $(N,g_{N})$ be a a closed scalar flat Riemannian manifold. A warped
product

$(M,g_{\varphi})=(N\times[a,b],\varphi^{2}(t)g_{N}+\mathrm{d}{t}^{2})$

is called a _model space_ if $\operatorname{scal}(M,g_{\varphi})$ is constant
and $\varphi$ is strictly $\log$-concave.

######  Remark 3.4.

This notion of a model space arises in the scalar and mean curvature
comparison theory surrounding Conjecture 1.3. In Cecchini-
Zeidler:ScalarMean,Rae21 a compact Riemannian band $(X,g)$ is compared, in
scalar curvature, mean curvature and width, to a warped product
$(M,g_{\varphi})$ over an arbitrary scalar flat manifold $(N,g_{N})$ with
strictly $\log$-concave warping function. It turns out that, if $X$ has
Property A, $\operatorname{scal}(X,g)\geq\operatorname{scal}(M,g_{\varphi})$
and $\operatorname{H}(\partial X,g)\geq\operatorname{H}(\partial
M,g_{\varphi})$, then
$\operatorname{width}(X,g)\leq\operatorname{width}(M,g_{\varphi})$.
Furthermore, Cecchini and Zeidler [3, Theorem 8.3, Theorem 9.1] showed that,
under further topological assumptions on $X$, equality of widths can only be
achieved if $(X,g)$ itself is isometric to such a warped product. In this
section we adapt these ideas and compare a compact partitioned Riemannian band
with Property A to a sequence of model spaces, whose scalar curvatures may
differ, but whose mean curvatures fit together pairwise.

For a compact Riemannian manifold with boundary $(N,h)$ and a connected
component of the boundary $\partial_{j}N$, we denote by
$\operatorname{H}(\partial_{j}N,h)$ the mean curvature of $\partial_{j}N$ with
respect to the interior unit normal field. According to our convention, the
boundary of the unit disk in $\mathbb{R}^{3}$ has mean curvature $2$. In the
following theorem, we give estimates on the widths of segments inside a
partitioned band based on scalar and mean curvature.

######  Theorem 3.5.

Let $n\leq 7$ and let $(X,\Sigma_{i},k)$ be an orientable partitioned
$n$-dimensional band with Property A. For $j\in\\{1,\ldots,k+1\\}$, let
$(M_{j},g_{\varphi_{j}})$ be model spaces over a fixed closed scalar flat
Riemannian manifold $(N,g)$. If

  * $\triangleright$

$\operatorname{scal}(V_{j},g)\geq\operatorname{scal}(M_{j},g_{\varphi_{j}})$
for all $j\in\\{1,\ldots,k+1\\}$,

  * $\triangleright$

$\operatorname{H}(\partial_{-}X,g)\geq\operatorname{H}(\partial_{-}M_{1},g_{\varphi_{1}})$
and
$\operatorname{H}(\partial_{+}X,g)\geq\operatorname{H}(\partial_{+}M_{k+1},g_{\varphi_{k+1}})$,

  * $\triangleright$

$\operatorname{H}(\partial_{+}M_{j},g_{\varphi_{j}})=-\operatorname{H}(\partial_{-}M_{j+1},g_{\varphi_{j+1}})$
for all $j\in\\{1,\ldots,k\\}$,

then
$\operatorname{width}(V_{j},g)\leq\operatorname{width}(M_{j},g_{\varphi_{j}})$
for at least one $j\in\\{1,\ldots,k+1\\}$.

The following result is more or less a direct application of Theorem 3.5:

######  Corollary 3.6.

Let $n\leq 7$ and let $(X,\Sigma_{i},2)$ be an orientable partitioned
$n$-dimensional band with Property A. Let $g$ be a metric on $X$ and let
$\kappa$ be a positive constant. If

  * $\triangleright$

$\operatorname{scal}(V_{2},g)\geq\kappa n(n-1)$,

  * $\triangleright$

$\operatorname{scal}(X,g)\geq 0$,

then
$\min\\{\operatorname{width}(V_{1},g),\operatorname{width}(V_{3},g)\\}<\ell=\frac{2}{\sqrt{\kappa}n}\cot\left(\frac{\sqrt{\kappa}nd}{4}\right)$,
where $d\coloneqq\operatorname{width}(V_{2},g)<\frac{2\pi}{\sqrt{\kappa}n}$.

If, instead of $\operatorname{scal}(X,g)\geq 0$, one assumes that the scalar
curvature of the partitioned band is bounded from below by a negative
constant, Theorem 3.5 provides the following estimate, which is very much in
the same spirit of Corollary 3.6 and should be compared with Zeidler’s result
[37, Theorem 1.4] in the spin setting.

######  Corollary 3.7.

Let $n\leq 7$ and let $(X,\Sigma_{i},2)$ be an orientable partitioned
$n$-dimensional band with Property A. Let $g$ be a metric on $X$ and let
$\kappa$ be a positive constant. If

  * $\triangleright$

$\operatorname{scal}(V_{2},g)\geq\kappa n(n-1)$,

  * $\triangleright$

$\operatorname{scal}(X,g)\geq-\sigma>-\kappa
n(n-1)\tan\left(\frac{\sqrt{\kappa}nd}{4}\right)^{2}$, where
$d\coloneqq\operatorname{width}(V_{2},g)<\frac{2\pi}{\sqrt{\kappa}n}$,

then
$\min\\{\operatorname{width}(V_{1},g),\operatorname{width}(V_{3},g)\\}<\ell$,
where $\ell$ is such that

$\sqrt{\kappa}(n-1)\tan\left(\frac{\sqrt{\kappa}nd}{4}\right)=\sqrt{\frac{\sigma(n-1)}{n}}\coth\left(\frac{\sqrt{\sigma
n}\ell}{2\sqrt{n-1}}\right).$

We postpone the proofs of Theorems 3.5, 3.6 and 3.7 to Section 6.

##  4 Obstructions on open bands

We want to use Corollary 3.6 or Corollary 3.7 to attack Conjecture 1.1. If
$X=Y\times\mathbb{R}$ and $g$ is a complete metric on $X$, we consider the
compact segment $Y\times[-C,C]$ for any $C>0$, which is partitioned into the
bands $Y\times[-C,-1]$, $Y\times[-1,1]$ and $Y\times[1,C]$. If the scalar
curvature of $(X,g)$ is assumed to be positive and $Y\times[-C,C]$ has
Property A, the minimum of the widths of $(Y\times[-C,-1],g)$ and
$(Y\times[1,C],g)$ is bounded from above in terms of
$\operatorname{width}(Y\times[-1,1],g)$ and the infimum of
$\operatorname{scal}(Y\times[-1,1],g)$. For $C>0$ large enough, this produces
a contradiction.

In this section, we formulate the most general result that one can prove in
this manner based on our notion of an open band from Definition 2.1. Indeed,
Corollary 3.6 or Corollary 3.7 will yield obstructions to the existence of a
complete metric of positive scalar curvature on an open band with Property A.
Moreover, since any open band $M$ has at least two ends by definition, we can
also establish a rigidity result using the Cheeger–Gromoll splitting theorem.

######  Theorem 4.1.

Let $n\leq 7$ and let $M$ be an open $n$-dimensional band with Property A. If
$g$ is a complete metric on $M$ with nonnegative scalar curvature, then
$(M,g)$ is isometric to $(Y\times\mathbb{R},g_{Y}+dt^{2})$, where $(Y,g_{Y})$
is a closed Ricci flat manifold.

###### Proof.

If $(M,g)$ is not Ricci flat, then $M$ admits a complete metric $\hat{g}$ of
positive scalar curvature by [Ka82, Theorem B]. Let $\Sigma\subset M$ be a
properly separating hypersurface which exists by Lemma 2.3. There is a
$\kappa>0$ such that $\operatorname{scal}(M,\hat{g})\geq\kappa n(n-1)$ in a
neighborhood of width $d<\frac{2\pi}{\sqrt{\kappa}n}$, which is bounded by two
properly separating hypersurfaces $\Sigma_{1}$ and $\Sigma_{2}$ such that
$\Sigma_{1}\prec\Sigma\prec\Sigma_{2}$. For every $C>0$, we can use Lemma 2.3
to find further properly separating hypersurfaces $\Sigma_{\pm}^{C}$ such that
$\Sigma_{-}^{C}\prec\Sigma_{1}\prec\Sigma_{2}\prec\Sigma_{+}^{C}$ and
$\operatorname{dist}_{\hat{g}}(\Sigma_{-}^{C},\Sigma_{1})\geq
C\leq\operatorname{dist}_{\hat{g}}(\Sigma_{2},\Sigma_{+}^{C})$. Let $X^{C}$ be
the compact band bounded by $\Sigma_{\pm}^{C}$. Then for $C$ large enough,
Corollary 3.6 or Corollary 3.7 applied to $X^{C}$ yields a contradiction.
Hence $(X,g)$ is Ricci flat.

Since $M$ is an open band it is disconnected at infinity and so admits a
geodesic line (see e.g. [24, Lemma 7.3.1]). Thus by the Cheeger–Gromoll
splitting theorem (see e.g. [24, Theorem 7.3.5]), $(M,g)$ is isometric to
$(Y\times\mathbb{R},g_{Y}+dt^{2})$, where $(Y,g_{Y})$ is a Ricci flat
manifold. Furthermore, $M$ has more than one end by the definition of an open
band, and so $Y$ must be compact. ∎

Using a similar (and somewhat simpler argument), we also obtain the following
statement:

######  Proposition 4.2.

Let $n\leq 7$ and let $M$ be an open $n$-dimensional band with Property A.
Then $M$ does not admit a complete metric which has uniformly positive scalar
curvature outside a compact subset.

###### Proof.

Assume, by contradiction, that $M$ admits a complete metric $g$ which has
scalar curvature $\operatorname{scal}_{g}\geq\kappa n(n-1)>0$ on $M\setminus
K$ for some compact subset $K\subset M$ and some $\kappa>0$. Then, applying
Lemma 2.3 twice, we can find a compact band $X\subset M\setminus K$ which is
bounded by two properly separating hypersurfaces
$\partial_{-}X\prec\partial_{+}X\subset M$ such that
$\operatorname{width}(X,g)>\frac{2\pi}{\sqrt{\kappa}n}$. By construction, we
also have $\operatorname{scal}_{g}\geq\kappa n(n-1)>0$ on $X$. But this
contradicts Corollary 3.7 or even the usual band width estimate on compact
bands [Section 3.6]gromovFourLecturesScalar2019v6raede2021scalar of dimension
$\leq 7$. ∎

In the rest of this section, we investigate topological conditions which imply
the existence of open bands with Property A and use them together with Theorem
4.1 to prove the main Theorems 1.5 and 1.9. Let us start with the notion of an
incompressible hypersurface.

######  Definition 4.3.

Let $X$ be a connected manifold and let $\Sigma\subset X$ be a connected
hypersurface. We say that $\Sigma$ is incompressible if the induced map
$\iota_{*}:\pi_{1}(\Sigma)\to\pi_{1}(X)$ is injective.

Our first goal is to provide the most general conditions under which the
existence of an incompressible hypersurface which does not admit psc shows
that the manifold is covered by an open band with Property A. The arguments
will be based on positive scalar curvature bordism and surgery techniques
which usually require a case distinction depending on the presence of a spin
structure. To treat this in a unified way, we use the language of tangential
structures, for details see e.g. [10, §5]. We briefly recall that an
$n$-dimensional tangential structure is given by a fibration $\theta\colon
B\to\mathrm{B}\mathrm{O}(n)$ and a $\theta$-structure on an $n$-manifold $M$
is a lift $M\to B$ of the classifying map of the tangent bundle
$M\to\mathrm{B}\mathrm{O}(n)$ along $\theta\colon
B\to\mathrm{B}\mathrm{O}(n)$. Then, given an $n$-dimensional tangential
structure $\theta\colon B\to\mathrm{B}\mathrm{O}(n)$, one may define a notion
of $\theta$-cobordism for $(n-1)$-dimensional $\theta$-manifolds444To be
precise, here one implicitly considers the $(n-1)$-dimensional tangential
structure obtained from $\theta$ via the pullback along
$\mathrm{BO}(n-1)\to\mathrm{BO}(n)$. and form the corresponding cobordism
group which we denote by $\Omega^{\theta}_{n-1}$. Now the positive scalar
curvature surgery principle can be phrased as follows: Let $\theta\colon
B\to\mathrm{BO}(n)$ be an $n$-dimensional tangential structure and fix an
$(n-1)$-dimensional $\theta$-manifold $N$ such that the map $N\to B$ is
$2$-connected. If $N$ is $\theta$-cobordant to a $\theta$-manifold which
admits a metric of psc, then $N$ itself already admits psc, see [8, Theorem
1.5].

We now connect this back to the study of open bands and first observe that an
open band endowed with a $\theta$-structure induces a well-defined
$\theta$-cobordism class of $(n-1)$-manifolds.

######  Lemma 4.4.

Let $\theta\colon B\to\mathrm{B}\mathrm{O}(n)$ be a tangential structure and
let $M$ be an open band endowed with a $\theta$-structure. Then any two
properly separating hypersurfaces $\Sigma_{1},\Sigma_{2}\subset M$ are
$\theta$-cobordant.

###### Proof.

First consider the case that $\Sigma_{1}\prec\Sigma_{2}$. Then the band in $M$
bounded in between $\Sigma_{1}$ and $\Sigma_{2}$ is a cobordism witnessing
that $\Sigma_{1}$ and $\Sigma_{2}$ are $\theta$-cobordant. In general, Lemma
2.3 implies the existence of a properly separating hypersurface
$\Sigma^{\prime}\subset M$ such that $\Sigma_{1}\prec\Sigma^{\prime}$ and
$\Sigma_{2}\prec\Sigma^{\prime}$. Thus, by the first case, both $\Sigma_{1}$
and $\Sigma_{2}$ are $\theta$-cobordant to $\Sigma^{\prime}$ which proves the
desired statement. ∎

Next we use the psc surgery principle as stated above to derive a sufficient
condition for the existence of open bands with Property A.

######  Lemma 4.5.

Let $n\geq 6$ and let $\theta\colon B\to\mathrm{B}\mathrm{O}(n)$ be a
tangential structure. Let $M$ be a connected $\theta$-manifold without
boundary and let $Y\subset M$ be a closed connected two-sided hypersurface.
Assume that $Y$ does not support a psc metric and that the composition
$Y\subset M\to B$ induces an injection $\pi_{1}Y\hookrightarrow\pi_{1}B$ and a
surjection $\pi_{2}Y\twoheadrightarrow\pi_{2}B$. Then the connected covering
$\hat{M}$ of $M$ with $\pi_{1}\hat{M}=\pi_{1}Y$ is an open band with Property
A.

###### Proof.

First let $\hat{B}\to B$ be the covering with $\pi_{1}\hat{B}=\pi_{1}Y$. Then
$\hat{M}\to M$ is the pullback of the covering $\hat{B}\to B$ along
$\theta\colon M\to B$. The embedding $Y\subset M$ lifts to an embedding
$Y\subset\hat{M}$ as a hypersurface which separates $\hat{M}$ into two
components (otherwise there existed a loop in $\hat{M}$ intersecting $Y$
transversally in precisely one point which would contradict
$\pi_{1}Y\cong\pi_{1}\hat{M}$). Moreover, we obtain a $\hat{\theta}$-structure
on $\hat{M}$, where $\hat{\theta}\colon\hat{B}\to
B\xrightarrow{\theta}\mathrm{B}\mathrm{O}(n)$.

${\hat{M}}$${\hat{B}}$${Y}$${M}$${B}$${\mathrm{B}\mathrm{O}(n)}$$\scriptstyle{\hat{\theta}}$$\scriptstyle{\theta}$

By assumption, the composition $Y\hookrightarrow\hat{M}\to\hat{B}$ is
$2$-connected. Let $U_{\pm}$ be the two components of $\hat{M}\setminus Y$. We
observe that $U_{\pm}$ are both non-compact because otherwise $Y$ would be
$\hat{\theta}$-nullbordant and thus support a psc metric by [8, Theorem 1.5].
We can now define the ends of $U_{-}$ to be $\mathcal{E}_{-}\hat{M}$ and the
ends of $U_{+}$ to be $\mathcal{E}_{+}\hat{M}$ in order to turn $\hat{M}$ into
an open band. Then, by construction, the open band $\hat{M}$ contains $Y$ as a
properly separating hypersurface.

Finally, we need to verify Property A. To this end, assume by contradiction
that $\Sigma\subset\hat{M}$ is a separating hypersurface that admits a psc
metric. By Lemma 2.2, we may assume without loss of generality that $\Sigma$
is properly separating. Then it follows from Lemma 4.4 that $\Sigma$ and $Y$
are $\hat{\theta}$-cobordant. But since the map $Y\to\hat{B}$ is $2$-connected
and $\Sigma$ admits a psc metric, [8, Theorem 1.5] implies that $Y$ also
admits a psc metric, a contradiction. ∎

To describe the possible concrete applications of Lemma 4.5, it turns out to
be enough to go through the possible tangential $2$-types of $M$. Recall that
for an arbitrary $n$-manifold, its tangential $2$-type is the (unique up to
homotopy) tangential structure $\theta_{M}$ which factors the tangent bundle
as $M\to B_{M}\xrightarrow{\theta_{M}}\mathrm{B}\mathrm{O}(n)$, where $M\to
B_{M}$ is $2$-connected and $\theta_{M}\colon B_{M}\to\mathrm{B}\mathrm{O}(n)$
is $2$-co-connected. In other words, $B_{M}$ is the second stage of the
Moore–Postnikov tower for the map $M\to\mathrm{B}\mathrm{O}(n)$. Now a simple
diagram chase shows that if the hypotheses of Lemma 4.5 are satisfied for some
tangential structure $\theta$ on $M$, then they are already satisfied for
$\theta=\theta_{M}$. Moreover, the tangential $2$-type can be described
algebraically in terms of the fundamental group $\pi=\pi_{1}M$, the first
Stiefel–Whitney class $w\colon\pi\to\mathbb{Z}/2$, and an extension
$\hat{\pi}\twoheadrightarrow\pi$ determined by the second Stiefel–Whitney
class whose kernel has order at most $2$, see [33, §2] for details. However,
for our purposes we do not need such a full description, and we only need to
distinguish two cases depending on wether $\pi_{2}B_{M}\cong\mathbb{Z}/2$ or
$\pi_{2}B_{M}=0$ as the following proposition demonstrates.

######  Proposition 4.6.

Let $n\geq 6$. Let $M$ be a connected $n$-dimensional manifold without
boundary and $Y\subset M$ a closed two-sided incompressible hypersurface that
does not admit a psc metric. Suppose that one of the following conditions
holds:

  1. (a)

$M$ is almost spin.

  2. (b)

$Y$ is totally non-spin.

Then there exists a covering $\hat{M}\to M$ which is an open band with
Property A.

###### Proof.

In light of the discussion in the previous paragraph, we need to check that in
either case we can apply Lemma 4.5 for the tangential structure
$\theta=\theta_{M}$ given by its $2$-type.

  1. (a)

If $M$ is almost spin, that is, the universal covering $\tilde{M}$ of $M$ is
spin, then the map $M\to\mathrm{B}\mathrm{O}(n)$ classifying the tangent
bundle induces the zero map
$\pi_{2}M\to\pi_{2}\mathrm{B}\mathrm{O}(n)=\mathbb{Z}/2$ because this map can
be identified with the second Stiefel–Whitney class of $\tilde{M}$ via the
Hurewicz isomorphism
$\operatorname{H}_{2}(\tilde{M})\cong\pi_{2}(\tilde{M})\cong\pi_{2}(M)$. Thus
it follows that $\pi_{2}(B_{M})=0$ and so LABEL:{lem:tangential_structure}
applies to every incompressible hypersurface $Y\subset M$ which does not admit
psc.

  2. (b)

On the other hand, if $M$ is totally non-spin, that is $\tilde{M}$ is non-
spin, then by an analogous consideration involving the second Stiefel–Whitney
class of $\tilde{M}$ we necessarily have $\pi_{2}B_{M}\cong\mathbb{Z}/2$ and
$\pi_{2}M\to\pi_{2}B_{M}=\mathbb{Z}/2$ is surjective. Now in this situation
the condition on the hypersurface $Y\subset M$ in Lemma 4.5 is that it is
incompressible, itself totally non-spin, and does not admit psc. ∎

We are now ready to deduce Theorem 1.5:

###### Proof of Theorem 1.5.

By Proposition 4.6, there exists a covering $\hat{M}$ which is an open band
with Property A. Now suppose that $M$ admits a complete metric of non-negative
scalar curvature $g$. Let $\hat{g}$ denote its lift to $\hat{M}$. Then Theorem
4.1 implies that $(\hat{M},\hat{g})$ must be isometric to
$(N\times\mathbb{R},g_{N}+\mathrm{d}{t}^{2})$ for a closed Ricci flat manifold
$(N,g_{N})$. ∎

We now turn to Theorem 1.9. Note that a proof of Theorem 1.9 has already
appeared recently in [5, Theorem 1.1], but we give a separate argument here
because it also fits directly into our topological setup. In fact, it is
simpler than Theorem 1.5 as it does not need surgery arguments and instead
only relies on Lemma 4.4 together with some homological considerations:

######  Lemma 4.7.

Let $M$ be an oriented open band and $Y\subseteq M$ a properly separating
hypersurface together with a map $\phi\colon M\to Y_{0}$ to an
$\mathrm{NPSC^{+}}$ manifold $Y_{0}$ such that the restriction
$\phi|_{Y}\colon Y\to Y_{0}$ has non-zero degree. Then $M$ has Property A.

###### Proof.

We define a tangential structure
$M\xrightarrow{l}B\xrightarrow{\theta}\mathrm{BO}(n)$, where we set
$B=Y_{0}\times\mathrm{B}\mathrm{SO}(n)$ and $l$ is induced by the map $M\to
Y_{0}$ together with the orientation of $M$. Note that a $\theta$-structure on
an $(n-1)$-manifold $N$ is the same as an orientation on $N$ together with a
map $N\to Y_{0}$, and thus
$\Omega^{\theta}_{n-1}=\Omega_{n-1}^{\mathrm{SO}}(Y_{0})$. In this picture,
the degree of the map $N\to Y_{0}$ can be read off from the transformation
$\Omega^{\mathrm{SO}}_{n-1}(Y_{0})\to\operatorname{H}_{n-1}(Y_{0};\mathbb{Z})\cong\mathbb{Z}$.
Suppose that $\Sigma\subset M$ is a separating hypersurface, and assume
without loss of generality that it is properly separating. Then, by Lemma 4.4,
$\Sigma\to Y_{0}$ and $Y\to Y_{0}$ represent the same class of
$\Omega_{n-1}^{\mathrm{SO}}(Y_{0})$. In particular, $\deg(\Sigma\to
Y_{0})=\deg(Y\to Y_{0})\neq 0$. Since $Y_{0}$ is $\mathrm{NPSC^{+}}$, this
proves that $\Sigma$ does not admit a psc metric and so $M$ must have Property
A. ∎

######  Proposition 4.8.

Let $M$ be an oriented connected $n$-dimensional manifold and let $\iota\colon
Y\hookrightarrow M$ be a two-sided closed connected hypersurface that admits a
map of non-zero degree $\phi\colon Y\to Y_{0}$ to an aspherical
$\mathrm{NPSC^{+}}$ manifold $Y_{0}$ and such that
$\ker(\pi_{1}Y\xrightarrow{\iota_{\ast}}\pi_{1}M)\subseteq\ker(\pi_{1}Y\xrightarrow{\phi_{\ast}}\pi_{1}Y_{0})$.
Then there exists a connected covering $\hat{M}\to M$ that is an open band
with Property A.

###### Proof.

Let $\Lambda\coloneqq\iota_{\ast}(\pi_{1}Y)\subseteq\pi_{1}M$ and let
$\hat{M}\to M$ be the covering with $\pi_{1}\hat{M}=\Lambda$. Then
$Y\hookrightarrow M$ lifts to an embedding $Y\hookrightarrow\hat{M}$ which
induces a surjection $\pi_{1}Y\twoheadrightarrow\Lambda$. By the assumption on
the kernels of the induced maps on fundamental groups, it follows that the
homomorphism $\phi_{\ast}\colon\pi_{1}Y\to\pi_{1}Y_{0}$ factors as a
composition $\pi_{1}Y\twoheadrightarrow\Lambda\to\pi_{1}Y_{0}$. Since $Y_{0}$
is aspherical and $\pi_{1}\hat{M}=\Lambda$, this implies that the map
$\phi\colon Y\to Y_{0}$ extends to a map $\hat{M}\to Y_{0}$. As in the proof
of Lemma 4.5, let $U_{\pm}$ be the two components of $\hat{M}\setminus Y$.
Then $U_{\pm}$ must be non-compact because otherwise $[Y\to
Y_{0}]=0\in\Omega_{n-1}^{\mathrm{SO}}(Y_{0})$ which would contradict the
hypothesis $\deg(Y\to Y_{0})\neq 0$ (compare the proof of Lemma 4.7 above).
Thus $\hat{M}$ can be turned into an open band such that $Y$ is a properly
separating hypersurface. Thus the proposition follows from Lemma 4.7. ∎

###### Proof of Theorem 1.9.

Combine Propositions 4.8 and 4.1 analogously as in the proof of Theorem 1.5
above. ∎

##  5 The codimension two obstruction

In this section, we prove our codimension two obstruction results. These are
based on a reduction to a codimension one problem essentially following
original ideas of [13] [13, Theorem 7.5] and their adaptation by [17] [17].

######  Lemma 5.1 (cf. [Theorem 4.3]HankePapeSchick:CodimensionTwoIndex[Lemma
4.1.4]Zeidler:SecondaryLargescaleIndex).

Let $X$ be a connected manifold without boundary. Let $Y\subset X$ be a
submanifold without boundary of codimension two whose normal bundle is trivial
and suppose that the pair $(X,Y)$ is $2$-connected. Consider the manifold
$W\coloneqq X\setminus\mathcal{U}$ obtained by deleting a small open tubular
neighborhood $Y\times\mathrm{B}^{2}\cong\mathcal{U}\subseteq X$. Then the map
$\partial W\cong
Y\times\mathrm{S}^{1}\xrightarrow{\operatorname{pr}_{2}}\mathrm{S}^{1}$ given
by projection onto the second factor extends to a continuous map
$W\to\mathrm{S}^{1}$. In particular, the map
$\pi_{1}Y\times\mathbb{Z}=\pi_{1}(\partial W)\to\pi_{1}W$ induced by the
inclusion $\partial W\hookrightarrow W$ is split-injective.

###### Proof.

It suffices to show that the induced homomorphism
$(\operatorname{pr}_{2})_{\ast}\colon\pi_{1}(\partial
W)\to\pi_{1}(\mathrm{S}^{1})=\mathbb{Z}$ extends to a homomorphism
$\pi_{1}(W)\to\mathbb{Z}$. The hypotheses imply that the pair
$(X,\mathcal{U})$ is also $2$-connected and so excision and the Hurewicz
theorem show that $\operatorname{H}_{k}(W,\partial
W)\cong\operatorname{H}_{k}(X,\mathcal{U})=0$ for $0\leq k\leq 2$. In
particular, the map $\operatorname{H}_{1}(\partial
W)\xrightarrow{\cong}\operatorname{H}_{1}(W)$ induced by the inclusion
$\partial W\hookrightarrow W$ is an isomorphism. The existence of the desired
extension now follows from the following diagram because the map
$(\operatorname{pr}_{2})_{\ast}\colon\pi_{1}(\partial
W)\to\pi_{1}(\mathrm{S}^{1})=\mathbb{Z}$ factors through the Hurewicz
homomorphism.

${\pi_{1}(\partial W)}$${\operatorname{H}_{1}(\partial
W)}$${\operatorname{H}_{1}(\mathrm{S}^{1})}$${\pi_{1}(\mathrm{S}^{1})=\mathbb{Z}}$${\pi_{1}(W)}$${\operatorname{H}_{1}(W)}$$\scriptstyle{\mathrm{hur}}$$\scriptstyle{\cong}$$\scriptstyle{(\operatorname{pr}_{2})_{\ast}}$$\scriptstyle{\mathrm{hur}}$$\scriptstyle{\mathrm{hur}}$

This also implies that the map $\pi_{1}Y\times\mathbb{Z}=\pi_{1}(\partial
W)\to\pi_{1}W$ is split injective because a retraction can be constructed
using the map $\pi_{1}W\to\mathbb{Z}$ from the previous paragraph together
with the map $\pi_{1}W\to\pi_{1}X\cong\pi_{1}Y$ induced by the inclusion
$W\hookrightarrow X$. ∎

######  Proposition 5.2.

Let $n\geq 6$. Let $X$ be a connected $n$-dimensional manifold and let
$Y\subset X$ be a closed connected submanifold of codimension two with trivial
normal bundle such that the pair $(X,Y)$ is $2$-connected. Consider the
manifold $W\coloneqq X\setminus\mathcal{U}$ obtained by deleting a small open
tubular neighborhood $Y\times\mathrm{B}^{2}\cong\mathcal{U}\subseteq X$. If
$Y\times\mathrm{S}^{1}$ does not admit a metric of positive scalar curvature,
then the double $\mathsf{d}{W}$of $W$ is an open band with Property A.

###### Proof.

We start with considering the tangential $2$-type $X\to
B_{X}\to\mathrm{B}\mathrm{O}(n)$ of $X$. By restriction to $W\subset X$ this
induces a tangential structure $l^{\prime}\colon W\to B_{X}$. Let $p\colon
W\to\mathrm{S}^{1}$ be a map extending the projection $\partial W\cong
Y\times\mathrm{S}^{1}\to\mathrm{S}^{1}$ which exists by Lemma 5.1. We obtain a
new tangential structure
$W\xrightarrow{l}B\coloneqq(B_{X}\times\mathrm{S}^{1})\xrightarrow{\theta}\mathrm{B}\mathrm{O}(n)$,
where $l=(l^{\prime},p)$, and $\theta$ is defined as projection to $B_{X}$
followed by $B_{X}\to\mathrm{B}\mathrm{O}(n)$. Furthermore, this tangential
structure $l$ extends to the double $\mathsf{d}{W}$ by reflection. Its
restriction to the hypersurface $\mathsf{d}{W}\supset\partial W\cong
Y\times\mathrm{S}^{1}$ is a $2$-connected map $Y\times\mathrm{S}^{1}\to
B=B_{X}\times\mathrm{S}^{1}$ because by construction it is homotopic to
$l_{Y}\times\operatorname{id}_{\mathrm{S}^{1}}$, where $l_{Y}$ denotes the
restriction of $X\to B_{X}$ to $Y\subset X$ and $l_{Y}$ is $2$-connected. Thus
Lemma 4.5 (applied to the trivial covering) proves the desired conclusion. ∎

###### Proof of Theorem 1.11.

Note that since $Y$ does not admit positive scalar curvature and $\dim(Y)=5$,
Theorem 1.5 implies that $Y\times\mathrm{S}^{1}$ does not admit positive
scalar curvature either. Then we let $X$ be the connected covering of $M$ with
$\pi_{1}X=\pi_{1}Y$. It follows $Y\hookrightarrow X$ and the pair $(X,Y)$
satisfies the hypotheses of Proposition 5.2. Thus, if we let $W$ be as in the
statement of Proposition 5.2, then its double $\mathsf{d}{W}$ is an open band
with Property A. Now assume by contradiction that $M$ admits a complete metric
of uniformly positive scalar curvature. Then by first lifting it to $X$ and
restricting it to $W$, we obtain a complete metric of uniformly positive
scalar curvature on $W$. After changing this metric in a compact neighborhood
of $\partial W$, we obtain a smooth complete metric on $\mathsf{d}W$ which has
uniformly positive scalar curvature outside a compact subset, a contradiction
to Proposition 4.2. ∎

###### Proof of Theorem 1.12.

We first verify that $Y_{0}\times\mathrm{S}^{1}$ is also $\mathrm{NPSC^{+}}$.
To this end, let $N$ be an oriented $(n-1)$-manifold and $\phi\colon N\to
Y_{0}\times\mathrm{S}^{1}$ a map of non-zero degree, where we assume $N$ to be
connected without loss of generality. Then let
$Z=\phi^{-1}(Y_{0}\times\\{\ast\\})$ be a transversal pre-image. It follows
that $\iota\colon Z\hookrightarrow N$ and $\phi\colon Z\to
Y_{0}\times\\{\ast\\}=Y_{0}$ satisfies the hypotheses of Theorem 1.9 and so
$N$ does not admit a psc metric. This proves that $Y_{0}\times\mathrm{S}^{1}$
is $\mathrm{NPSC^{+}}$.

To prove the theorem, we again consider the connected covering of $X\to M$
with $\pi_{1}X=\pi_{1}Y$, and we let $W$ be as in the statement of Proposition
5.2. By Lemma 5.1, the map $\pi_{1}Y\times\mathbb{Z}=\pi_{1}(\partial
W)\to\pi_{1}W$ induced by the inclusion $Y\times\mathrm{S}^{1}=\partial
W\hookrightarrow W$ is injective and admits a retraction
$r\colon\pi_{1}W\twoheadrightarrow\pi_{1}Y\times\mathbb{Z}$. Since
$Y_{0}\times\mathrm{S}^{1}$ is aspherical, this implies that the map
$Y\times\mathrm{S}^{1}\to Y_{0}\times\mathrm{S}^{1}$ extends to a map $W\to
Y_{0}\times\mathrm{S}^{1}$ and subsequently to a map $\mathsf{d}W\to
Y_{0}\times\mathrm{S}^{1}$ on the double. In summary, $\mathsf{d}W$ is an open
band that contains $Y\times\mathrm{S}^{1}$ as a properly separating
hypersurface and it satisfies the hypotheses of Lemma 4.7 because
$Y_{0}\times\mathrm{S}^{1}$ is $\mathrm{NPSC^{+}}$. Thus $\mathsf{d}W$ has
Property A and the theorem now follows again from Proposition 4.2 as in the
proof of Theorem 1.11 above. ∎

##  6 Proof of the Partitioned Comparison Principle

We will prove Theorem 3.5 by contradiction. We stress that all bands
considered in this section are compact. Under the assumption that
$\operatorname{width}(V_{j},g)>\operatorname{width}(M_{j},g_{\varphi_{j}})$
for all $j\in\\{1,\ldots,k+1\\}$, we will produce a closed embedded
hypersurface $\Sigma\subset X$ which separates $\partial_{-}X$ and
$\partial_{+}X$ and admits a metric of positive scalar curvature.

This hypersurface $\Sigma$ will appear as the boundary of a $\mu$-bubble. The
key ingredient for the corresponding functional is the potential function
$h\colon X\to\mathbb{R}$. We use the ideas from [Rae21, Section 3] for each
band $(V_{j},g)$ and model space $(M_{j},g_{\varphi_{j}})$ separately to
produce $h_{j}\colon V_{j}\to\mathbb{R}$ as the concatenation of a strictly
1-Lipschitz band map $(V_{j},g)\to(M_{j},g_{\varphi_{j}})$ and the function
$h_{\varphi_{j}}:M_{j}\to\mathbb{R}$. Subsequently, we use a gluing
construction to paste all of the $h_{j}$ together to obtain a smooth function
$h\colon X\to\mathbb{R}$ which is suitable for our purposes.

The idea to combine potential functions in this way, was already used in
CeZeiPMT,ZeidlerWidthLargeness. The gluing construction is based on the
following result:

######  Lemma 6.1.

Let $h\colon[a,b]\to\mathbb{R}$ be a smooth strictly monotonously decreasing
function such that

$-\frac{n}{n-1}h^{2}-2h^{\prime}=\sigma,$

for some constant $\sigma\in\mathbb{R}$. Then, for every sufficiently small
$\varepsilon>0$, there exists a function
$\hat{h}\colon[a-\varepsilon,b+\varepsilon]\to\mathbb{R}$ such that:

  * $\triangleright$

$\hat{h}(t)=h(t)$ for $t\in[a+\varepsilon,b-\varepsilon]$,

  * $\triangleright$

$\hat{h}(t)=h(a)$ in a neighborhood of $a-\varepsilon$ and $\hat{h}(t)=h(b)$
in a neighborhood of $b+\varepsilon$,

  * $\triangleright$

$\hat{h}^{\prime}\leq 0$,

  * $\triangleright$

$-\frac{n}{n-1}\hat{h}^{2}-2\hat{h}^{\prime}\leq\sigma$ and
$-\frac{n}{n-1}\hat{h}^{2}(t)-2\hat{h}^{\prime}(t)<\sigma$ if
$\hat{h}^{\prime}(t)=0$.

###### Proof.

Let $\rho\colon\mathbb{R}\to[a,b]$ be a smooth function with:

  * $\triangleright$

$\rho(t)=a$ for $t\in(-\infty,a-\frac{\varepsilon}{2}]$, $\rho(t)=t$ for
$t\in[a+\frac{\varepsilon}{2},b-\frac{\varepsilon}{2}]$ and $\rho(t)=b$ for
$t\in[b+\frac{\varepsilon}{2},\infty)$.

  * $\triangleright$

$0<\rho^{\prime}(t)<1$ for
$t\in(a-\frac{\varepsilon}{2},a+\frac{\varepsilon}{2})$ and
$t\in(b-\frac{\varepsilon}{2},b+\frac{\varepsilon}{2})$.

Then the function $\hat{h}\colon[a-\varepsilon,b+\varepsilon]\to\mathbb{R}$
defined by $\hat{h}=h\circ\rho$ has all of the desired properties. The first
two are immediate from the definition. The third one holds since
$\hat{h}^{\prime}(t)=h^{\prime}(\rho(t))\rho^{\prime}(t)$ and $h^{\prime}<0$
while $\rho^{\prime}\geq 0$. To check the last property we point out that

$-\frac{n}{n-1}\hat{h}^{2}(t)-2\hat{h}^{\prime}(t)=-\frac{n}{n-1}h(\rho(t))-2h^{\prime}(\rho(t))\rho^{\prime}(t)=\sigma+2h^{\prime}(\rho(t))(1-\rho^{\prime}(t)).$

Since $h^{\prime}(\rho(t))<0$ and $0\leq\rho^{\prime}\leq 1$, the above is
always $\leq\sigma$ and it is $<\sigma$ if $\rho^{\prime}(t)<1$. This holds
true in particular when $\hat{h}^{\prime}(t)=0$, that is,
$\rho^{\prime}(t)=0$. ∎

In order to construct our functions $h_{j}\colon V_{j}\to\mathbb{R}$, we make
use of the following basic existence result of band maps; for the proof, we
refer to [Lemma 4.1]Zhu:WidthEstimates[Lemma 7.2]Cecchini-Zeidler:ScalarMean.

######  Lemma 6.2.

Let $(V,g)$ be a Riemannian band and let $a<b$ be two real numbers. If
$\operatorname{width}(V,g)>b-a$, then there exists a smooth function
$\beta\colon V\to[a,b]$ with $\beta(\partial_{-}V)=a$,
$\beta(\partial_{+}V)=b$ and $\mathrm{Lip}(\beta)<1$.

Next, for a Riemannian band $(X,g)$, we give conditions on the scalar
curvature, the mean curvature of $\partial X$ and on the potential function
$h$ so that the $\mu$-bubble associated to $h$ produces a separating closed
hypersurface admitting a metric of positive scalar curvature. For more details
on $\mu$-bubbles, we refer the reader to [15, Section 5].

######  Proposition 6.3.

Let $n\leq 7$ and let $(X,g)$ be an $n$-dimensional oriented Riemannian band.
Let $h\colon X\to\mathbb{R}$ be a smooth function with the property that

$\operatorname{scal}(X,g)+\frac{n}{n-1}h^{2}-2|\nabla h|>0.$ (6.1)

Furthermore, suppose that the mean curvature satisfies

$\operatorname{H}(\partial_{\pm}X,g)>\pm h\bigr{|}_{\partial_{\pm}X}.$ (6.2)

Then there exist a closed embedded hypersurface $\Sigma$ which separates
$\partial_{-}X$ and $\partial_{+}X$ and a constant $b>0$ such that

$\int_{\Sigma}\Bigl{(}|\nabla_{\Sigma}\psi|^{2}+\frac{1}{2}\operatorname{scal}(\Sigma,g)\psi^{2}\Bigr{)}\,d\mathrm{vol}_{\Sigma}\geq
b\int_{\Sigma}\psi^{2}\,d\mathrm{vol}_{\Sigma},\qquad\forall\psi\in
C^{\infty}(\Sigma).$ (6.3)

###### Proof.

Denote by $\mathcal{C}(X)$ the set of all Caccioppoli sets in $X$ which
contain an open neighborhood of $\partial_{-}X$ and are disjoint from
$\partial_{+}X$. For $\hat{\Omega}\in\mathcal{C}(X)$ consider the functional

$\mathcal{A}_{h}(\hat{\Omega})=\mathcal{H}^{n-1}(\partial^{*}\hat{\Omega}\cap\mathring{X})-\int_{\hat{\Omega}}h\,d\mathcal{H}^{n},$

where $\partial^{*}\hat{\Omega}$ is the reduced boundary [Giu84, Chapters 3,4]
of $\hat{\Omega}$. By Condition (6.2) and [Rae21, Lemma 4.2], there exists a
smooth $\mu$-_bubble_ $\Omega\in\mathcal{C}(X)$, that is, a smooth Caccioppoli
set with

$\mathcal{A}_{h}(\Omega)=\mathcal{I}\coloneqq\inf\\{\mathcal{A}_{h}(\hat{\Omega})\bigr{|}\hat{\Omega}\in\mathcal{C}(X)\\}.$

Then $\Sigma\coloneqq\partial\Omega\cap\mathring{X}$ is a closed embedded
hypersurface that separates $\partial_{-}X$ and $\partial_{+}X$. Let $\nu$ be
the outward pointing unit normal vector field to $\Sigma$. By the first
variation formula for $\mathcal{A}_{h}$ (see [Rae21, Lemma 4.3]) the mean
curvature of $\Sigma$ (computed with respect to $-\nu$) is equal to
$h\bigr{|}_{\Sigma}$. By stability, from the second variation formula (see
[Rae21, Lemma 4.4]) we deduce

$\displaystyle\int_{\Sigma}\Bigl{(}2|\nabla_{\Sigma}\psi|^{2}+\operatorname{scal}(\Sigma,g)\psi^{2}\Bigr{)}\,d\mathrm{vol}_{\Sigma}\geq$
$\displaystyle\int_{\Sigma}\Bigl{(}\operatorname{scal}(X,g)+\frac{n}{n-1}h^{2}+2g(\nabla_{X}h,\nu)\Bigr{)}\psi^{2}\,d\mathrm{vol}_{\Sigma}$
$\displaystyle\geq$
$\displaystyle\min_{\Sigma}\Bigl{\\{}\operatorname{scal}(X,g)+\frac{n}{n-1}h^{2}-2|\nabla
h|\Bigr{\\}}\int_{\Sigma}\psi^{2}\,d\mathrm{vol}_{\Sigma}$

for all $\psi\in C^{\infty}(\Sigma)$. By Condition (6.1), the previous
inequality yields a constant $b>0$ such that Inequality (6.3) holds. ∎

We have gathered all the ingredients we need to prove Theorem 3.5.

###### Proof of Theorem 3.5.

Assume, by contradiction, that
$\operatorname{width}(V_{j},g)>\operatorname{width}(M_{j},g_{\varphi_{j}})$
for all $j\in\\{1,\ldots,k+1\\}$. Consider the functions

$h_{\varphi_{j}}(t)=(n-1)\frac{\varphi_{j}^{\prime}(t)}{\varphi_{j}(t)}\colon[a_{j},b_{j}]\to\mathbb{R}.$

Since the $\varphi_{j}$ are strictly $\log$-concave, the functions
$h_{\varphi_{j}}$ are strictly monotonously decreasing and the scalar
curvature of $(M_{j},g_{\varphi_{j}})$ is given by

$\sigma_{j}=\operatorname{scal}(M_{j},g_{\varphi_{j}})=-\frac{n}{n-1}h_{\varphi_{j}}^{2}-2h^{\prime}_{\varphi_{j}}.$
(6.4)

For $j\in\\{2,\ldots,k\\}$, we apply Lemma 6.1 to $h_{\varphi_{j}}$ and obtain
smooth functions

$\hat{h}_{\varphi_{j}}\colon[a_{j}-\varepsilon,b_{j}+\varepsilon]\to\mathbb{R}$

with the aforementioned properties.

For $j=1$ we extend the domain of $h_{\varphi_{j}}$ to
$[a_{1}-\varepsilon,b_{1}]$ and apply the interpolation procedure of Lemma 6.1
only on the right hand side of the interval to produce
$\hat{h}_{\varphi_{1}}\colon[a_{1}-\varepsilon,b_{1}+\varepsilon]\to\mathbb{R}$.
For $j=k+1$ we extend the domain of $h_{\varphi_{k+1}}$ to
$[a_{k+1},b_{k+1}+\varepsilon]$ and apply the interpolation procedure of Lemma
6.1 only on the left hand side of the interval to produce
$\hat{h}_{\varphi_{k+1}}\colon[a_{k+1}-\varepsilon,b_{k+1}+\varepsilon]\to\mathbb{R}$.

By Lemma 6.2, there are smooth maps

$\beta_{j}\colon V_{j}\to[a_{j}-\varepsilon,b_{j}+\varepsilon]$

such that $\beta_{j}(\partial_{-}V_{j})=a_{j}-\varepsilon$,
$\beta_{j}(\partial_{+}V_{j})=b_{j}+\varepsilon$ and
$\mathrm{Lip}(\beta_{j})<1$. Define $h\colon X\to\mathbb{R}$ by
$h(x)=\hat{h}_{\varphi_{j}}\circ\beta_{j}(x)$ if $x\in V_{j}$. The function
$h$ is continuous since

$\hat{h}_{\varphi_{j}}(b_{j}+\varepsilon)=\operatorname{H}(\partial_{+}M_{j},g_{\varphi_{j}})=-\operatorname{H}(\partial_{-}M_{j+1},g_{\varphi_{j+1}})=\hat{h}_{\varphi_{j+1}}(a_{j+1}-\varepsilon)$

for all $j\in\\{1,\ldots,k\\}$. It is smooth since the
$\hat{h}_{\varphi_{j}}\circ\phi_{j}$ are constant in a neighborhood of the
separating hypersurfaces $\Sigma_{i}$, which partition the band.

Note that $h$ satisfies Condition (6.1) by the chain rule, the fourth property
of the $\hat{h}_{\varphi_{j}}$ from Lemma 6.1, and since
$\mathrm{Lip}(\beta_{j})<1$. For the mean curvature of the boundary, the
following holds true:

$\operatorname{H}(\partial_{-}X,g)\geq\operatorname{H}(\partial_{-}M_{1},g_{\varphi_{1}})=-\hat{h}_{\varphi_{1}}(a_{1})>-\hat{h}_{\varphi_{1}}(a_{1}-\varepsilon)=-h\bigr{|}_{\partial_{-}X}$

and

$\operatorname{H}(\partial_{+}X,g)\geq\operatorname{H}(\partial_{+}M_{k+1},g_{\varphi_{k+1}})=\hat{h}_{\varphi_{k+1}}(b_{k+1})>\hat{h}_{\varphi_{k+1}}(b_{k+1}+\varepsilon)=h\bigr{|}_{\partial_{+}X}.$

Hence, Condition (6.2) is satisfied as well. By Proposition 6.3, there exist a
closed embedded hypersurface $\Sigma$ which separates $\partial_{-}X$ and
$\partial_{+}X$ and a constant $b>0$ such that Inequality (6.3) holds.

If $n=2$, this yields an immediate contradiction by choosing $\psi=1$ in
(6.3). If $n=3$, we again choose $\psi=1$ and use Gauß-Bonnet to see that
$\Sigma$ admits a psc metric. If $n\geq 4$, then $\Sigma$ admits a metric of
positive scalar curvature by the conformal change argument of Schoen and Yau
[31]. This contradicts the fact that $X$ has Property A. ∎

###### Proof of Corollary 3.6.

Consider the function

$\varphi_{2}\colon\left(-\frac{\pi}{\sqrt{\kappa}n},\frac{\pi}{\sqrt{\kappa}n}\right)\to\mathbb{R}_{+}\qquad
t\mapsto\cos\left(\frac{\sqrt{\kappa}nt}{2}\right)^{\frac{2}{n}},$

which is strictly $\log$-concave and has

$h_{\varphi_{2}}(t)=-\sqrt{\kappa}(n-1)\tan\left(\frac{\sqrt{\kappa}nt}{2}\right).$

Consider the function

$\varphi_{1}\colon\mathbb{R}_{+}\to\mathbb{R}_{+}\qquad t\mapsto
t^{\frac{2}{n}},$

which is strictly $\log$-concave and has

$h_{\varphi_{1}}(t)\colon=\frac{2(n-1)}{nt}.$

Since $h_{\varphi_{1}}(t)\to\infty$ as $t\to 0$, there is a value $t_{-}>0$
such that $H(\partial_{-}X,g)\geq-h_{\varphi_{1}}(t_{-})$. By continuity,
there are $\delta_{1},\delta_{2}>0$ small enough such that

$h_{\varphi_{2}}\left(\frac{-d+\delta_{1}}{2}\right)=h_{\varphi_{1}}(\ell+\delta_{2}),$

while $\delta_{2}<t_{-}$ and hence $\ell+\delta_{2}-t_{-}<\ell$. Let
$(N,g_{N})$ be a closed scalar flat Riemannian manifold. We fix the model
space

$(M_{1},g_{\varphi_{1}})=(N\times[t_{-},\ell+\delta_{2}],\varphi_{1}^{2}(t)g_{N}+dt^{2})$

with scalar curvature equal to zero and $\operatorname{width}<\ell$.

Let $\varphi_{3}\colon\mathbb{R}_{-}\to\mathbb{R}_{+}$ be defined by
$\varphi_{3}(t)=\varphi_{1}(-t)$. This function is strictly $\log$-concave and
$h_{\varphi_{3}}(t)=-h_{\varphi_{1}}(-t)$. Since
$h_{\varphi_{3}}(t)\to-\infty$ as $t\to 0$, there is a value $t_{+}<0$ such
that $H(\partial_{X},g)\geq h_{\varphi_{3}}(t_{+})$. Similarly as before, we
find $\delta_{3},\delta_{4}>0$ such that

$h_{\varphi_{2}}\left(\frac{d-\delta_{3}}{2}\right)=h_{\varphi_{3}}(-\ell-\delta_{4}),$

while $\delta_{4}<-t_{+}$ and hence $\ell+\delta_{4}+t_{+}<\ell$. We fix the
model space

$(M_{3},g_{\varphi_{3}})=(N\times[t_{-},\ell+\delta_{4}],\varphi_{3}^{2}(t)g_{N}+dt^{2})$

with scalar curvature equal to zero and $\operatorname{width}<\ell$.

Finally, we fix the model space

$(M_{2},g_{\varphi_{2}})=\left(N\times\left[\frac{-d+\delta_{1}}{2},\frac{d-\delta_{3}}{2}\right],\varphi_{2}^{2}(t)g_{N}+dt^{2}\right)$

with scalar curvature equal to $\kappa n(n-1)$ and $\operatorname{width}<d$.

It follows from Theorem 3.5 that
$\operatorname{width}(V_{j},g)\leq\operatorname{width}(M_{j},g_{\varphi_{j}})$
for at least one $i\in\\{1,2,3\\}$. Since
$d=\operatorname{width}(V_{2},g)>\operatorname{width}(M_{2},g_{\varphi_{2}})$,
we conclude that

$\min\\{\operatorname{width}(V_{1},g),\operatorname{width}(V_{3},g)\\}<\ell,$

which is what we wanted to prove. ∎

###### Proof of Corollary 3.7.

Consider the function

$\varphi_{2}\colon\left(-\frac{\pi}{\sqrt{\kappa}n},\frac{\pi}{\sqrt{\kappa}n}\right)\to\mathbb{R}_{+}\qquad
t\mapsto\cos\left(\frac{\sqrt{\kappa}nt}{2}\right)^{\frac{2}{n}},$

which is strictly $\log$-concave and has

$h_{\varphi_{2}}(t)=-\sqrt{\kappa}(n-1)\tan\left(\frac{\sqrt{\kappa}nt}{2}\right).$

Consider the function

$\varphi_{1}\colon\mathbb{R}_{+}\to\mathbb{R}_{+}\qquad
t\mapsto\sinh\left(\frac{\sqrt{\sigma n}t}{2\sqrt{n-1}}\right)^{\frac{2}{n}},$

which is strictly $\log$-concave and has

$h_{\varphi_{1}}(t)\coloneqq\sqrt{\frac{\sigma(n-1)}{n}}\coth\left(\frac{\sqrt{\sigma
n}t}{2\sqrt{n-1}}\right).$

Since $h_{\varphi_{1}}(t)\to\infty$ as $t\to 0$, there is a value $t_{-}>0$
such that the mean curvature $H(\partial_{-}X,g)$ is greater or equal to
$-h_{\varphi_{1}}(t_{-})$. By continuity, there are $\delta_{1},\delta_{2}>0$
small enough such that $-\sigma>-\kappa
n(n-1)\tan\left(\frac{\sqrt{\kappa}n(d-2\delta_{1})}{4}\right)^{2}$ and

$h_{\varphi_{2}}\left(\frac{-d+\delta_{1}}{2}\right)=h_{\varphi_{1}}(\ell+\delta_{2}),$

while $\delta_{2}<t_{-}$ and hence $\ell+\delta_{2}-t_{-}<\ell$. Let
$(N,g_{N})$ be a closed scalar flat Riemannian manifold. We fix the model
space

$(M_{1},g_{\varphi_{1}})=(N\times[t_{-},\ell+\delta_{2}],\varphi_{1}^{2}(t)g_{N}+dt^{2})$

with scalar curvature equal to $-\sigma$ and $\operatorname{width}<\ell$.

Let $\varphi_{3}\colon\mathbb{R}_{-}\to\mathbb{R}_{+}$ be defined by
$\varphi_{3}(t)=\varphi_{1}(-t)$. This function is strictly $\log$-concave and
$h_{\varphi_{3}}(t)=-h_{\varphi_{1}}(-t)$. Since
$h_{\varphi_{3}}(t)\to-\infty$ as $t\to 0$, there is a value $t_{+}<0$ such
that $H(\partial_{X},g)\geq h_{\varphi_{3}}(t_{+})$. Similarly as before, we
find $\delta_{3},\delta_{4}>0$ such that $-\sigma>-\kappa
n(n-1)\tan\left(\frac{\sqrt{\kappa}n(d-2\delta_{3})}{4}\right)^{2}$ and

$h_{\varphi_{2}}\left(\frac{d-\delta_{3}}{2}\right)=h_{\varphi_{3}}(-\ell-\delta_{4}),$

while $\delta_{4}<-t_{+}$ and hence $\ell+\delta_{4}+t_{+}<\ell$. We fix the
model space

$(M_{3},g_{\varphi_{3}})=(N\times[-\ell-\delta_{4},t_{+}],\varphi_{3}^{2}(t)g_{N}+dt^{2})$

with scalar curvature equal to $-\sigma$ and $\operatorname{width}<\ell$.

Finally we fix the model space

$(M_{2},g_{\varphi_{2}})=\left(N\times\left[\frac{-d+\delta_{1}}{2},\frac{d-\delta_{3}}{2}\right],\varphi_{2}^{2}(t)g_{N}+dt^{2}\right)$

with scalar curvature equal to $\kappa n(n-1)$ and $\operatorname{width}<d$.

It follows from Theorem 3.5 that
$\operatorname{width}(V_{j},g)\leq\operatorname{width}(M_{j},g_{\varphi_{j}})$
for at least one $i\in\\{1,2,3\\}$. Since
$d=\operatorname{width}(V_{2},g)>\operatorname{width}(M_{2},g_{\varphi_{2}})$,
we conclude that

$\min\\{\operatorname{width}(V_{1},g),\operatorname{width}(V_{3},g)\\}<\ell,$

which is what we wanted to prove. ∎

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Giu84Giusti1984:Minimal Gro18Gromov:MetricInequalitiesScalar
Ka82Kazdan1982:Deformations Rae21raede2021scalar RoS94Rosenberg-
Stolz:Manifolds-of-psc