# Complex hyperbolic Gromov-Thurston metrics and almost $\frac{1}{4}$-pinched
Kähler manifolds

Barry Minemyer  Department of Mathematical and Digital Sciences, Commonwealth
University - Bloomsburg Campus, Bloomsburg, Pennsylvania 17815
<EMAIL_ADDRESS>
(Date: .)

###### Abstract.

In this paper we construct an almost negatively $\frac{1}{4}$-pinched
Riemannian metric on a class of compact manifolds that, via previous work, was
already known to be Kähler and not locally symmetric. This is the first known
example of such manifolds and, via the result of Hernandez [7] and Yau and
Zheng [14], these manifolds cannot admit a negatively quarter-pinched
Riemannian metric. This metric is also interesting because it is the complex
hyperbolic analogue to the famous pinched metric constructed by Gromov and
Thurston in [6].

##  1\. Introduction

Via Mostow rigidity [10] one knows that any compact Kähler manifold with
negatively $\frac{1}{4}$-pinched holomorphic sectional curvature (meaning the
curvatures of all complex lines lie in the interval $[-1,-1/4]$) must be
isometric to a quotient of the complex hyperbolic space
$\mathbb{C}\mathbb{H}^{n}$. Even more surprisingly, by Hernandez [7] and Yau
and Zheng [14] it holds that any Kähler manifold with a negatively
$\frac{1}{4}$-pinched Riemannian metric must likewise be isometric to a
locally symmetric manifold.

Mostow rigidity of course holds in the hyperbolic case, but there are now
various examples of Riemannian manifolds whose sectional curvatures are
pinched arbitrarily close to $-1$, but which are not homotopy equivalent to a
locally symmetric manifold. The first known examples were the branched cover
manifolds constructed by Gromov and Thurston in [6], but one now obtains many
more examples from Ontaneda’s smooth hyperbolization [12].

Very little is known about pinched metrics in the Kähler setting. If a Kähler
metric has holomorphic sectional curvatures that are $\varepsilon$-close to
being negatively $\frac{1}{4}$-pinched, meaning that all curvatures lie in the
$\varepsilon$ neighborhood of $[-1,-1/4]$ for some sufficiently small
$\varepsilon>0$, then the underlying manifold must be “close” to being locally
symmetric in a sense made explicit by Dereax and Seshadri in [4]. But it is
unknown if any such manifolds exist (outside of the locally symmetric ones).
In light of the results in [7] and [14], it is natural to ask if there are
even any Kähler manifolds which admit an almost negatively
$\frac{1}{4}$-pinched Riemannian metric but are not locally symmetric? The
main result of this paper is the construction of the first known examples in
this latter category.

######  Theorem 1.1.

For any $\varepsilon>0$, there exist Kähler manifolds that admit a Riemannian
metric that is $\varepsilon$-close to being negatively $\frac{1}{4}$-pinched
and are not homotopy equivalent to a locally symmetric manifold.

A few quick remarks about the Theorem. As mentioned in the abstract, the
Riemannian metric in Theorem 1.1 is the complex hyperbolic analogue to the
famous Gromov-Thurston metric in [6]. The Riemannian metric from the Theorem
is generally not Kähler. Also, for the remainder of this paper we use the
interval $(-4-\varepsilon,-1+\varepsilon)$ instead of
$(-1-\varepsilon,-1/4+\varepsilon)$ purely for notational convenience. One can
adjust the curvatures between these two intervals by appropriately scaling the
metric.

The new construction in this paper is the almost negatively
$\frac{1}{4}$-pinched metric in Theorem 1.1. The remainder of the Theorem can
be obtained from the literature as follows. Following Stover and Toledo in
[13], let $\Gamma<\text{Isom}(\mathbb{C}\mathbb{H}^{n})$ be a cocompact
congruence arithmetic lattice of simple type, and let
$M=\Gamma\setminus\mathbb{C}\mathbb{H}^{n}$. It is known that such a manifold
$M$ contains an immersed (complex) codimension one totally geodesic
submanifold $N$. One can then find a finite congruence cover
$(M^{\prime},N^{\prime})$ of $(M,N)$ with $N^{\prime}$ embedded in
$M^{\prime}$. Let
$M^{\prime}=\Gamma^{\prime}\setminus\mathbb{C}\mathbb{H}^{n}$. Our almost
$\frac{1}{4}$-pinched Riemannian metric will require a large normal
injectivity radius about the totally geodesic submanifold. Suppose that, for a
fixed $\varepsilon>0$, we require a normal injectivity radius of $R$. By [5]
$M^{\prime}$ contains only finitely many homotopy classes of closed geodesics
with length less than $R$. Since $\Gamma^{\prime}$ is residually finite, we
can pass to a finite index subgroup $\Gamma^{\prime\prime}$ of
$\Gamma^{\prime}$ where we remove elements of $\Gamma^{\prime}$ that
correspond to each of these closed geodesics. In the resulting pair
$(M^{\prime\prime},N^{\prime\prime})$, $N^{\prime\prime}$ will still be
embedded and will now have normal injectivity radius of at least $R$. Let us
quickly note that this is the only place where we require $M$ to be compact.
If the submanifold $N$ can be constructed to have a sufficiently large normal
injectivity radius, then the ambient manifold $M$ in this construction need
not be compact.

By Theorem 5.1 of [13] there exists an integer $d>2$ and a further finite
cover $(\tilde{M},\tilde{N})$ of $(M^{\prime\prime},N^{\prime\prime})$ such
that the $d$-fold ramified branched cover $X$ of $\tilde{M}$ about $\tilde{N}$
is a smooth manifold. This manifold $X$ is the manifold described in Theorem
1.1. Note that $\tilde{N}$ will still have a normal injectivity radius of at
least $R$ within $\tilde{M}$, and therefore the ramification locus will still
have a normal injectivity radius of at least $R$ in $X$. Stover and Toledo in
[13] also prove that $X$ is not homotopy equivalent to a locally symmetric
manifold (Theorem 1.5 in [13]). The fact that $X$ admits a Kähler metric is
due to Zheng in [15].

The metric constructed by Zheng in [15] actually has negative holomorphic
sectional curvature (and thus all sectional curvatures are negative), but this
metric generally cannot be altered to be almost $\frac{1}{4}$-pinched. Since
this metric is Kähler, if it were close to $\frac{1}{4}$-pinched then the
ratio of the Chern numbers of manifolds which contain this metric would be
close to the same ratios of complex hyperbolic manifolds by [4]. But Zheng’s
metric can be applied to the surfaces developed by Hirzebruch in [8].
Hirzebruch computes formulas for the ratio of the two Chern numbers for these
surfaces, and for $n\neq 1$ they differ from that of a complex hyperbolic
manifold (see also Section 3 of [15]). Zheng’s metric is the regular complex
hyperbolic metric outside of a small tubular neighborhood of the ramification
locus of the branched cover, and has a small additive “fix” within this
tubular neighborhood. But, for the degree $d$ large, any hope of constructing
a metric with pinched curvature would require one to fix a large angle defect
in a small interval while keeping tight control of the sectional curvature
tensor. As such, the author believes that any almost $\frac{1}{4}$-pinched
metric on $X$ will require a modification on a large normal injectivity radius
of the ramification locus.

Lastly, Stover and Toledo do not compute the Chern numbers of $X$ when proving
that $X$ is not homotopy equivalent to a locally symmetric manifold. They
instead give an argument based on the normal bundle of the ramification locus.
So it is unknown if the bounds on the difference between the ratio of Chern
numbers of $X$ and of a complex hyperbolic manifold developed in [4] are
satisfied. It seems likely to the author that there exists an example of a
branched cover $X$ which satisfies Theorem 1.1 but whose ratio of chern
numbers is not close to that of a complex hyperbolic manifold, but again this
is not known.

### Outline of the construction of the metric $g$

In what follows, all constructions are at the level of the universal cover.
The constructed metric will descend to the manifold $X$ as long as we end up
with the pullback metric by the end of the normal injectivity radius of the
ramification locus. We also assume that the ramification locus is connected,
and thus is a copy of $\mathbb{C}\mathbb{H}^{n-1}$ in the universal cover. If
it is disconnected, then we just apply the constructed metric about each
component.

Let us quickly review the metric constructed by Gromov and Thurston in [6].
Let

$\lambda=h^{2}\mathfrak{h}_{n-2}+v^{2}d\theta^{2}+dr^{2}.$

where $\mathfrak{h}_{n-2}$ denotes the hyperbolic metric on the core copy of
$\mathbb{H}^{n-2}$. The $h^{2}\mathfrak{h}_{n-2}$ component of $\lambda$ is
the horizontal fiber and is tangent to the $r$-tube about the base copy of
$\mathbb{H}^{n-2}$. It is well-known that $\lambda$ is the hyperbolic metric
$\mathfrak{h}_{n}$ on $\mathbb{H}^{n}$ written in polar coordinates about a
totally geodesic codimension two copy of $\mathbb{H}^{n-2}$ if we choose the
warping functions $h(r)=\cosh(r)$ and $v(r)=\sinh(r)$. If we let
$h(r)=\cosh(r)$ and $v(r)=d\sinh(r)$ then $\lambda$ will be the pullback
metric on the entire branched cover, which will be everywhere Riemannian
except on the ramification locus. At these points the metric will have a cone
angle of $2d\pi$.

If one chooses an orthonormal basis for $\lambda$ at any point and computes
the components of the curvature tensor (Theorem 2.1 below), one first sees
that all mixed terms vanish. In the curvature formulas for the coordinate
planes, every appearance of $v$ happens in the denominator of a fraction.
There is then a $v^{\prime}$ or a $v^{\prime\prime}$ in the numerator of each
of these fractions. So whether $v$ is equal to $\sinh(r)$ or $d\sinh(r)$, the
curvatures are still identically equal to $-1$. Gromov and Thurston then
define $v(r)=\sigma(r)\sinh(r)$ where $\sigma(r):\mathbb{R}\to\mathbb{R}$ is a
function that very slowly increases from $1$ to $d$. For $\sigma^{\prime}$ and
$\sigma^{\prime\prime}$ sufficiently small this produces a pinched negatively
curved metric which is hyperbolic about a tube of the ramification locus, and
which slowly warps to the full hyperbolic metric on each page of the branched
cover.

The metric that we construct on the $d$-fold branched cover of
$\mathbb{C}\mathbb{H}^{n}$ about a copy of $\mathbb{C}\mathbb{H}^{n-1}$ is
similar, but the situation is now much more complicated. The associated warped
product metric is

$\mu=h^{2}\mathfrak{c}_{n-1}+\frac{1}{4}v^{2}d\theta^{2}+dr^{2}.$

where $\mathfrak{c}_{n-1}$ denotes the complex hyperbolic metric on the core
copy of $\mathbb{C}\mathbb{H}^{n-1}$. A major difference between the
hyperbolic and complex hyperbolic metrics is that the horizontal fiber is now
not integrable, meaning that it is not everywhere tangent to the $r$-tube
about $\mathbb{C}\mathbb{H}^{n-1}$. One obtains the complex hyperbolic metric
$\mathfrak{c}_{n}$ with the substitutions $h(r)=\cosh(r)$ and $v(r)=\sinh(2r)$
in the definition of $\mu$. The non-integrability of this metric leads to
nonzero structure constants (equation 2.3 below), which create (among other
issues) non-zero mixed terms for the sectional curvature tensor (Theorem 2.2
below).

There now exist terms in various components of the curvature tensor where $v$
appears without a corresponding $v^{\prime}$ or $v^{\prime\prime}$. But each
of these terms contains a structure constant. So the first part of our warping
process (Theorem 3.2 below) is to slowly “unwind” the complex hyperbolic
metric making all structure constants zero. This is equivalent to slowly
turning the horizontal fiber of the metric until it is tangent to the $r$-tube
about the core copy of $\mathbb{C}\mathbb{H}^{n-2}$. By turning this fiber
slowly we can keep the curvatures pinched near $[-4,-1]$. The almost
$\frac{1}{4}$-pinched metric in Theorem 1.1 is not Kähler, and this is the
part of the process that is the reason for this.

The resulting metric has an integrable horizontal fiber like the hyperbolic
metric, but the horizontal fiber is still a warped copy of
$\mathfrak{c}_{n-1}$ and the vertical fiber is still $\sinh(2r)$-warped. We
call this metric the integrable complex hyperbolic metric $g_{I}$. We can then
increase the angle with this metric in an almost identical way as to what was
described above with the Gromov-Thurston metric while keeping the curvature
pinched (Theorem 3.5). This leaves us with a copy of the metric $g_{I}$ on
each page of the branched cover.

The final step is to “rewind” the metric $g_{I}$ back to the complex
hyperbolic metric $\mathfrak{c}_{n}$ on each page of the branched cover
(Corollary 3.3). This process is exactly backwards of the unwinding procedure
discussed above. This results in a copy of the original complex hyperbolic
metric on each page of the branched cover, which will descend to a pinched
negatively curved metric on $X$.

### Acknowledgments

The author would first and foremost like to thank J.F. Lafont for mentioning
this project to the author many years ago and for pointing out several
references in the Introduction. The author has also worked on a related
project in the past with J. Meyer and B. Tshishiku and would like to thank
them for various comments during our group meetings that have been helpful
with writing this paper.

##  2\. Hyperbolic and complex hyperbolic metrics in polar coordinates

In this Section we review curvature formulas for $\mathbb{H}^{n}$ written in
terms of polar coordinates about a copy of $\mathbb{H}^{n-2}$, and we give a
slightly new version of the curvature formulas for $\mathbb{C}\mathbb{H}^{n}$
written in polar coordinates about a copy of $\mathbb{C}\mathbb{H}^{n-1}$. We
then prove a key “Pinching Lemma” which will be very important when we want to
prove that our metric that “unwinds” the complex hyperbolic metric has almost
$\frac{1}{4}$-pinched curvature. The $(\mathbb{H}^{n},\mathbb{H}^{n-2})$ case
below is well known, but can certainly be found in [2] and [11]. Curvature
formulas for $(\mathbb{C}\mathbb{H}^{n},\mathbb{C}\mathbb{H}^{n-1})$ were
derived in [1] for curvatures in $[-1,-1/4]$, and converted to curvatures in
$[-4,-1]$ in [11]. For this work we will need a slightly different version of
these formulas, which we derive below.

###  2.1. The metric on $\mathbb{H}^{n}$ in polar coordinates about
$\mathbb{H}^{n-2}$

Let $\mathbb{H}^{n-2}$ denote a totally geodesic codimension two submanifold
in $\mathbb{H}^{n}$, let $r$ denote the distance to $\mathbb{H}^{n-2}$ within
$\mathbb{H}^{n}$, and let $\mathfrak{h}_{n}$, $\mathfrak{h}_{n-2}$ denote the
hyperbolic metrics on $\mathbb{H}^{n}$ and $\mathbb{H}^{n-2}$, respectively.
Let $\phi_{h}:\mathbb{H}^{n}\to\mathbb{H}^{n-2}$ denote the orthogonal
projection onto $\mathbb{H}^{n-2}$, and let $E_{h}(r)$ denote the $r$-tube
about $\mathbb{H}^{n-2}$. Since $\mathbb{H}^{n-2}$ is contractible we know
that topologically $E_{h}(r)\cong\mathbb{R}^{n-2}\times\mathbb{S}^{1}$. The
metric for $\mathbb{H}^{n}$ in polar coordinates about $\mathbb{H}^{n-2}$ is
given by

(2.1)
$\mathfrak{h}_{n}=\cosh^{2}(r)\mathfrak{h}_{n-2}+\sinh^{2}(r)d\theta^{2}+dr^{2}$

where $d\theta^{2}$ denotes the standard metric on the unit circle
$\mathbb{S}^{1}$. Note that the metric in equation (2.1) is defined on
$E\times(0,\infty)$, where $E$ is an arbitrary $r$-tube as defined above.

A key feature of the real hyperbolic metric is the following. Let
$p\in\mathbb{H}^{n-2}$ and let
$\check{X}_{1},\check{X}_{2},\ldots,\check{X}_{n-2}$ be a local orthonormal
frame near $p$ in $\mathbb{H}^{n-2}$ satisfying that
$[\check{X}_{i},\check{X}_{j}]_{p}=0$ for all $i,j$. Let $q\in\mathbb{H}^{n}$
be such that $\phi_{h}(q)=p$. Extend the collection
$(\check{X}_{i})_{i=1}^{n-2}$ to vector fields $X_{1},X_{2},\ldots,X_{n-2}$
defined near $q$ in $\mathbb{H}^{n}$ via $d\phi_{h}^{-1}$, which are
orthogonal to both $\frac{\partial}{\partial\theta}$ and
$\frac{\partial}{\partial r}$. Then the key property is that
$[X_{i},X_{j}]_{q}=0$ for all $i,j$, or equivalently that the distribution
determined by $(X_{i})_{i=1}^{n-2}$ is integrable.

Let $v(r)$ and $h(r)$ be positive real-valued functions of $r$. Define the
warped-product metric $\lambda:=\lambda_{v,h}$ on $E\times(0,\infty)$ by

$\lambda=h^{2}\mathfrak{h}_{n-2}+v^{2}d\theta^{2}+dr^{2}.$

Of course, when $v=\sinh(r)$ and $h=\cosh(r)$ we recover the hyperbolic metric
$\mathfrak{h}_{n}$. Fix vector fields $(X_{i})_{i=1}^{n-2}$ as above. Let
$X_{n-1}=\frac{\partial}{\partial\theta}$ and $X_{n}=\frac{\partial}{\partial
r}$. Define the following orthonormal frame for $\lambda$:

$Y_{i}=\frac{1}{h}X_{i}\;\text{ for }1\leq i\leq n-2\hskip
30.0ptY_{n-1}=\frac{1}{v}X_{n-1}\hskip 30.0ptY_{n}=X_{n}.$

Formulas for the components of the $(4,0)$ curvature tensor $R_{\lambda}$ for
$\lambda$ are given by the following Theorem.

######  Theorem 2.1 (c.f. Section 2 of [2]).

Let $R^{\lambda}_{i,j,k,l}:=\lambda(R_{\lambda}(Y_{i},Y_{j})Y_{k},Y_{l})$. Up
to the symmetries of the curvature tensor, the only nonzero components of the
(4,0) curvature tensor $R_{\lambda}$ are the following:

$\displaystyle
R^{\lambda}_{i,j,i,j}=-\frac{1}{h^{2}}-\left(\frac{h^{\prime}}{h}\right)^{2}\hskip
40.0ptR^{\lambda}_{i,n-1,i,n-1}=-\frac{h^{\prime}v^{\prime}}{hv}$
$\displaystyle R^{\lambda}_{i,n,i,n}=-\frac{h^{\prime\prime}}{h}\hskip
80.0ptR^{\lambda}_{n-1,n,n-1,n}=-\frac{v^{\prime\prime}}{v}$

where $1\leq i,j\leq n-2$.

One easily checks that plugging in the values $v(r)=\sinh(r)$ and
$h(r)=\cosh(r)$ gives all sectional curvatures of $-1$.

###  2.2. The metric on $\mathbb{C}\mathbb{H}^{n}$ in polar coordinates about
$\mathbb{C}\mathbb{H}^{n-1}$

Let $\mathbb{C}\mathbb{H}^{n-1}$ denote a codimension one (real codimension
two) complex submanifold in $\mathbb{C}\mathbb{H}^{n}$, let $r$ denote the
distance to $\mathbb{C}\mathbb{H}^{n-1}$ within $\mathbb{C}\mathbb{H}^{n}$,
and let $\mathfrak{c}_{n}$ and $\mathfrak{c}_{n-1}$ denote the metrics on
$\mathbb{C}\mathbb{H}^{n}$ and $\mathbb{C}\mathbb{H}^{n-1}$ normalized to have
constant holomorphic curvature $-4$. Let
$\phi_{c}:\mathbb{C}\mathbb{H}^{n}\to\mathbb{C}\mathbb{H}^{n-1}$ denote the
orthogonal projection onto $\mathbb{C}\mathbb{H}^{n-1}$, and let $E_{c}(r)$
denote the $r$-tube about $\mathbb{C}\mathbb{H}^{n-1}$. Again, since
$\mathbb{C}\mathbb{H}^{n-1}$ is contractible, topologically one has that
$E_{c}(r)\cong\mathbb{R}^{2n-2}\times\mathbb{S}^{1}$. The metric in
$\mathbb{C}\mathbb{H}^{n}$ in polar coordinates about
$\mathbb{C}\mathbb{H}^{n-1}$ is then given by

(2.2)
$\mathfrak{c}_{n}=\cosh^{2}(r)\mathfrak{c}_{n-1}+\frac{1}{4}\sinh^{2}(2r)d\theta^{2}+dr^{2}.$

Note that the presence of the $1/4$ in the $d\theta^{2}$ term is so that
$\mathfrak{c}_{n}$ is complete, or equivalently so that $\mathfrak{c}_{n}$ has
total angle of $2\pi$ about the core copy of $\mathbb{C}\mathbb{H}^{n-1}$.

Let $p\in\mathbb{C}\mathbb{H}^{n-1}$. We define a special basis
$(\check{X}_{1},\check{X}_{2},\ldots,\check{X}_{2n-2})$ of
$T_{p}\mathbb{C}\mathbb{H}^{n-1}$, which we call a holomorphic basis near $p$,
as follows. We first define $\check{X}_{1}$ to be any unit vector in
$T_{p}\mathbb{C}\mathbb{H}^{n-1}$. We then define
$\check{X}_{2}=J\check{X}_{1}$, where $J$ denotes the complex structure on
$\mathbb{C}\mathbb{H}^{n}$. We call such a pair $\\{\check{X},J\check{X}\\}$ a
holomorphic pair. Let $\check{X}_{3}$ be any unit vector in
$T_{p}\mathbb{C}\mathbb{H}^{n-1}$ which is orthogonal to
$\text{span}(\check{X}_{1},\check{X}_{2})$, and let
$\check{X}_{4}=J\check{X}_{3}$. We continue in this way to construct the
entire orthonormal basis
$(\check{X}_{1},\check{X}_{2},\ldots,\check{X}_{2n-2})$ of
$T_{p}\mathbb{C}\mathbb{H}^{n-1}$ in such a way that, for $i$ odd, the pair
$(\check{X}_{i},\check{X}_{i+1})$ is a holomorphic pair. Via a standard
construction, we can extend this basis to a neighborhood of $p$ in
$\mathbb{C}\mathbb{H}^{n-1}$ in such a way that
$[\check{X}_{i},\check{X}_{j}]_{p}=0$ for all $i$ and $j$. Let us note that if
$i$ is an odd integer then
$\text{exp}_{p}(\text{span}(\check{X}_{i},\check{X}_{i+1}))$ is a complex
line, whereas if $\\{\check{X}_{i},\check{X}_{j}\\}$ do not form a holomorphic
pair then $\text{exp}_{p}(\text{span}(\check{X}_{i},\check{X}_{j}))$ is a
totally real totally geodesic subspace of $\mathbb{C}\mathbb{H}^{n-1}$.

Let $q\in\mathbb{C}\mathbb{H}^{n}$ be such that $\phi_{c}(q)=p$. Extend the
collection $(\check{X}_{i})_{i=1}^{2n-2}$ to vector fields
$X_{1},X_{2},\ldots,X_{2n-2}$ defined near $q$ in $\mathbb{C}\mathbb{H}^{n}$
via $d\phi_{c}^{-1}$ in such a way that they are invariant with respect to
both $\theta$ and $r$. We will call such a frame a holomorphic frame near $q$.
Just as above, we need to understand the Lie brackets of this frame associated
to the metric in (2.2). It is proved in [1] that there exist structure
constants $c_{i}$ such that

(2.3) $[X_{i},X_{i+1}]=c_{i}\frac{\partial}{\partial\theta}\qquad\text{for
}i\text{ odd}$

and that $[X_{i},X_{j}]=0$ whenever $\\{X_{i},X_{j}\\}$ is not a holomorphic
pair. Moreover, in [1] it is actually proved111Belegradek obtains structure
constants of $\pm\dfrac{1}{2}$, but the structure constants here are $\pm 2$
due to the metric being scaled to have sectional curvatures in $[-4,-1]$
instead of $\displaystyle{\left[-1,\frac{-1}{4}\right]}$ that

(2.4) $c_{i}=\pm 2\qquad\forall i.$

In our curvature calculations later in the paper, we will need to understand
the complex hyperbolic metric with arbitrary structure constants. This is why
we cannot use the curvature calculations in [1] and why we keep $c_{i}$ in
equation (2.3).

Let $v(r)$ and $h(r)$ be positive real-valued functions of $r$. Define the
metric $\mu:=\mu_{v,h}$ on $E\times(0,\infty)$ by

(2.5) $\mu=h^{2}\mathfrak{c}_{n-1}+\frac{1}{4}v^{2}d\theta^{2}+dr^{2}.$

Note that when $v=\sinh(2r)$ and $h=\cosh(r)$ we recover the complex
hyperbolic metric $\mathfrak{c}_{n}$. Fix a holomorphic frame
$(X_{i})_{i=1}^{2n-2}$ as above. Let
$X_{2n-1}=\frac{\partial}{\partial\theta}$ and
$X_{2n}=\frac{\partial}{\partial r}$. Define the following orthonormal frame
for $\mu$:

(2.6) $Y_{i}=\frac{1}{h}X_{i}\;\text{ for }1\leq i\leq 2n-2\hskip
30.0ptY_{2n-1}=\frac{1}{\frac{1}{2}v}X_{2n-1}\hskip 30.0ptY_{2n}=X_{2n}.$

Formulas for the components of the $(4,0)$ curvature tensor $R_{\mu}$ of $\mu$
are given by the following Theorem.

######  Theorem 2.2 (compare Sections 7 and 8 of [1] with Theorem 4.3 of
[11]).

Let $R^{\mu}_{i,j,k,l}:=\mu(R_{\mu}(Y_{i},Y_{j})Y_{k},Y_{l})$, and let
$(X_{i})_{i=1}^{2n-2}$ be a holomorphic basis. Then, up to the symmetries of
the curvature tensor, we have the following formulas for the components of the
(4,0) curvature tensor $R_{\mu}$ with respect to the corresponding orthonormal
basis $(Y_{i})_{i=1}^{2n}$:

$\displaystyle
R^{\mu}_{i,j,i,j}=-\frac{1}{h^{2}}-\left(\frac{h^{\prime}}{h}\right)^{2}\hskip
40.0ptR^{\mu}_{i,2n-1,i,2n-1}=-\frac{h^{\prime}v^{\prime}}{hv}+\frac{c_{i}^{2}v^{2}}{16h^{4}}$
$\displaystyle
R^{\mu}_{i,i+1,i,i+1}=-\left(\frac{h^{\prime}}{h}\right)^{2}-\frac{4}{h^{2}}-\frac{3c_{i}^{2}v^{2}}{16h^{4}}$
$\displaystyle R^{\mu}_{i,2n,i,2n}=-\frac{h^{\prime\prime}}{h}\hskip
70.0ptR^{\mu}_{2n-1,2n,2n-1,2n}=-\frac{v^{\prime\prime}}{v}$ $\displaystyle
R^{\mu}_{i,i+1,2n-1,2n}=2R^{\mu}_{i,2n-1,i+1,2n}=-2R^{\mu}_{i,2n,i+1,2n-1}=-c_{i}\frac{v}{2h^{2}}\left(\ln\frac{v}{h}\right)^{\prime}$
$\displaystyle
R^{\mu}_{i,i+1,k,k+1}=2R^{\mu}_{i,k,i+1,k+1}=-2R^{\mu}_{i,k+1,i+1,k}=-\frac{2}{h^{2}}-\frac{c_{i}c_{k}v^{2}}{8h^{4}}$

where $1\leq i,j,k\leq 2n-2$, $k$ is an odd integer different from $i$, and
$j\neq i,i+1$. Also, any equations using both $i$ and $i+1$ assume that $i$ is
an odd integer.

###### Proof.

The proof here is analogous to Section 6 of [1]. We compute the $A$ and $T$
tensors of the metric

$\mu_{r}=h^{2}\mathfrak{c}_{n-1}+\frac{1}{4}v^{2}d\theta^{2}.$

We then use Theorem 9.28 of [3] to compute formulas for the components of the
sectional curvature tensor $R_{\mu_{r}}$ of $\mu_{r}$. Finally, we use the
equations in Appendix B of [1] to obtain the components of $R_{\mu}$.

First, by identical reasoning to [1], the fibers of the Riemannian submersion
are totally geodesic. Thus, the $T$ tensor is identically zero. We now compute
the $A$ tensor. For this, we fix the notation that $i$ is an odd integer and
$j\neq i,i+1$.

By ([3] eqn. 9.24) we have that

$A_{X_{i}}X_{i+1}=\frac{1}{2}\mathscr{V}[X_{i},X_{i+1}]=\frac{c_{i}}{2}X_{2n-1}\hskip
20.0pt\text{and}\hskip 20.0ptA_{X_{i}}X_{j}=0.$

Then by ([3] eqn. 9.21d), we have

$\displaystyle\mu_{r}(A_{X_{i}}X_{2n-1},X_{i+1})$
$\displaystyle=-\mu_{r}(A_{X_{i}}X_{i+1},X_{2n-1})$
$\displaystyle=-\frac{c_{i}}{2}\mu_{r}(X_{2n-1},X_{2n-1})=-\frac{c_{i}}{8}v^{2}.$

In a similar manner we see that all other components of $A_{X_{i}}X_{2n-1}$
are zero. Thus, we have that

$A_{X_{i}}X_{2n-1}=-\frac{c_{i}v^{2}}{8h^{2}}X_{i+1}\hskip
20.0pt\text{and}\hskip
20.0ptA_{X_{i+1}}X_{2n-1}=\frac{c_{i}v^{2}}{8h^{2}}X_{i}.$

We are now ready to compute the components of $R_{\mu_{r}}$ in terms of the
basis $(Y_{i})_{i=1}^{2n-1}$. In the following calculations we use Theorem
9.28 from [3], as well as the linearity of the curvature tensor. Also, in the
fourth bullet point, $j$ denotes an odd integer different from $i$.

$\displaystyle\bullet\;\mu_{r}($ $\displaystyle
R_{\mu_{r}}(X_{i},X_{2n-1})X_{i},X_{2n-1})=\mu_{r}(A_{X_{i}}X_{2n-1},A_{X_{i}}X_{2n-1})=\frac{c_{i}^{2}v^{4}}{64h^{4}}\cdot
h^{2}=\frac{c_{i}^{2}v^{4}}{64h^{2}}$
$\displaystyle\implies\;\mu_{r}(R_{\mu_{r}}(Y_{i},Y_{2n-1})Y_{i},Y_{2n-1})=\frac{4}{h^{2}v^{2}}\cdot\frac{c_{i}^{2}v^{4}}{64h^{2}}=\frac{c_{i}^{2}v^{2}}{16h^{4}}.$
$\displaystyle\bullet\;\mu_{r}($ $\displaystyle
R_{\mu_{r}}(X_{i},X_{i+1})X_{i},X_{i+1})=\mu_{r}(\check{R}_{\mu_{r}}(\check{X}_{i},\check{X}_{i+1})\check{X}_{i},\check{X}_{i+1})-3\mu_{r}(A_{X_{i}}X_{i+1},A_{X_{i}}X_{i+1})$
$\displaystyle=-4h^{2}-\frac{3c_{i}^{2}v^{2}}{16}$
$\displaystyle\implies\;\mu_{r}(R_{\mu_{r}}(Y_{i},Y_{i+1})Y_{i},Y_{i+1})=\frac{1}{h^{4}}\cdot\left(-4h^{2}-\frac{3c_{i}^{2}v^{2}}{16}\right)=-\frac{4}{h^{2}}-\frac{3c_{i}^{2}v^{2}}{16h^{4}}.$
$\displaystyle\bullet\;\mu_{r}($ $\displaystyle
R_{\mu_{r}}(X_{i},X_{j})X_{i},X_{j})=\mu_{r}(\check{R}_{\mu_{r}}(\check{X}_{i},\check{X}_{j})\check{X}_{i},\check{X}_{j})-3\mu_{r}(A_{X_{i}}X_{j},A_{X_{i}}X_{j})=-h^{2}$
$\displaystyle\implies\;\mu_{r}(R_{\mu_{r}}(Y_{i},Y_{j})Y_{i},Y_{j})=\frac{1}{h^{4}}\cdot(-h^{2})=-\frac{1}{h^{2}}.$
$\displaystyle\bullet\;\mu_{r}($ $\displaystyle
R_{\mu_{r}}(X_{i},X_{i+1})X_{j},X_{j+1})=\mu_{r}(\check{R}_{\mu_{r}}(\check{X}_{i},\check{X}_{i+1})\check{X}_{j},\check{X}_{j+1})-2\mu_{r}(A_{X_{i}}X_{i+1},A_{X_{j}}X_{j+1})$
$\displaystyle=-2h^{2}-2\left(\frac{c_{i}c_{j}}{4}\cdot\frac{v^{2}}{4}\right)=-2h^{2}-\frac{c_{i}c_{j}v^{2}}{8}$
$\displaystyle\implies\;\mu_{r}(R_{\mu_{r}}(Y_{i},Y_{i+1})Y_{j},Y_{j+1})=\frac{1}{h^{4}}\cdot\left(-2h^{2}-\frac{c_{i}c_{j}v^{2}}{8}\right)=-\frac{2}{h^{2}}-\frac{c_{i}c_{j}v^{2}}{8h^{4}}.$

A few quick remarks about the above calculations. All of the curvatures with a
“hat” above the symbols occur in the horizontal fiber, which is isometric to
the $h^{2}$-multiple of $\mathfrak{c}_{n-1}$. One can use Proposition IX.7.3
from [9] to verify that the corresponding curvatures above in
$\mathfrak{c}_{n-1}$ are indeed $-4$, $-1$, and $-2$. One can also deduce from
Theorem 9.28 in [3] that
$R^{\mu_{r}}_{i+1,2n-1,i+1,2n-1}=R^{\mu_{r}}_{i,2n-1,i,2n-1}$, and that
$R^{\mu_{r}}_{i,i+1,j,j+1}=2R^{\mu_{r}}_{i,j,i+1,j+1}=-2R^{\mu_{r}}_{i,j+1,i_{1},j}$
(and recall that this subscript notation is just the components of the
curvature tensor with respect to the ON basis $(Y_{i})$).

Combining the above computations with the equations in Appendix B of [1]
proves the Theorem.

∎

###  2.3. The Pinching Lemma

The following Lemma is our main technical result which ensures that the metric
constructed below in Theorem 3.2 and Corollary 3.3 is almost negatively
$\frac{1}{4}$-pinched. In the statement below, $K_{g}(\sigma)$ denotes the
sectional curvature of the $2$-plane $\sigma$ with respect to the metric $g$.

######  Lemma 2.3 (The Pinching Lemma).

Let $g$ denote the metric $\mu$ from equation (2.5) on
$X=\mathbb{R}^{n-2}\times\mathbb{S}^{1}\times(0,\infty)$ with $h(r)=\cosh(r)$
and $v(r)=\sinh(2r)=2\sinh(r)\cosh(r)$. Let $q=(p,\theta,r)\in X$. Then for
all $\varepsilon>0$, there exists $R>0$ large and $\eta>0$ small such that for
all 2-planes $\sigma\subseteq T_{q}(X)$ where $r>R$, we have that
$K_{g}(\sigma)\in(-4-\varepsilon,-1+\varepsilon)$ provided that all structure
constants $c_{i}$ from equation (2.3) satisfy $c_{i}\in(-2-\eta,2+\eta)$.

###### Proof.

Let $\sigma\subseteq T_{q}(X)$ be a 2-plane. Note that, for almost all such
2-planes $\sigma$, we have that $d\phi_{c}(\sigma)$ has dimension 2 in the
base $T_{p}\mathbb{C}\mathbb{H}^{n-1}$. We will prove Lemma 2.3 in the case
when $d\phi_{c}(\sigma)$ has dimension 2, and the result then extends to all
2-planes by continuity. Also, we will prove the statement for
$c_{i}\in[-2,2]$. The result will also then extend to $(-2-\eta,2+\eta)$ for
some $\eta>0$ by continuity.

In a similar way as in Section 9 of [1], we will choose our holomorphic frame
near $q$ depending on the position of $\sigma$. Since $d\phi_{c}(\sigma)$ has
dimension 2, $\sigma$ must contain a unit vector $B$ that is tangent at $q$ to
the $r$-tube about $\mathbb{C}\mathbb{H}^{n-1}$. Choose $\check{X}_{1}$ to be
a unit-vector in $T_{p}\mathbb{C}\mathbb{H}^{n-1}$ that is parallel to
$d\phi_{c}(B)$. Let $\check{X}_{2}=J\check{X}_{1}$. Finally, let $A$ be a unit
vector in $T_{q}X$ perpendicular to $B$ such that $\sigma=\text{span}(A,B)$,
and choose a unit vector $\check{X}_{3}\in T_{p}\mathbb{C}\mathbb{H}^{n-1}$
perpendicular to both $\check{X}_{1}$ and $\check{X}_{2}$ so that
$d\phi_{c}(A)\in\text{span}(\check{X}_{1},\check{X}_{2},\check{X}_{3})$. We
then extend $\\{\check{X}_{1},\check{X}_{2},\check{X}_{3}\\}$ to a holomorphic
frame $\\{\check{X}_{i}\\}_{i=1}^{2n-2}$ near $p$ as in Subsection 2.2, and
pull this back via $d\phi^{-1}$ to a holomporphic frame $\\{X_{i}\\}$ near
$q$. Let $\displaystyle{X_{2n-1}=\frac{\partial}{\partial\theta}}$ and
$\displaystyle{X_{2n}=\frac{\partial}{\partial r}}$, and define the
corresponding orthonormal basis $(Y_{i})$ as in equation (2.6). Note that we
can write

$A=a_{1}Y_{1}+a_{2}Y_{2}+a_{3}Y_{3}+a_{4}Y_{2n-1}+a_{5}Y_{2n}\qquad
B=b_{1}Y_{1}+b_{4}Y_{2n-1}$

where

(2.7)
$a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}=1=b_{1}^{2}+b_{4}^{2}\quad\text{
and }\quad a_{1}b_{1}+a_{4}b_{4}=0.$

Letting $h(r)=\cosh(r)$ and $v(r)=\sinh(2r)=2\sinh(r)\cosh(r)$ in Theorem 2.2,
and using the subscripts for the basis chosen above gives the following
formulas for the components of the curvature tensor $R^{\mu}:=R$.

$\displaystyle R_{1313}=R_{2323}=-1\hskip
40.0ptR_{1,2n,1,2n}=R_{2,2n,2,2n}=R_{3,2n,3,2n}=-1$ $\displaystyle
R_{1212}=-\left(\frac{\sinh(r)}{\cosh(r)}\right)^{2}-\frac{4}{\cosh^{2}(r)}-\frac{3c_{1}^{2}\sinh^{2}(r)}{4\cosh^{2}(r)}\approx-1-\frac{3c_{1}^{2}}{4}$
$\displaystyle
R_{1,2n-1,1,2n-1}=R_{2,2n-1,2,2n-1}=-1-\frac{\sinh^{2}(r)}{\cosh^{2}(r)}+\frac{c_{1}^{2}\sinh^{2}(r)}{4\cosh^{2}(r)}\approx-2+\frac{c_{1}^{2}}{4}$
$\displaystyle R_{3,2n-1,3,2n-1}\approx-2+\frac{c_{3}^{2}}{4}\hskip
70.0ptR_{2n-1,2n,2n-1,2n}=-4$ $\displaystyle
R_{1,2,2n-1,2n}=2R_{1,2n-1,2,2n}=-2R_{1,2n,2,2n-1}=-c_{1}$

where “$\approx"$ is for $r$ sufficiently large.

The sectional curvature $K(\sigma)$ is $\sigma$ is then computed by

$\displaystyle K(\sigma)=$ $\displaystyle g(R(A,B)A,B)$ $\displaystyle=$
$\displaystyle
b_{1}^{2}g(R(A,Y_{1})A,Y_{1})+b_{4}^{2}g(R(A,Y_{2n-1})A,Y_{2n-1})+2b_{1}b_{4}g(R(A,Y_{1})A,Y_{2n-1})$
$\displaystyle=$ $\displaystyle
b_{1}^{2}\left(a_{2}^{2}R_{1212}+a_{3}^{2}R_{1313}+a_{4}^{2}R_{1,2n-1,1,2n-1}+a_{5}^{2}R_{1,2n,1,2n}\right)$
$\displaystyle+b_{4}^{2}\left(a_{1}^{2}R_{1,2n-1,1,2n-1}+a_{2}^{2}R_{2,2n-1,2,2n-1}+a_{3}^{2}R_{3,2n-1,3,2n-1}+a_{5}^{2}R_{2n-1,2n,2n-1,2n}\right)$
$\displaystyle+2b_{1}b_{4}\left(a_{2}a_{5}R_{1,2,2n-1,2n}-a_{1}a_{4}R_{1,2n-1,1,2n-1}-a_{2}a_{5}R_{1,2n,2,2n-1}\right)$

Plugging in the above curvature formulas gives

$\displaystyle K(\sigma)\approx$ $\displaystyle
b_{1}^{2}\left(a_{2}^{2}\left(-1-\frac{3c_{1}^{2}}{4}\right)-a_{3}^{2}+a_{4}^{2}\left(-2+\frac{c_{1}^{2}}{4}\right)-a_{5}^{2}\right)$
$\displaystyle+b_{4}^{2}\left(a_{1}^{2}\left(-2+\frac{c_{1}^{2}}{4}\right)+a_{2}^{2}\left(-2+\frac{c_{1}^{2}}{4}\right)+a_{3}^{2}\left(-2+\frac{c_{3}^{2}}{4}\right)-4a_{5}^{2}\right)$
$\displaystyle+2b_{1}b_{4}\left(-c_{1}a_{2}a_{5}-a_{1}a_{4}\left(-2+\frac{c_{1}^{2}}{4}\right)-a_{2}a_{5}\cdot\frac{c_{1}}{2}\right).$

Rearranging terms and applying the fact that $b_{1}^{2}+b_{4}^{2}=1$ yields

$\displaystyle K(\sigma)\approx$ $\displaystyle-
a_{2}^{2}-a_{3}^{2}-a_{5}^{2}-3a_{5}^{2}b_{4}^{2}-(a_{1}b_{4}-a_{4}b_{1})^{2}-a_{2}^{2}b_{1}^{2}\left(\frac{3c_{1}^{2}}{4}\right)$
$\displaystyle+\left(-1+\frac{c_{1}^{2}}{4}\right)\left[\left(a_{1}b_{4}+a_{4}b_{1}\right)^{2}+a_{2}^{2}b_{4}^{2}\right]-3c_{1}a_{2}a_{5}b_{1}b_{4}+a_{3}^{2}b_{4}^{2}\left(-1+\frac{c_{3}^{2}}{4}\right).$

Now, notice that since $0=a_{1}b_{1}+a_{4}b_{4}$, we have

$(a_{1}b_{4}-a_{4}b_{1})^{2}=(a_{1}b_{4}-a_{4}b_{1})^{2}+(a_{1}b_{1}+a_{4}b_{4})^{2}=a_{1}^{2}b_{4}^{2}+a_{4}^{2}b_{1}^{2}+a_{1}^{2}b_{1}^{2}+a_{4}^{2}b_{4}^{2}=a_{1}^{2}+a_{4}^{2}.$

Substituting this above and using that $a_{1}^{2}+\ldots+a_{5}^{2}=1$ gives

$\displaystyle K(\sigma)\approx$
$\displaystyle-1-3\left(a_{5}^{2}b_{4}^{2}+c_{1}a_{2}a_{5}b_{1}b_{4}+\frac{1}{4}c_{1}^{2}a_{2}^{2}b_{1}^{2}\right)+\left(-1+\frac{c_{1}^{2}}{4}\right)\left[\left(a_{1}b_{4}+a_{4}b_{1}\right)^{2}+a_{2}^{2}b_{4}^{2}\right]$
$\displaystyle+a_{3}^{2}b_{4}^{2}\left(-1+\frac{c_{3}^{2}}{4}\right)$
$\displaystyle=$
$\displaystyle-1-3\left(a_{5}b_{4}+\frac{1}{2}c_{1}a_{2}b_{1}\right)^{2}+\left(-1+\frac{c_{1}^{2}}{4}\right)\left[\left(a_{1}b_{4}+a_{4}b_{1}\right)^{2}+a_{2}^{2}b_{4}^{2}\right]$
$\displaystyle+a_{3}^{2}b_{4}^{2}\left(-1+\frac{c_{3}^{2}}{4}\right).$

This last expression is clearly negative. One can check that, under the
conditions in equation (2.7) and if $|c_{i}|\leq 2$ for $i=1,3$, this last
expression

  * •

is maximized when $c_{1}=\pm 2$, $c_{3}=\pm 2$, and
$\displaystyle{a_{5}b_{4}+\frac{1}{2}c_{1}a_{2}b_{1}=0}$. In this case, we
have that $K(\sigma)\approx-1$.

  * •

is minimized when $c_{1}=c_{3}=0$, $b_{4}=\pm 1$, and $a_{5}=\pm 1$. Note that
we then must have all other $a$’s and $b$’s equal to $0$. This then yields
$K(\sigma)\approx-4$.

∎

######  Remark 2.4.

Note that the condition “for $r$ sufficiently large” in Lemma 2.3 is really
only used in the proof above in order to simplify calculations. This is also
the only situation where we will need to use Lemma 2.3. But an inspection of
the formulas above shows that it is possible that this Theorem holds for all
positive values of $r$ and, at the very least, the value of $R$ in the
statement of the Theorem need not be very large.

##  3\. Two special metrics and almost pinched metrics that interpolate
between them

Fix $E=\mathbb{R}^{2n-2}\times\mathbb{S}^{1}$. In this Section we define two
metrics on the product $E\times(0,\infty)$ and prove that we can interpolate
between various metrics in an almost $\frac{1}{4}$-pinched way as discussed in
the Introduction (see Theorem 3.2, Corollary 3.3, and Theorem 3.5 below).

###  3.1. The integrable complex hyperbolic metric $g_{I}$

The integrable complex hyperbolic metric $g_{I}$ is defined by setting
$h(r)=\cosh(r)$ and $v(r)=\sinh(2r)=2\sinh(r)\cosh(r)$ in equation (2.5), and
setting all structure constants defined in (2.3) identically equal to zero.
This is the metric one would obtain if the complex hyperbolic metric were
integrable, that is, if the horizontal fiber
$d\phi_{c}^{-1}(T_{p}\mathbb{C}\mathbb{H}^{n-1})$ were always tangent to
$E_{c}(r)$ and orthogonal to $\phi_{c}^{-1}(p)$ as is the case with the
hyperbolic metric. So, as a metric, we have that

$g_{I}=\cosh^{2}(r)\mathfrak{c}_{n-1}+\frac{1}{4}\sinh^{2}(2r)d\theta^{2}+dr^{2}$

but with different structure constants than the metric $\mathfrak{c}_{n}$.

Setting $c_{i}=c_{k}=0$ in Theorem 2.2, as well as inserting $h=\cosh(r)$ and
$v=2\sinh(r)\cosh(r)$, proves the following Lemma.

######  Lemma 3.1.

Let $R$ denote the sectional curvature tensor with respect to the integrable
complex hyperbolic metric $g_{I}$. Then, using the notation of Theorem 2.2, we
have:

$\displaystyle R_{i,j,i,j}=-1\hskip
40.0ptR_{i,2n-1,i,2n-1}=-1-\tanh^{2}(r)\hskip 40.0ptR_{i,2n,i,2n}=-1$
$\displaystyle R_{i,i+1,i,i+1}=-1-\frac{3}{\cosh^{2}(r)}\hskip
40.0ptR_{2n-1,2n,2n-1,2n}=-4$ $\displaystyle
R_{i,i+1,2n-1,2n}=R_{i,2n-1,i+1,2n}=R_{i,2n,i+1,2n-1}=0$ $\displaystyle
R_{i,i+1,k,k+1}=2R_{i,k,i+1,k+1}=-2R_{i,k+1,i+1,k}=-\frac{2}{\cosh^{2}(r)}$

where $1\leq i,j,k\leq 2n-2$, $k$ is an odd integer different from $i$, and
$j\neq i,i+1$. Also, any equations using both $i$ and $i+1$ assume that $i$ is
an odd integer.

By Lemma 2.3 we know that, given $\varepsilon>0$, all sectional curvatures of
$g_{I}$ lie in $(-4-\varepsilon,-1+\varepsilon)$ for $r$ sufficiently large.

The next Theorem proves that there exists an almost $\frac{1}{4}$-pinched
metric that interpolates between the complex hyperbolic metric
$\mathfrak{c}_{n}$ and the metric $g_{I}$. This Theorem and the following
Corollary provide two of the three steps in the construction of the almost
$\frac{1}{4}$-pinched Riemannian metric in Theorem 1.1.

######  Theorem 3.2.

Let $\varepsilon>0$. Then there exist positive constants $r_{1}<r_{2}$ and a
Riemannian metric $g$ defined on $E\times(0,\infty)$ such that

  1. (1)

$g$ restricted to $E\times(0,r_{1})$ is the metric $\mathfrak{c}_{n}$.

  2. (2)

$g$ restricted to $E\times(r_{2},\infty)$ is the metric $g_{I}$.

  3. (3)

all sectional curvatures of $g$ lie in the interval
$(-4-\varepsilon,-1+\varepsilon)$.

###### Proof.

From Section 2 we know that both
$\mathbb{C}\mathbb{H}^{n}/\mathbb{C}\mathbb{H}^{n-1}$ and
$\mathbb{H}^{2n}/\mathbb{H}^{2n-2}$ are diffeomorphic to $E\times(0,\infty)$
where, for any
$p\in\mathbb{R}^{2n-2}\cong\mathbb{C}\mathbb{H}^{n}\cong\mathbb{H}^{2n}$ and
$r>0$, we have that $(p,\mathbb{S}^{1},r)$ corresponds to the intersection
$E_{h}(r)\cap\phi^{-1}_{h}(p)$ or $E_{c}(r)\cap\phi^{-1}_{c}(p)$. So, by
fixing diffeomorphisms $f_{h}:\mathbb{H}^{2n-2}\to\mathbb{R}^{2n-2}$ and
$f_{c}:\mathbb{C}\mathbb{H}^{n-1}\to\mathbb{R}^{2n-2}$, we can naturally
identify both $\mathbb{C}\mathbb{H}^{n}/\mathbb{C}\mathbb{H}^{n-1}$ and
$\mathbb{H}^{2n}/\mathbb{H}^{2n-2}$ with $E\times(0,\infty)$.

Via the diffeomorphisms $f_{h}$ and $f_{c}$ we will henceforth abuse notation
and apply all functions and metrics from Section 2 to the space
$E\times(0,\infty)$. For example, the metric $\mathfrak{c}_{n}$ will really
mean $(f_{c}^{-1})^{*}\mathfrak{c}_{n}$, and so on.

Let $p\in\mathbb{C}\mathbb{H}^{n-1}$ be arbitrary, and let
$(\check{X}_{1},\check{X}_{2},\ldots,\check{X}_{2n-2})$ be a holomorphic basis
at $p$. Extend this basis to a local frame near $p$ which satisfies that
$[\check{X}_{i},\check{X}_{j}]_{p}=0$ for all $i$ and $j$. Let $U$ be a
neighborhood of $p$ in $\mathbb{C}\mathbb{H}^{n-1}$ on which this frame is
defined.

Via the orthogonal projections $\phi_{h}$ and $\phi_{c}$ we will define two
sets of vector fields on the space $U\times\mathbb{S}^{1}\times(0,\infty)$.
Using $\phi_{h}^{-1}$ we pull the frame $(\check{X}_{i})$ back to obtain the
frame $(X^{h}_{i})$, and using $\phi_{c}^{-1}$ we similarly obtain the frame
$(X^{c}_{i})$.

Define a smooth function $\alpha:(0,\infty)\to[0,1]$ which satisfies

  * •

$\alpha(r)=1$ for all $r\leq r_{1}$.

  * •

$\alpha(r)=0$ for all $r\geq r_{2}$.

  * •

over the interval $(r_{1},r_{2})$ we have $\alpha^{\prime}(r)<\delta$ and
$\alpha^{\prime\prime}(r)<\delta$ for some prescribed $\delta>0$ which depends
on $\varepsilon$.

For any given $\delta>0$, such a function $\alpha$ clearly exists provided
that the distance $r_{2}-r_{1}$ is sufficiently large.

For $1\leq i\leq 2n-2$ we define the vector field $X_{i}$ on
$U\times\mathbb{S}^{1}\times(0,\infty)$ by

$X_{i}=\alpha X_{i}^{c}+(1-\alpha)X_{i}^{h}.$

Note that, since $\alpha$ is only a function of $r$ and since $X_{i}^{c}$ and
$X_{i}^{h}$ are both defined via the same vector field $\check{X}_{i}$, this
“unbending” process is independent of $p$ and $\theta$. We define the metric
$g$ to be

(3.1)
$g=\cosh^{2}(r)\left(dX_{1}^{2}+dX_{2}^{2}+\ldots+dX_{2n-2}^{2}\right)+\frac{1}{4}\sinh^{2}(2r)d\theta^{2}+dr^{2}$

where $dX_{i}$ is the covector field dual to $X_{i}$ for all $i$. Note that
the horizontal fiber of $g$ deforms from
$\text{span}(X_{1}^{c},X_{2}^{c},\ldots,X_{2n-2}^{c})$ to
$\text{span}(X_{1}^{h},X_{2}^{h},\ldots,X_{2n-2}^{h})$ in a manner that is
independent of the choice of our original frame since $\alpha$ only depends on
$r$. Thus, this metric is really well-defined on all of $E\times(0,\infty)$ .
Also, note that $g=\mathfrak{c}_{n}$ when $r<r_{1}$ and $g=g_{I}$ for
$r>r_{2}$.

To show that the metric $g$ is almost $\frac{1}{4}$-pinched, we need to
compute the Lie brackets of the basis $(X_{i})$ of the horizontal fiber. We
will show that, for $\delta$ sufficiently small and $r_{1}$ and $r_{2}-r_{1}$
sufficiently large, we can $C^{2}$-approximate $g$ by a metric which satisfies
Lemma 2.3. Then, since sufficiently $C^{2}$-small changes in a Riemannian
metric cause arbitrarily small changes to its sectional curvature, the proof
will be complete.

To start, note that all vector fields are invariant with respect to $\theta$,
and therefore $[X_{i},\frac{\partial}{\partial\theta}]=0$ for all $i$. Next,
consider all pairs $(\check{X}_{i},\check{X}_{j})$ that are not a holomorphic
pair (that is, where $\\{i,j\\}\neq\\{k,k+1\\}$ for some odd $k$). These
vector fields were chosen independently of each other, and so
$[X^{h}_{i},X^{c}_{j}]=0=[X^{c}_{i},X^{h}_{j}]$. From this it follows that
$[X_{i},X_{j}]=0$.

We now consider a holomorphic pair $(\check{X}_{i},\check{X}_{i+1})$. In what
follows we assume that the structure constant for this pair is $+2$ instead of
$-2$. If the sign is reversed, then we just swap the order of the indices. We
have

$\displaystyle[X_{i},$ $\displaystyle X_{i+1}]=[\alpha
X_{i}^{c}+(1-\alpha)X_{i}^{h},\alpha X_{i+1}^{c}+(1-\alpha)X_{i+1}^{h}]$
$\displaystyle=\alpha^{2}[X_{i}^{c},X_{i+1}^{c}]+\alpha(1-\alpha)\left([X_{i}^{c},X_{i+1}^{h}]+[X_{i}^{h},X_{i+1}^{c}]\right)+(1-\alpha)^{2}[X_{i}^{h},X_{i+1}^{h}]$
$\displaystyle=2\alpha^{2}\frac{\partial}{\partial\theta}+\alpha(1-\alpha)\left([X_{i}^{c},X_{i+1}^{h}]+[X_{i}^{h},X_{i+1}^{c}]\right)$

To analyze the mixed Lie brackets in this equation, write
$X_{i}^{h}=\sum_{j=1}^{2n-1}a_{i,j}X_{j}^{c}$ for some locally defined
coefficient functions $a_{i,j}$ and where
$X_{2n-1}^{c}=\frac{\partial}{\partial\theta}$. When we plug this sum into the
Lie bracket $[X_{i}^{h},X_{i+1}^{c}]$ and evaluate at $q=(p,\theta,r)$, the
only term that does not zero out is $[a_{i,i}X_{i}^{c},X_{i+1}^{c}]$ since we
chose our frame so that $[\check{X}_{i},\check{X}_{i+1}]_{p}=0$. Therefore,

$[X_{i}^{h},X_{i+1}^{c}]=a_{i,i}[X_{i}^{c},X_{i+1}^{c}]=2a_{i,i}\frac{\partial}{\partial\theta}.$

Note that, for $r$ large, we have

$\displaystyle\frac{1}{4}e^{2r}$
$\displaystyle\approx\mathfrak{h}_{2n}(X_{i}^{h},X_{i}^{h})\geq\mathfrak{h}_{2n}(a_{i,i+1}X_{i+1}^{c},a_{i,i+1}X_{i+1}^{c})$
$\displaystyle=(a_{i,i+1})^{2}\mathfrak{h}_{2n}(X_{i+1}^{c},X_{i+1}^{c})\approx(a_{i,i+1})^{2}\frac{1}{4}e^{2r}.$

and therefore $|a_{i,i}|\leq 1+\gamma$ for some $\gamma>0$ small. Analogously,
this all also holds for the Lie bracket $[X_{i}^{c},X_{i+1}^{h}]$ where we
write $X_{i+1}^{h}=\sum_{j=1}^{2n-1}b_{i+1,j}X_{j}^{c}$. We therefore have

$\displaystyle\bigr{|}[X_{i},X_{i+1}]\bigr{|}$
$\displaystyle=\biggr{|}2\alpha^{2}\frac{\partial}{\partial\theta}+\alpha(1-\alpha)\left(2b_{i+1,i}\frac{\partial}{\partial\theta}+2a_{i,i+1}\frac{\partial}{\partial\theta}\right)\biggr{|}$
$\displaystyle\leq|2\alpha^{2}+4\alpha(1-\alpha)+\eta|\biggr{|}\frac{\partial}{\partial\theta}\biggr{|}$

where $\eta$ is a small error term due to $\gamma$ above that can be made
arbitrarily small by choosing $r$ sufficiently large.

Note that, ignoring $\eta$, the coefficient within the absolute values on the
right-hand side of the inequality reduces to $4\alpha-2\alpha^{2}$. If we let
$b(r)=4\alpha(r)-2\alpha^{2}(r)$, we see that $b(r_{1})=2$, $b(r_{2})=0$, and
$b$ is decreasing over $[r_{1},r_{2}]$. Thus, the metric in equation (3.1)
satisfies the condition on the structure constants for Lemma 2.3. One can
check that, if the values for both $a_{i,i+1}$ and $b_{i+1,i}$ are both $-1$
(or any values for each between $-1$ and $1$), this coefficient still falls in
the interval $(-2-\eta,2+\eta)$ for sufficiently large $r$ and sufficiently
small $\gamma$.

The last Lie brackets that we need to consider are the Lie brackets of the
form $[X_{i},\partial r]$ where $\partial r=\frac{\partial}{\partial r}$. This
is easier to consider with respect to an orthonormal basis $(Y_{i})$ where
$Y_{i}=\frac{1}{\cosh(r)}X_{i}$ for all $1\leq i\leq 2n-2$. We then compute

$\displaystyle[Y_{i},\partial r]$ $\displaystyle=-\left[\partial
r,\frac{\alpha}{\cosh(r)}X_{i}^{c}+\frac{1-\alpha}{\cosh(r)}X_{i}^{h}\right]$
$\displaystyle=-\left(\frac{\alpha^{\prime}}{\cosh(r)}-\frac{\alpha\sinh(r)}{\cosh^{2}(r)}\right)X_{i}^{c}-\left(\frac{-\alpha^{\prime}}{\cosh(r)}-\frac{(1-\alpha)\sinh(r)}{\cosh^{2}(r)}\right)X_{i}^{h}$
$\displaystyle=\frac{\alpha^{\prime}}{\cosh(r)}(X_{i}^{h}-X_{i}^{c})+\tanh(r)Y_{i}.$

With respect to both the hyperbolic and complex hyperbolic metric one has that
$[Y_{i},\partial r]=\tanh(r)Y_{i}$. So the “error term” in the above summand
is $\frac{\alpha^{\prime}}{\cosh(r)}(X_{i}^{h}-X_{i}^{c})$. If this term were
equal to $0$, then Lemma 2.3 would apply and our proof would be complete. But
the vectors $X_{i}^{h}$ and $X_{i}^{c}$ grow at a rate of approximately
$\cosh(r)$ with respect to all of the metrics here. So this error term can be
made as small as possible by requiring that $\delta$ be chosen sufficiently
small. We can then $C^{2}$-approximate this metric by one which satisfies
Lemma 2.3 to complete the proof. ∎

######  Corollary 3.3.

Let $\varepsilon>0$. Then there exist positive constants $r_{3}<r_{4}$ and a
Riemannian metric $g$ defined on $E\times[r_{3},\infty)$ such that

  1. (1)

$g$ restricted to $E\times\\{r_{3}\\}$ is the metric $g_{I}$.

  2. (2)

$g$ restricted to $E\times(r_{4},\infty)$ is the metric $\mathfrak{c}_{n}$.

  3. (3)

all sectional curvatures of $g$ lie in the interval
$(-4-\varepsilon,-1+\varepsilon)$.

###### Proof.

Just apply the procedure in the proof of Theorem 3.2 backwards. Note that this
metric is only defined on $E\times[r_{3},\infty)$ for some $r_{3}>0$ large
since $g_{I}$ may not have almost pinched negative curvature for small values
of $r$. ∎

###  3.2. The $d$-fold integrable complex hyperbolic metric ${\bf g_{d}}$

Let $d\geq 2$ be a positive integer. The $d$-fold integrable complex
hyperbolic metric $g_{d}$ is defined as $h(r)=\cosh(r)$ and
$v(r)=d\sinh(2r)=2d\sinh(r)\cosh(r)$ in equation (2.5), and setting all
structure constants defined in (2.3) identically equal to zero. So as a
metric, we have that

$g_{d}=\cosh^{2}(r)\mathfrak{c}_{n-1}+\frac{d^{2}}{4}\sinh^{2}(2r)d\theta^{2}+dr^{2}.$

Note that this metric has a cone angle of $2d\pi$ about the core copy of
$\mathbb{C}\mathbb{H}^{n-1}$. For the metric $g_{d}$ to be complete, we need
to restrict it to $E\times(r_{0},\infty)$ for some $r_{0}>0$.

######  Remark 3.4.

The curvature formulas for $g_{d}$ are exactly the same as the curvature
formulas for $g_{I}$. When all structure constants are $0$ in Theorem 2.2,
then the extra “d” that is inserted in any denominator with a $v$ will cancel
with the same $d$ in the numerator that comes from either a $v^{\prime}$ or a
$v^{\prime\prime}$.

Therefore, by Lemma 2.3 we know that, given $\varepsilon>0$, all sectional
curvatures of $g_{d}$ lie in $(-4-\varepsilon,-1+\varepsilon)$ for $r$
sufficiently large.

######  Theorem 3.5.

Let $\varepsilon>0$. Then there exist positive constants $r_{2}<r_{3}$ and a
Riemannian metric $g$ defined on $E\times[r_{2},\infty)$ such that

  1. (1)

$g$ restricted to $E\times\\{r_{2}\\}$ is the metric $g_{I}$.

  2. (2)

$g$ restricted to $E\times(r_{3},\infty)$ is the metric $g_{d}$.

  3. (3)

all sectional curvatures of $g$ lie in the interval
$(-4-\varepsilon,-1+\varepsilon)$.

###### Proof.

This proof is very similar to the construction in Section 2 of [6]. Define a
function $\sigma:\mathbb{R}\to[1,d]$ which satisfies

  * •

$\sigma(r)=1$ for all $r\leq r_{2}$.

  * •

$\sigma(r)=d$ for all $r\geq r_{3}$.

  * •

$\sigma^{\prime}(r)<\delta$ and $\sigma^{\prime\prime}(r)<\delta$ for all $r$.

Given any fixed $\delta>0$, such a function $\sigma$ clearly exists provided
$r_{2}$ and $r_{3}-r_{2}$ are sufficiently large.

We now define the metric $g$ on $E\times[r_{2},\infty)$ to be the metric in
Theorem 2.2 with the substitutions $h(r)=\cosh(r)$ and
$v(r)=\sigma(r)\sinh(2r)=2\sigma(r)\sinh(r)\cosh(r)$. Using the notation of
Theorem 2.2, up to the symmetries of the curvature tensor, the nonzero
components of the sectional curvature tensor with respect to $g$ are

$\displaystyle R_{i,j,i,j}=-1\hskip 40.0ptR_{i,2n,i,2n}=-1$ $\displaystyle
R_{i,2n-1,i,2n-1}=-1-\tanh^{2}(r)-\frac{\sigma^{\prime}}{\sigma}\tanh(r)$
$\displaystyle R_{i,i+1,i,i+1}=-1-\frac{3}{\cosh^{2}(r)}\hskip
40.0ptR_{2n-1,2n,2n-1,2n}=-4-\frac{4\sigma^{\prime}}{\sigma}\coth(r)-\frac{\sigma^{\prime\prime}}{\sigma}$
$\displaystyle R_{i,i+1,2n-1,2n}=R_{i,2n-1,i+1,2n}=R_{i,2n,i+1,2n-1}=0$
$\displaystyle
R_{i,i+1,k,k+1}=2R_{i,k,i+1,k+1}=-2R_{i,k+1,i+1,k}=-\frac{2}{\cosh^{2}(r)}$

Since $\tanh(r)$ and $\coth(r)$ both approach $1$ as $r$ approaches $\infty$,
one sees that these curvature formulas approach those of the metric $g_{I}$
for $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ sufficiently small. That is,
for $\delta$ chosen sufficiently small. Thus, all sectional curvatures of $g$
will lie in the interval $(-4-\varepsilon,-1+\varepsilon)$ for appropriately
chosen $r_{2}$, $r_{3}$, and $\delta$. ∎

###### Proof of Theorem 1.1.

Via the discussion in the Introduction, all that is left to prove is the
construction of the metric $g$ on the $d$-fold ramified cover of
$\mathbb{C}\mathbb{H}^{n}$ about $\mathbb{C}\mathbb{H}^{n-1}$ with all
curvatures almost $\frac{1}{4}$-pinched. This really follows immediately from
Theorem 3.2, Theorem 3.5, and Corollary 3.3, but we give a quick rigorous
argument here.

Define $r_{1}$ as in Theorem 3.2. Choose $r_{2}$ to be the larger of those
from Theorem 3.2 and Theorem 3.5 and $r_{3}$ to be the larger of those from
Theorem 3.5 and Corollary 3.3. Finally, choose $r_{4}$ as in Corollary 3.3 as
needed for the chosen $r_{3}$. Let $g_{1}$ be the metric from Theorem 3.2, let
$g_{2}$ be the metric from Theorem 3.5, and let $g_{3}$ be the metric from
Corollary 3.3.

Define a metric $g$ on $E\times(0,r_{3}]$ by

$g=\begin{cases}\mathfrak{c}_{n}&\text{if }r\leq r_{1}\\\ g_{1}&\text{if
}r_{1}\leq r\leq r_{2}\\\ g_{2}&\text{if }r_{2}\leq r\leq r_{3}\end{cases}$

and note that $g$ is well-defined when $r=r_{1},r_{2}$, and that all sectional
curvatures of $g$ lie in the interval $(-4-\varepsilon,-1+\varepsilon)$.

At this point, the metric $g$ restricted to $E\times\\{r_{3}\\}$ has total
angle of $2d\pi$ about the core copy of $\mathbb{R}^{2n-2}$. Subdivide
$\mathbb{S}^{1}$ into $d$ equidistant arcs, and label them
$A_{1},\ldots,A_{d}$. Then for each $1\leq\ell\leq d$, the metric $g$
restricted to $\mathbb{R}^{2n-2}\times A_{\ell}\times\\{r_{3}\\}$ has total
angle $2\pi$ about $\mathbb{R}^{2n-2}$. Moreover, $g$ restricted to a sector
$A_{\ell}$ is equal to $g_{I}$ over all of $\mathbb{S}^{1}$. So to finish our
construction of $g$ we restrict our attention to $\mathbb{R}^{2n-2}\times
A_{\ell}\times[r_{3},\infty)$.

On $\mathbb{R}^{2n-2}\times A_{\ell}\times[r_{3},\infty)$ define $g=g_{3}$ for
all $1\leq\ell\leq d$. Note that this agrees with the previous definition for
$g$ on $E\times\\{r_{3}\\}$, and all sectional curvatures of $g$ remain in
$(-4-\varepsilon,-1+\varepsilon)$. As long as the ramification locus of $X$
has a normal injectivity radius of at least $r_{3}$, this metric $g$ will
descend to a well-defined Riemannian metric on $X$ that is $\varepsilon$-close
to being negatively $\frac{1}{4}$-pinched.

∎

## References

  * [1] I. Belegradek, Complex hyperbolic hyperplane complements, Math. Ann. 353 (2012), no. 2, 545-579.
  * [2] I. Belegradek, Rigidity and relative hyperbolicity of real hyperbolic hyperplane complements, Pure Appl. Math. Q. 8 (2012), no. 1, 15-51.
  * [3] A. L. Besse,  _Einstein manifolds_.  Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10, Springer-Verlag, Berlin (1987).
  * [4] M. Deraux and H. Seshadri, Almost quarter-pinched Kähler metrics and chern numbers, Proc. Amer. Math. Soc. 139 (2011), no. 7, 2571-2576.
  * [5] J.J. Duistermaat, J.A.C. Kolk, and V.S. Varadarajan, Spectra of compact locally symmetric manifolds of negative curvature, Invent. math. 52 (1979), no. 1, 27-93.
  * [6] M. Gromov and W. Thurston, Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), 1-12.
  * [7] L. Hernandez, Kähler manifolds and $\frac{1}{4}$-pinching, Duke Math. J. 62 (1991), 601-611.
  * [8] F. Hirzebruch, Arrangements of lines and algebraic surfaces, Arithmetic and Geometry vol II, Progress in Math 36 (1983), 113-140.
  * [9] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. II, Wiley Classics Library, Wiley, New York (1996). Reprint of the 1969 original.
  * [10] G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, Princeton University Press, Princeton (1973).
  * [11] B. Minemyer, Warped product metrics on hyperbolic and complex hyperbolic manifolds, preprint, preliminary version available on the arXiv.
  * [12] P. Ontaneda, Riemannian hyperbolization, Publ. math. IHES 131 (2020), 1-72.
  * [13] M. Stover and D. Toledo, Residual finiteness for central extensions of lattices in PU($n$,1) and negatively curved projective varieties, Pure Appl. Math. Q. 18 (2022), no. 4, 1771-1797.
  * [14] S.T. Yau and F. Zheng, Negatively $\dfrac{1}{4}$-pinched Riemannian metric on a compact Kähler manifold, Invent. math. 103 (1991), 527-535.
  * [15] F. Zheng, Examples of non-positively curved Kähler manifolds, Comm. Anal. Geom. 4 (1996), no. 1-2, 129-160.


$K_{a}\psi_{i}\rightarrow K_{a}K_{a}\psi_{i}$ for each $i=1,\dots,m$ axiom 4

  2. 2.

$K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m}\rightarrow
K_{a}K_{a}\psi_{1}\land\cdots\land K_{a}K_{a}\psi_{m}$ Lemma A.1(1.)

  3. 3.

$K_{a}K_{a}\psi_{1}\land\cdots\land K_{a}K_{a}\psi_{m}\rightarrow
K_{a}(K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m})$ Lemma A.3

  4. 4.

$K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m}\rightarrow
K_{a}(K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m})$ Lemma 4.6(2.,3.) as
$K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m}\rightarrow
K_{a}(K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m})\mathrel{\ltimes}K_{a}K_{a}\psi_{1}\land\cdots\land
K_{a}K_{a}\psi_{m}$

See 4.15

######  Proof B.8.

  1. 1.

$\lnot(\varphi\land K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m})$ derivable by
assumption

  2. 2.

$K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m}\rightarrow\lnot\varphi$
equidefinable and equivalent to 1.

  3. 3.

$K_{a}\bigl{(}K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m}\rightarrow\lnot\varphi\bigr{)}$ N(2.)

  4. 4.

$K_{a}\bigl{(}K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m}\rightarrow\lnot\varphi\bigr{)}\rightarrow\bigl{(}K_{a}(K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m})\rightarrow\widehat{K}_{a}\lnot\varphi\bigr{)}$ axiom
$\mathbf{K\widehat{K}}$

  5. 5.

$K_{a}(K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m})\rightarrow\widehat{K}_{a}\lnot\varphi$ MP⋈(3.,4.) as, by Lemma
2.15(2.15),
$K_{a}(K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m})\rightarrow\widehat{K}_{a}\lnot\varphi\mathrel{\ltimes}K_{a}\bigl{(}K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m}\rightarrow\lnot\varphi\bigr{)}$

  6. 6.

$K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m}\rightarrow
K_{a}(K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m})$ Lemma 4.14

  7. 7.

$K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m}\rightarrow\widehat{K}_{a}\lnot\varphi$ Lemma 4.6(5.,6.) as, by
Lemma 2.15(2.15),
$K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m}\rightarrow\widehat{K}_{a}\lnot\varphi\mathrel{\ltimes}K_{a}(K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m})$

  8. 8.

$K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m}\rightarrow\lnot
K_{a}\lnot\lnot\varphi$ from 7. by definition of $\widehat{K}_{a}$

  9. 9.

$\lnot\lnot
K_{a}\lnot\lnot\varphi\rightarrow\lnot(K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m})$ Lemma 4.8(8.)

  10. 10.

$K_{a}\lnot\lnot\varphi\rightarrow\lnot(K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m})$ equidefinable and equivalent to 9.

  11. 11.

$\varphi\to\lnot\lnot\varphi$ tautology

  12. 12.

$K_{a}\varphi\to K_{a}\lnot\lnot\varphi$ Lemma 4.9(11.) as
$\lnot\lnot\varphi\mathrel{\ltimes}\varphi$

  13. 13.

$K_{a}\varphi\rightarrow\lnot(K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m})$
Lemma 4.6(12.,10.) as
$K_{a}\varphi\rightarrow\lnot(K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m})\mathrel{\ltimes}K_{a}\lnot\lnot\varphi$

  14. 14.

$\lnot(K_{a}\varphi\land K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m})$
equidefinable and equivalent to 13.

See 4.16

######  Proof B.9.

  1. 1.

$\lnot(\lnot\varphi\land K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m})$
derivable by assumption

  2. 2.

$K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m}\rightarrow\varphi$ equidefinable
and equivalent to 1.

  3. 3.

$K_{a}(K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m}\rightarrow\varphi)$ N(2.)

  4. 4.

$K_{a}(K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m}\rightarrow\varphi)\rightarrow\bigl{(}K_{a}(K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m})\rightarrow K_{a}\varphi\bigr{)}$ K⋈ as
$K_{a}(K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m})\mathrel{\ltimes}K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m}$ by
Lemma 2.15(2.15)

  5. 5.

$K_{a}(K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m})\rightarrow K_{a}\varphi$
MP⋈(3.,4.) as, by Lemma 2.15(2.15),
$K_{a}(K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m})\rightarrow
K_{a}\varphi\mathrel{\ltimes}K_{a}(K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m}\rightarrow\varphi)$

  6. 6.

$K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m}\rightarrow
K_{a}(K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m})$ Lemma 4.14

  7. 7.

$K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m}\rightarrow K_{a}\varphi$ Lemma
4.6(5.,6.) as, by Lemma 2.15(2.15),
$K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m}\rightarrow
K_{a}\varphi\mathrel{\ltimes}K_{a}(K_{a}\psi_{1}\land\cdots\land
K_{a}\psi_{m})$

  8. 8.

$\lnot(\lnot K_{a}\varphi\land K_{a}\psi_{1}\land\cdots\land K_{a}\psi_{m})$
equidefinable and equivalent to 7.

See 5.6

######  Proof B.10.

  1. 1.

$\lnot\bigwedge\Gamma\rightarrow\lnot\left(\varphi\land\bigwedge\Gamma\right)$
tautology

  2. 2.

$\lnot\bigwedge\Gamma$ derivable by assumption

  3. 3.

$\lnot\left(\varphi\land\bigwedge\Gamma\right)$ MP⋈(1.,2.) as
$\lnot\left(\varphi\land\bigwedge\Gamma\right)\mathrel{\ltimes}\lnot\bigwedge\Gamma$

  1. 1.

$\varphi\land\bigwedge\Gamma\rightarrow\bigwedge\Gamma$ tautology

  2. 2.

$\bigwedge\Gamma\to\psi$ derivable by assumption

  3. 3.

$\varphi\land\bigwedge\Gamma\to\psi$ Lemma 4.6(1.,2.) as
$\varphi\land\bigwedge\Gamma\to\psi\mathrel{\ltimes}\bigwedge\Gamma$

See 5.17

######  Proof B.11.

Since any face of any simplicial model yields a DMC set by Lemma 5.9,
$\mathbb{G}\neq\varnothing$. DMC sets are not empty by definition. Since
$\lnot\varphi\mathrel{\ltimes}\varphi$ and
$\varphi_{1}\land\varphi_{2}\mathrel{\ltimes}\varphi_{i}$ for $i=1,2$, it
follows that, for each $\Gamma\in\mathbb{G}$, either $p_{a}\in^{*}\Gamma$ or
$K_{a}\psi\in^{*}\Gamma$ for some formula $\psi$ and $a\in A$ because $\Gamma$
is DM. Since $p_{a}\mathrel{\ltimes}K_{a}p_{a}$ by Lemma 2.15(2.15) and,
hence, also $\lnot p_{a}\mathrel{\ltimes}K_{a}p_{a}$, we conclude that
$K_{a}\psi\in^{*}\Gamma$ for some formula $\psi$ and $a\in A$ in either case,
from which it follows that $K_{a}\Gamma\neq\varnothing$ for this $a$. Hence,
$X_{\Gamma}\neq\varnothing$ for all $\Gamma\in\mathbb{G}$. As a consequence,
$C^{c}\neq\varnothing$ and every DMC set $\Gamma$ is represented in the
canonical model by $X_{\Gamma}\in C^{c}$. Set $C^{c}$ is closed under taking
non-empty subsets and all its faces are non-empty by construction. The faces
are finite because for $X\subseteq X_{\Gamma}$, we have
$|X|\leq|X_{\Gamma}|\leq n+1$. Chromatic map $\chi^{c}$ is defined correctly
because all vertices $K_{a}\Gamma\in\mathcal{V}^{c}$ are non-empty by
construction and all formulas in $K_{a}\Gamma$ start with $K_{a}$. Thus, it
cannot happen that $K_{a}\Gamma=K_{b}\Delta\neq\varnothing$ for $a\neq b$, and
the definition of $\chi^{c}$ is unambiguous. Finally, by construction, each
$X_{\Gamma}$ (and, consequently, any $X\subseteq X_{\Gamma}$) has at most one
$K_{a}\Gamma$ per agent $a$. Thus, the chromatic map condition is fulfilled.
The correctness of $\ell^{c}$ relies on the same argument as that of
$\chi^{c}$, but in addition, by construction, $\ell^{c}(K_{a}\Gamma)\subseteq
P_{a}$ and $\ell^{c}(K_{a}\Gamma)=\ell^{c}(K_{a}\Delta)$ in case
$K_{a}\Gamma=K_{a}\Delta\neq\varnothing$ for $\Gamma\neq\Delta$.