# The Yamabe invariants of Inoue Surfaces, Kodaira Surfaces, and their blowups

Michael Albanese  Université du Québec à Montréal
Département de mathématiques<EMAIL_ADDRESS>
###### Abstract.

Shortly after the introduction of Seiberg-Witten theory, LeBrun showed that
the sign of the Yamabe invariant of a compact Kähler surface is determined by
its Kodaira dimension. In this paper, we show that LeBrun’s Theorem is no
longer true for non-Kähler surfaces. In particular, we show that the Yamabe
invariants of Inoue surfaces and their blowups are all zero. We also take this
opportunity to record a proof that the Yamabe invariants of Kodaira surfaces
and their blowups are all zero, as previously indicated by LeBrun.

##  1\. Introduction

The complex geometric properties of a compact Riemann surface are intimately
related to its Riemannian geometry. By the Gauss-Bonnet theorem, the total
scalar curvature of any Riemannian metric is a positive multiple of the Euler
characteristic. It follows that the cases of Kodaira dimension $-\infty$, $0$,
and $1$, corresponds precisely to the total scalar curvature being positive,
zero, and negative.

In higher dimensions, the total scalar curvature of a Riemannian metric is no
longer independent of the metric, and hence forms a functional on the space of
all Riemannian metrics. After normalising the functional, one can use it to
define the Yamabe invariant $Y(M)$ of a closed manifold $M$ via a min-max
definition (see section 3), which coincides with the total scalar curvature
when $M$ is a Riemann surface. In [33], LeBrun showed that the relationship
between the trichotomies for Riemann surfaces carries over to Kähler surfaces
with the Yamabe invariant taking the place of total scalar curvature.

######  Theorem.

(LeBrun) Let $M$ be a connected compact complex surface which admits a Kähler
metric. Then $Y(M)$ is positive if $\kappa(M)=-\infty$, zero if $\kappa(M)=0$
or $1$, and negative if $\kappa(M)=2$.

As LeBrun pointed out, there is no higher dimensional analogue of this
trichotomy; in particular, manifolds of general type can have positive Yamabe
invariant in higher dimensions.

Unlike the case of Riemann surfaces, there exist complex manifolds in complex
dimension two which do not admit Kähler metrics. With this in mind, a natural
question to ask is whether the Kähler hypothesis of the above theorem is
necessary. The main result of this paper is the following:

######  Theorem.

Let $M$ be an Inoue surface or a blownup Inoue surface. Then $Y(M)=0$.

As Inoue surfaces are non-Kähler surfaces with Kodaira dimension $-\infty$, we
immediately conclude the following:

######  Corollary.

The Kähler hypothesis of LeBrun’s Theorem is necessary.

In addition, we compute the Yamabe invariant of Kodaira surfaces and their
blowups, as previously indicated by LeBrun.

Disclaimer. The results of this paper were proved during the author’s PhD and
were conceived prior to some recent results which give rise to different
proofs of Theorem 4.5 and Theorem 4.8 which we outline here. Inoue surfaces
and Kodaira surfaces are solvmanifolds as shown by Hasegawa [16], so they are
enlargeable in the sense of Gromov and Lawson [13]. Using Schoen and Yau’s
upgraded minimal hypersurface technique [46], Cecchini and Schick [8] have
proved that a compact enlargeable manifold cannot admit a metric of positive
scalar curvature – prior to this, one needed the additional assumption that
the manifold was spin (note that blowups of Inoue surfaces and Kodaira
surfaces are enlargeable, but not spin). The result of Cecchini and Schick in
low dimensions (namely, at most $8$) can be deduced from Gromov and Lawson
[14, Theorem 12.1], at least for compactly enlargeable manifolds. None of
these results are employed here, instead we use a different argument which may
be of independent interest.

Acknowledgements. First and foremost, I would like to thank my former advisor
Claude LeBrun for introducing me to this problem and guiding me during this
work. I would also like to thank Blaine Lawson for answering my questions
regarding his work on metrics of positive scalar curvature. The exposition has
benefited from illuminating conversations with Georges Dloussky and Sönke
Rollenske, many of which took place at the Banff International Research
Station for Mathematical Innovation and Discovery where I presented these
results during the workshop “Bridging the Gap between Kähler and non-Kähler
Complex Geometry”. Finally, thanks to the anonymous referee for their helpful
suggestions.

##  2\. Compact Complex Surfaces

Recall that for a connected compact complex $n$-dimensional manifold $X$, its
Kodaira dimension, denoted by $\kappa(X)$, is defined to be $-\infty$ if
$h^{0}(X,K_{X}^{\ell})=\dim H^{0}(X,K_{X}^{\ell})=0$ for all $\ell>0$; here
$K_{X}$ denotes the canonical bundle of $X$. Otherwise, the Kodaira dimension
of $X$ is given by

$\kappa(X)=\limsup\frac{\log h^{0}(X,K_{X}^{\ell})}{\log\ell}.$

The Kodaira dimension takes values in $\\{-\infty,0,1,\dots,n\\}$, and if $X$
has maximal Kodaira dimension, i.e. $\kappa(X)=n$, we say that $X$ is of
general type.

Now suppose $n=2$. By the Kodaira-Enriques classification, the surface $X$ has
a minimal model in precisely one111$X$ need not have a unique minimal model,
but any two minimal models belong to the same class. of the following 10
classes, organised in the table below.

$\kappa(X)$ |  $b_{1}(X)$ even |  $b_{1}(X)$ odd
---|---|---
$-\infty$ |  | Rational
---
Ruled
Class $\mathrm{VII}_{0}$
$0$ |  | $K3$
---
Enriques
Tori
Hyperelliptic
Kodaira
$1$ | Properly Elliptic | Properly Elliptic
$2$ | General Type |

The parity of $b_{1}(X)$ is of interest because it determines when $X$ admits
a Kähler metric: a connected compact complex surface admits a Kähler metric if
and only if $b_{1}(X)$ is even. This was initially proved using the Kodaira-
Enriques classification with the final case of $K3$ surfaces completed by Siu
[47], while a proof without using the classification was obtained
independently by Buchdahl [6] and Lamari [27].

We now turn our attention to the two classes of interest for this paper,
namely class $\mathrm{VII}_{0}$ surfaces and Kodaira surfaces.

###  2.1. Class $\mathrm{VII}_{0}$ Surfaces

A connected compact complex surface $X$ is said to be a class $\mathrm{VII}$
surface if $\kappa(X)=-\infty$ and $b_{1}(X)=1$; furthermore, if $X$ is not
the blowup of another complex surface (i.e. $X$ is minimal), then it is said
to be a class $\mathrm{VII}_{0}$ surface.

The following theorem regarding class $\mathrm{VII}_{0}$ surfaces with
$b_{2}=0$ first appeared in work by Bogomolov [4], [5]. It was later reproved
by Teleman [53], also see work by Li, Yau, and Zheng [35].

######  Theorem 2.1.

A class $\mathrm{VII}_{0}$ surface with $b_{2}=0$ is biholomorphic to a Hopf
surface or an Inoue surface.

In the following sections, we review the constructions of Hopf and Inoue
surfaces, before moving on to class $\mathrm{VII}_{0}$ surfaces with
$b_{2}>0$.

####  2.1.1. Hopf Surfaces

A Hopf manifold is a compact complex manifold whose universal cover is
biholomorphic to $\mathbb{C}^{n}\setminus\\{0\\}$. A Hopf manifold with
fundamental group isomorphic to $\mathbb{Z}$ is called a primary Hopf
manifold, otherwise it is called a secondary Hopf manifold. Primary Hopf
manifolds are sometimes defined to be quotients of
$\mathbb{C}^{n}\setminus\\{0\\}$ by the infinite cyclic group generated by a
contraction; Kodaira showed that these two definitions coincide. Moreover, he
showed that every secondary Hopf manifold is finitely covered by a primary
Hopf manifold, see [26, Theorem 30]222Kodaira only proves these statements in
the case of surfaces, but the proofs apply for complex manifolds of arbitrary
dimension..

The structure of Hopf surfaces is well understood – this is in part due to a
normal form for contractions on $\mathbb{C}^{2}$. Up to an automorphism, every
contraction on $\mathbb{C}^{2}$ is of the form
$(z_{1},z_{2})\mapsto(\alpha_{1}z_{1}+\lambda z_{2}^{m},\alpha_{2}z_{2})$
where $m$ is a positive integer and
$\alpha_{1},\alpha_{2},\lambda\in\mathbb{C}$ are subject to the conditions
$(\alpha_{1}-\alpha_{2}^{m})\lambda=0$ and $0<|\alpha_{1}|\leq|\alpha_{2}|<1$;
see [48] and [28]. It follows that all primary Hopf surfaces are deformation
equivalent, and hence diffeomorphic; namely, there are all diffeomorphic to
$S^{1}\times S^{3}$. The diffeomorphism types of secondary Hopf surfaces were
determined by Kato, see [21, Theorem 9] and [23]:

######  Theorem 2.2.

Every secondary Hopf surface is diffeomorphic to

  1. (1)

a product of $S^{1}$ and a quotient of $S^{3}$, or

  2. (2)

the mapping torus of a quotient of $S^{3}$ by a diffeomorphism of order two or
three.

Of course, one could view the first family as mapping tori of a diffeomorphism
of order one, namely the identity map. In particular, a secondary Hopf surface
in the second family is double or triple covered by a secondary Hopf surface
in the first family.

####  2.1.2. Inoue Surfaces

Inoue surfaces were introduced by Inoue in [17], and further explained in
[18]. We outline the construction of the four families of Inoue surfaces
$S^{+}_{M}$, $S^{-}_{M}$, $S^{+}_{N,p,q,r,t}$, and $S^{-}_{N,p,q,r}$, all of
which have universal cover biholomorphic to $\mathbb{H}\times\mathbb{C}$. As
Inoue surfaces are the main focus of this paper, we take this opportunity to
give a relatively detailed description of the underlying smooth topology.

Let $M=[m_{ij}]\in SL(3,\mathbb{Z})$ be a matrix with one real eigenvalue
$\alpha>0$ and two complex eigenvalues $\beta$, $\overline{\beta}$ where
$\operatorname{Im}(\beta)>0$. Let $(a_{1},a_{2},a_{3})$,
$(b_{1},b_{2},b_{3})$, and
$(\overline{b_{1}},\overline{b_{2}},\overline{b_{3}})$ be eigenvectors of $M$
corresponding to the eigenvalues $\alpha$, $\beta$, and $\overline{\beta}$
respectively. By replacing $(a_{1},a_{2},a_{3})$ with its real part if
necessary, we can assume the entries are real.

Consider the following biholomorphisms of $\mathbb{H}\times\mathbb{C}$:

$\displaystyle g_{0}(w,z)$ $\displaystyle=(\alpha w,\beta z)$ $\displaystyle
g_{i}(w,z)$ $\displaystyle=(w+a_{i},z+b_{i}),\quad i=1,2,3.$

The group generated by $g_{0},g_{1},g_{2},g_{3}$, which we denote by
$G_{M}^{+}$, acts freely, properly discontinuously, and cocompactly on
$\mathbb{H}\times\mathbb{C}$. The resulting quotient
$S_{M}^{+}:=(\mathbb{H}\times\mathbb{C})/G_{M}^{+}$ is a compact complex
surface. In order to better understand the topology of $S_{M}^{+}$, it will be
useful to consider the subgroup of $G_{M}^{+}$ generated by
$g_{1},g_{2},g_{3}$, which we denote by $\Gamma_{M}^{+}$; as
$g_{1},g_{2},g_{3}$ commute, we see that $\Gamma_{M}^{+}\cong\mathbb{Z}^{3}$.

Note that the action of $\Gamma_{M}^{+}$ on $(w,z)$ preserves
$\operatorname{Im}(w)$ and the quotient of
$\\{(w,z)\in\mathbb{H}\times\mathbb{C}\mid\operatorname{Im}(w)=d\\}$ by
$\Gamma_{M}^{+}$ can be identified with
$(\mathbb{R}\times\mathbb{C})/\Gamma_{M}^{+}$ by forgetting
$\operatorname{Im}(w)$. The vectors
$(a_{1},b_{1}),(a_{2},b_{2}),(a_{3},b_{3})\in\mathbb{R}\times\mathbb{C}$ are
linearly independent over $\mathbb{R}$, and hence generate a lattice.
Therefore $(\mathbb{R}\times\mathbb{C})/\Gamma_{M}^{+}$ is diffeomorphic to a
three-torus $T^{3}$and $(\mathbb{H}\times\mathbb{C})/\Gamma_{M}^{+}$ is
diffeomorphic to $(0,\infty)\times T^{3}$.

As $(a_{1},a_{2},a_{3})$ and $(b_{1},b_{2},b_{3})$ are eigenvectors of $M$ for
the eigenvalues $\alpha$ and $\beta$ respectively, a simple calculation shows
that $g_{0}g_{i}g_{0}^{-1}=g_{1}^{m_{i1}}g_{2}^{m_{i2}}g_{3}^{m_{i3}}$ for
$i=1,2,3$. So $g_{0}$ is in the normaliser of $\Gamma_{M}^{+}$ and therefore
descends to a diffeomorphism
$(\mathbb{H}\times\mathbb{C})/\Gamma_{M}^{+}\to(\mathbb{H}\times\mathbb{C})/\Gamma_{M}^{+}$.
Under the identification with $(0,\infty)\times T^{3}$, the map induced by
$g_{0}$ restricts to a diffeomorphism between $\\{1\\}\times T^{3}$ and
$\\{\alpha\\}\times T^{3}$. So
$S_{M}^{+}=(\mathbb{H}\times\mathbb{C})/G_{M}^{+}=((\mathbb{H}\times\mathbb{C})/\Gamma_{M}^{+})/\langle
g_{0}\rangle$ is diffeomorphic to a mapping torus of a self-diffeomorphism of
$T^{3}$. By tracing through the identifications, it is not hard to show that
the self-diffeomorphism is just the map on $T^{3}$ induced by $M^{T}$, see
[11, Proposition 2.9] for example.

The Inoue surface $S_{M}^{-}$ is defined similarly to $S_{M}^{+}$. Consider
the following biholomorphisms of $\mathbb{H}\times\mathbb{C}$:

$\displaystyle g_{0}(w,z)$ $\displaystyle=(\alpha w,\overline{\beta}z)$
$\displaystyle g_{i}(w,z)$ $\displaystyle=(w+a_{i},z+\overline{b_{i}}),\quad
i=1,2,3.$

Let $G^{-}_{M}$ be the group generated by $g_{0},g_{1},g_{2},g_{3}$ and define
$S^{-}_{M}:=(\mathbb{H}\times\mathbb{C})/G^{-}_{M}$. The arguments above
concerning $S^{+}_{M}$ can be made analagously for $S^{-}_{M}$. More directly,
note that the map $\mathbb{H}\times\mathbb{C}\to\mathbb{H}\times\mathbb{C}$
given by $(w,z)\mapsto(w,\overline{z})$ descends to a diffeomorphism
$S^{+}_{M}\to S^{-}_{M}$.

######  Remark 2.1.

There is some confusion in the literature regarding the Inoue surfaces
$S^{+}_{M}$ and $S^{-}_{M}$. For a matrix $M\in SL(3,\mathbb{Z})$ with one
real eigenvalue $\alpha>1$ and two complex conjugate eigenvalues $\beta$,
$\overline{\beta}$, Inoue defined in [17] a complex surface $S_{M}$ in the
same way as $S^{+}_{M}$ was defined above. However, Inoue did not indicate how
to distinguish between $\beta$ and $\overline{\beta}$; note that we required
$\operatorname{Im}(\beta)>0$. So, depending on the naming of the eigenvalues,
the surface $S_{M}$ could be $S^{+}_{M}$ or $S^{-}_{M}$. In [18], Inoue
separated the two cases and showed that they are not biholomorphic, or even
deformation equivalent. In the literature which followed, the distinction
between $S^{+}_{M}$ and $S^{-}_{M}$ was not always observed, furthermore, the
notation $S^{0}$ is sometimes used to indicate the Inoue surfaces which Inoue
denoted by $S_{M}$.

Now let $N=[n_{ij}]\in SL(2,\mathbb{Z})$ be a matrix with two real eigenvalues
$\alpha$ and $\frac{1}{\alpha}$ where $\alpha>1$; this is equivalent to
$\operatorname{tr}(N)>2$. Let $(a_{1},a_{2})$ and $(b_{1},b_{2})$ be real
eigenvectors of $N$ for $\alpha$ and $\frac{1}{\alpha}$ respectively. Let
$p,q,r\in\mathbb{Z}$ with $r\neq 0$ and $t\in\mathbb{C}$. Consider the
following equation for $c_{1}$ and $c_{2}$:

(2.1) $(I-N)\begin{bmatrix}c_{1}\\\ c_{2}\end{bmatrix}=\begin{bmatrix}e_{1}\\\
e_{2}\end{bmatrix}+\frac{1}{r}(a_{2}b_{1}-a_{1}b_{2})\begin{bmatrix}p\\\
q\end{bmatrix}$

where
$e_{i}=\frac{1}{2}n_{i1}(n_{i1}-1)a_{1}b_{1}+\frac{1}{2}n_{i2}(n_{i2}-1)a_{2}b_{2}+n_{i1}n_{i2}a_{2}b_{1}$;
the reason for the seemingly peculiar form of equation (2.1) will become clear
later. As $1$ is not an eigenvalue of $N$, the matrix $I-N$ is invertible, so
there is a unique solution to the above equation. Moreover, $c_{1}$ and
$c_{2}$ are real.

Consider the following biholomorphisms of $\mathbb{H}\times\mathbb{C}$:

$\displaystyle g_{0}(w,z)$ $\displaystyle=(\alpha w,z+t)$ $\displaystyle
g_{i}(w,z)$ $\displaystyle=(w+a_{i},z+b_{i}w+c_{i}),\quad i=1,2$
$\displaystyle g_{3}(w,z)$
$\displaystyle=\left(w,z+\frac{1}{r}(a_{2}b_{1}-a_{1}b_{2})\right).$

The group generated by $g_{0},g_{1},g_{2},g_{3}$, which we denote by
$G^{+}_{N,p,q,r,t}$, acts freely, properly discontinuously, and cocompactly on
$\mathbb{H}\times\mathbb{C}$. The resulting quotient
$S_{N,p,q,r,t}^{+}:=(\mathbb{H}\times\mathbb{C})/G_{N,p,q,r,t}^{+}$ is a
compact complex surface. In order to better understand the topology of
$S_{N,p,q,r,t}^{+}$, it will be useful to consider the subgroup of
$G_{N,p,q,r,t}^{+}$ generated by $g_{1},g_{2},g_{3}$, which we denote by
$\Gamma_{N}^{+}$ for brevity.

Note that $\Gamma_{N}^{+}$ acting on $(w,z)$ preserves $\operatorname{Im}(w)$
and the quotient of
$\\{(w,z)\in\mathbb{H}\times\mathbb{C}\mid\operatorname{Im}(w)=d\\}$ by
$\Gamma^{+}_{N}$ can be identified with
$(\mathbb{R}\times\mathbb{C})/\Gamma_{N}^{+}$ by forgetting
$\operatorname{Im}(w)$. There is a diffeomorphism from
$\mathbb{R}\times\mathbb{C}$ to $H(3,\mathbb{R})$, the three-dimensional real
Heisenberg group, given by

$(\operatorname{Re}(w),z)\mapsto\begin{bmatrix}1&\zeta&\gamma\\\ 0&1&\eta\\\
0&0&1\end{bmatrix}$

where $(\zeta,\eta)$ are the coordinates of
$(\operatorname{Re}(w),\operatorname{Im}(z)/\operatorname{Im}(w))$ in the
basis $\\{(a_{1},b_{1}),(a_{2},b_{2})\\}$, and
$\gamma=\operatorname{Re}(z)/(a_{2}b_{1}-a_{1}b_{2})$. Under this
diffeomorphism, the action of $g_{1}$, $g_{2}$, and $g_{3}$ correspond to left
multiplication by the matrices

$\begin{bmatrix}1&1&c_{1}^{\prime}\\\ 0&1&0\\\
0&0&1\end{bmatrix},\quad\begin{bmatrix}1&0&c_{2}^{\prime}\\\ 0&1&1\\\
0&0&1\end{bmatrix},\quad\ \text{and}\ \quad\begin{bmatrix}1&0&\frac{1}{r}\\\
0&1&0\\\ 0&0&1\end{bmatrix}$

which generate a lattice $\Lambda$ in $H(3,\mathbb{R})$; here
$c_{i}^{\prime}=c_{i}/(a_{2}b_{1}-a_{1}b_{2})$. In particular, the quotient
$(\mathbb{R}\times\mathbb{C})/\Gamma_{N}^{+}$ is diffeomorphic to a
nilmanifold. Mapping an element of $H(3,\mathbb{R})/\Lambda$ to
$(\zeta,\eta)+\mathbb{Z}^{2}$ defines a submersion to
$T^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$ which endows $H(3,\mathbb{R})/\Lambda$
with the structure of a circle bundle over $T^{2}$.

As the commutators of the matrices above do not depend on $c_{1}^{\prime}$ and
$c_{2}^{\prime}$, the lattice $\Lambda$ is isomorphic to $\Lambda_{r}$, the
lattice generated by the above matrices with zeroes in place of
$c_{1}^{\prime}$ and $c_{2}^{\prime}$. As $H(3,\mathbb{R})/\Lambda$ and
$H(3,\mathbb{R})/\Lambda_{r}$ are compact nilmanifolds with isomorphic
fundamental groups, they are diffeomorphic [2, page 235]. The latter is a
quotient of the Heisenberg manifold $H(3,\mathbb{R})/H(3,\mathbb{Z})$ by
$\mathbb{Z}_{r}=\mathbb{Z}/r\mathbb{Z}$ where $\mathbb{Z}_{r}$ acts on the
$\gamma$ entry by translation. The Heisenberg manifold is the total space of
the Chern class 1 circle-bundle over $T^{2}$, with the same projection map as
described above. The $\mathbb{Z}_{r}$ action preserves the fibers, so it
follows that $H(3,\mathbb{R})/\Lambda_{r}$ is the total space of the Chern
class $r$ circle-bundle over $T^{2}$.

From the above, we see that $(\mathbb{H}\times\mathbb{C})/\Gamma_{N}^{+}$ is
diffeomorphic to $(0,\infty)\times F$ where $F$ is the total space of the
Chern class $r$ circle-bundle over $T^{2}$. By direct computation, one can
show that $g_{0}g_{1}g_{0}^{-1}=g_{1}^{n_{11}}g_{2}^{n_{12}}g_{3}^{p}$ and
$g_{0}g_{2}g_{0}^{-1}=g_{1}^{n_{21}}g_{2}^{n_{22}}g_{3}^{q}$. In fact, these
two relations are equivalent to the constants $c_{1},c_{2}$ satisfying
equation (2.1), which explains its peculiar form. In addition, the
biholomorphism $g_{0}$ commutes with $g_{3}$, so $g_{0}$ is in the normaliser
of $\Gamma_{N}^{+}$ and therefore descends to a diffeomorphism
$(\mathbb{H}\times\mathbb{C})/\Gamma_{N}^{+}\to(\mathbb{H}\times\mathbb{C})/\Gamma_{N}^{+}$.
Under the identification with $(0,\infty)\times F$, the map induced by $g_{0}$
restricts to a diffeomorphism between $\\{1\\}\times F$ and
$\\{\alpha\\}\times F$. So
$S_{N,p,q,r,t}^{+}=(\mathbb{H}\times\mathbb{C})/G_{N,p,q,r,t}^{+}=((\mathbb{H}\times\mathbb{C})/\Gamma_{N}^{+})/\langle
g_{0}\rangle$ is diffeomorphic to a mapping torus of a self-diffeomorphism of
$F$.

Finally, let $N=[n_{ij}]\in GL(2,\mathbb{Z})$ be a matrix with determinant
$-1$ and real eigenvalues $\alpha$ and $-\frac{1}{\alpha}$ such that
$\alpha>1$. Let $(a_{1},a_{2})$ and $(b_{1},b_{2})$ be real eigenvectors of
$N$ for $\alpha$ and $-\frac{1}{\alpha}$ respectively. Let
$p,q,r\in\mathbb{Z}$ with $r\neq 0$. Consider the following equation for
$c_{1}$ and $c_{2}$:

(2.2) $(-I-N)\begin{bmatrix}c_{1}\\\
c_{2}\end{bmatrix}=\begin{bmatrix}e_{1}\\\
e_{2}\end{bmatrix}+\frac{1}{r}(a_{2}b_{1}-a_{1}b_{2})\begin{bmatrix}p\\\
q\end{bmatrix}$

where
$e_{i}=\frac{1}{2}n_{i1}(n_{i1}-1)a_{1}b_{1}+\frac{1}{2}n_{i2}(n_{i2}-1)a_{2}b_{2}+n_{i1}n_{i2}a_{2}b_{1}$
as before. As $-1$ is not an eigenvalue of $N$, the matrix $-I-N$ is
invertible, so there is a unique solution to the above equation. Moreover,
$c_{1}$ and $c_{2}$ are real.

Consider the following biholomorphisms of $\mathbb{H}\times\mathbb{C}$:

$\displaystyle g_{0}(w,z)$ $\displaystyle=(\alpha w,-z)$ $\displaystyle
g_{i}(w,z)$ $\displaystyle=(w+a_{i},z+b_{i}w+c_{i}),\quad i=1,2$
$\displaystyle g_{3}(w,z)$
$\displaystyle=\left(w,z+\frac{1}{r}(a_{2}b_{1}-a_{1}b_{2})\right).$

The group generated by $g_{0},g_{1},g_{2},g_{3}$, which we denote by
$G_{N,p,q,r}^{-}$, acts freely, properly discontinuously, and cocompactly on
$\mathbb{H}\times\mathbb{C}$. The resulting quotient
$S^{-}_{N,p,q,r}:=(\mathbb{H}\times\mathbb{C})/G_{N,p,q,r}^{-}$ is a compact
complex surface. Letting $\Gamma_{N}^{-}$ denote the subgroup generated by
$g_{1},g_{2},g_{3}$, then, as before, the quotient
$(\mathbb{H}\times\mathbb{C})/\Gamma_{N}^{-}$ is diffeomorphic to
$(0,\infty)\times F$ where $F$ denotes the total space of the Chern class $r$
circle-bundle over $T^{2}$. As in the previous case, we have
$g_{0}g_{1}g_{0}^{-1}=g_{1}^{n_{11}}g_{2}^{n_{12}}g_{3}^{p}$ and
$g_{0}g_{2}g_{0}^{-1}=g_{1}^{n_{21}}g_{2}^{n_{22}}g_{3}^{q}$, which is
equivalent to $c_{1},c_{2}$ satisfying (2.2), but now
$g_{0}g_{3}g_{0}^{-1}=g_{3}^{-1}$. Still, the biholomorphism $g_{0}$ is in the
normaliser of $\Gamma_{N}^{-}$ and therefore descends to a diffeomorphism
$(\mathbb{H}\times\mathbb{C})/\Gamma_{N}^{-}\to(\mathbb{H}\times\mathbb{C})/\Gamma_{N}^{-}$.
Under the identification with $(0,\infty)\times F$, the map induced by $g_{0}$
restricts to a diffeomorphism between $\\{1\\}\times F$ and
$\\{\alpha\\}\times F$. So
$S_{N,p,q,r}^{-}=(\mathbb{H}\times\mathbb{C})/G_{N,p,q,r}^{-}=((\mathbb{H}\times\mathbb{C})/\Gamma_{N}^{-})/\langle
g_{0}\rangle$ is diffeomorphic to a mapping torus of a self-diffeomorphism of
$F$.

Note that $N^{2}\in SL(2,\mathbb{Z})$ and has real eigenvalues $\alpha^{2}>1$
and $\frac{1}{\alpha^{2}}$ with real eigenvectors $(a_{1},a_{2})$ and
$(b_{1},b_{2})$. So if one considers the group of biholomorphisms of
$\mathbb{H}\times\mathbb{C}$ generated by $g_{0}^{2},g_{1},g_{2},g_{3}$, they
appear to be the same as the biholomorphisms which would generate
$G_{N^{2},p_{1},q_{1},r,0}^{+}$ for some $p_{1},q_{1}\in\mathbb{Z}$. However,
for that to be the case, one must check that the $c_{1},c_{2}$ given by
equation (2.2) satisfy equation (2.1) for the matrix $N^{2}$ and integers
$p_{1}$, $q_{1}$. This can be achieved by expanding
$(g_{0}^{2})g_{1}(g_{0}^{2})^{-1}$ and $(g_{0})^{2}g_{2}(g_{0}^{2})^{-1}$
carefully, which will dictate the values of $p_{1},q_{1}$. Therefore, the
surface $S_{N^{2},p_{1},q_{1},r,0}^{+}$ double covers $S_{N,p,q,r}^{-}$. If
particular, if $S_{N,p,q,r}^{-}$ is diffeomorphic to the mapping torus of
$f\colon F\to F$, then $S_{N^{2},p_{1},q_{1},r,0}^{+}$ is diffeomorphic to the
mapping torus of $f\circ f$.

We now record the following fact which follows from the constructions above.

######  Proposition 2.3.

Every Inoue surface is diffeomorphic to a mapping torus of a compact three-
dimensional nilmanifold $F$.

For those Inoue surfaces of the form $S_{M}^{+}$ or $S_{M}^{-}$, the fiber is
$F=T^{3}$, while for those of the form $S_{N,p,q,r,t}^{+}$ or
$S_{N,p,q,r}^{-}$, the fiber $F$ is the total space of the Chern class $r$
circle-bundle over $T^{2}$. Hasegawa has shown that all Inoue surfaces
themselves are diffeomorphic to solvmanifolds, see [16, Theorem 1]. In
general, a compact solvmanifold $S$ is the total space of a fiber bundle with
base a torus of dimension $b_{1}(S)$ and fiber a nilmanifold.

####  2.1.3. Class $\mathrm{VII}_{0}$ surfaces with $b_{2}>0$

A spherical shell in an $n$-dimensional complex manifold $X$ is an open subset
$U$ which is diffeomorphic to a connected open subset of $S^{2n-1}$ in
$\mathbb{C}^{n}\setminus\\{0\\}$. If $X\setminus U$ is connected, then $U$ is
called a global spherical shell.

While every complex manifold contains a spherical shell, the existence of a
global spherical shell is quite restrictive. Kato [22, Theorem 1] showed that,
for $n\geq 2$, if a connected complex manifold contains a global spherical
shell, then it is a deformation of a modification of a primary Hopf manifold
at finitely many points. In the case of surfaces, every modification is a
composition of blowups of points [29, Theorem 5.7]. Therefore, a connected
compact complex surface $X$ which contains a global spherical shell, often
called a Kato surface, is a deformation of a blownup primary Hopf surface.
Note that such a surface has $\kappa(X)=-\infty$ and $b_{1}(X)=1$, and is
therefore a class $\mathrm{VII}$ surface; the following conjecture asserts a
partial converse is true.

######  Global Spherical Shell Conjecture.

A class $\mathrm{VII}_{0}$ surface with $b_{2}>0$ contains a global spherical
shell, i.e. is a Kato surface.

Note that the assumption $b_{2}>0$ is necessary as Inoue surfaces do not
contain global spherical shells (they do not have fundamental group isomorphic
to $\mathbb{Z}$). Similarly, the minimality condition is necessary, otherwise
blownup Inoue surfaces would provide counterexamples.

Theorem 2.1 states that the only class $\mathrm{VII}_{0}$ surfaces with
$b_{2}=0$ are Hopf surface and Inoue surfaces. The global spherical shell
conjecture asserts that class $\mathrm{VII}_{0}$ surfaces with $b_{2}>0$ are
deformations of blownup primary Hopf surfaces. One might expect to be able to
construct counterexamples by considering deformations of secondary Hopf
surfaces or Inoue surfaces. This has been ruled out by work of Dloussky. In
[9], he shows that if a class $\mathrm{VII}_{0}$ surface with $b_{2}>0$ admits
a global spherical shell, then any finite quotient also admits a global
spherical shell; this rules out counterexamples which are deformations of
blownup secondary Hopf surfaces. While counterexamples arising from
deformations of blownup Inoue surfaces are prohibited by the arguments in the
proof of [10, Theorem 1.13], namely case 5.

By analysing the moduli space of stable holomorphic bundles on class
$\mathrm{VII}_{0}$ surfaces, Teleman has verified the global spherical shell
conjecture for small values of $b_{2}$, namely $b_{2}=1$ and $b_{2}=2$, see
[54] and [55] respectively. In [56], Teleman has announced the case $b_{2}=3$
with the “long and technical” details to follow in an upcoming paper.

If the global spherical shell conjecture were indeed true, then every class
$\mathrm{VII}_{0}$ surface with $b_{2}>0$ would be diffeomorphic to
$(S^{1}\times S^{3})\\#b_{2}\overline{\mathbb{CP}^{2}}$.

###  2.2. Kodaira Surfaces

A connnected compact complex surface $X$ is called a primary Kodaira surface
if $K_{X}$ is trivial and $b_{1}(X)=3$. A secondary Kodaira surface is a
surface which admits a primary Kodaira surface as a covering.

We recall the following elementary fact:

######  Proposition 2.4.

Primary Kodaira surfaces are symplectic.

###### Proof.

Let $X$ be a primary Kodaira surface. As $K_{X}$ is holomorphically trivial,
there is a nowhere-zero holomorphic two-form $\alpha$; let
$\omega=2\operatorname{Re}(\alpha)=\alpha+\overline{\alpha}$. As $\alpha$ is
holomorphic, $\overline{\partial}\alpha=0$ while $\partial\alpha=0$ for
bidegree reasons, so $\alpha$ is closed and hence so is $\omega$. In local
holomorphic coordinates $(U,(z^{1},z^{2}))$, we have
$\alpha|_{U}=fdz^{1}\wedge dz^{2}$ where $f\colon U\to\mathbb{C}$ is a
nowhere-zero holomorphic function. So
$\omega^{2}|_{U}=2\alpha\wedge\overline{\alpha}=|f|^{2}dz^{1}\wedge
dz^{2}\wedge d\overline{z}^{1}\wedge d\overline{z}^{2}$ which is nowhere-zero,
hence $\omega$ is a non-degenerate closed two-form, i.e. a symplectic form. ∎

The proof only uses the fact that Kodaira surfaces are compact complex
surfaces with trivial canonical bundle, so applies equally well to tori and
$K3$ surfaces. In higher dimensions, triviality of the canonical bundle does
not imply the existence of a symplectic form. For example, the connected sum
of $m\geq 2$ copies of $S^{3}\times S^{3}$ admits a complex structure with
trivial canonical bundle [36], but has $b_{2}=0$. Also note that secondary
Kodaira surfaces are not symplectic as they have $b_{2}=0$.

##  3\. Yamabe Invariant

Let $M$ be a closed smooth $n$-dimensional manifold. The Einstein-Hilbert
functional on the space of Riemannian metrics on $M$ is given by
$g\mapsto\mathcal{E}(g)$ where

$\mathcal{E}(g)=\frac{\displaystyle\int_{M}s_{g}d\mu_{g}}{\operatorname{Vol}(M,g)^{\frac{n-2}{n}}}$

is the normalised total scalar curvature; here $s_{g}$ denotes the scalar
curvature of $g$ and $d\mu_{g}$ is the Riemannian volume density. The exponent
in the denominator is such that $\mathcal{E}(cg)=\mathcal{E}(g)$ for any
positive constant $c$, and hence $\mathcal{E}(g)=\mathcal{E}(g_{1})$ where
$g_{1}$ is the unique unit-volume metric homothetic to $g$.

Hilbert first considered the unnormalised total scalar curvature functional
and showed that when $n\geq 3$, the critical points are precisely Ricci-flat
metrics, see [3, Proposition 4.17]. In the normalised case, the critical
points, again for $n\geq 3$, are instead Einstein metrics (the vanishing of
the Ricci tensor is replaced with the vanishing of the trace-free Ricci
tensor) see [3, Theorem 4.21]. When $n=2$, both the normalised and
unnormalised total scalar curvatures are independent of the metric and is
given by $4\pi\chi(M)$ due to the Gauss-Bonnet Theorem, hence the need for the
restriction $n\geq 3$ in the previous statements.

When restricted to a conformal class $\mathcal{C}$, the critical points of the
Einstein-Hilbert functional are constant scalar curvature metrics [3,
Proposition 4.25]. By Hölder’s inequality, the restricted functional is
bounded below, so we define the Yamabe constant of $\mathcal{C}$ to be the
real number $Y(M,\mathcal{C})=\inf_{g\in\mathcal{C}}Y(M,\mathcal{C})$. Yamabe
[58] claimed to prove that this infimum is always realised and hence every
conformal class contains a constant scalar curvature metric. Trudinger [57]
pointed out a critical flaw in Yamabe’s argument which was later rectified by
works by Aubin [1] and Schoen [43]. It follows that there is a unit-volume
metric in $\mathcal{C}$ with constant scalar curvature equal to
$Y(M,\mathcal{C})$; if $Y(M,\mathcal{C})\leq 0$, then this is the only unit-
volume constant scalar curvature metric in $\mathcal{C}$, but this need not be
the case if $Y(M,\mathcal{C})>0$ as the conformal class $[g_{\text{round}}]$
on $S^{n}$ demonstrates.

In his work on the Yamabe problem, Aubin showed that $Y(M,\mathcal{C})\leq
Y(S^{n},[g_{\text{round}}])$. We define
$Y(M)=\sup_{\mathcal{C}}Y(M,\mathcal{C})$ to be the Yamabe invariant of $M$;
note that $Y(M)\leq Y(S^{n})$. We say the Yamabe invariant is realised if
there is a conformal class $\mathcal{C}$ for which $Y(M,\mathcal{C})=Y(M)$,
and hence there is a unit-volume metric with constant scalar curvature $Y(M)$.
Note that $Y(M)$ is positive if and only if $M$ admits a metric of positive
scalar curvature. If $Y(M)$ is non-positive, then $Y(M)$ can be characterised
as the supremum of the values of the unit-volume constant scalar curvature
metrics on $M$.

###  3.1. Yamabe Invariants of Complex Surfaces

We now review what is known about the values of the Yamabe invariant for
complex surfaces.

In the Kähler case, LeBrun [33] proved the following:

######  Theorem 3.1.

Let $M$ be a a connected compact complex surface which admits a Kähler metric.
Then $Y(M)$ is positive if $\kappa(M)=-\infty$, zero if $\kappa(M)=0,1$, and
negative if $\kappa(M)=2$.

While the value of the Yamabe invariant itself is not enough to distinguish
between complex surfaces of Kodaira dimensions $0$ and $1$, LeBrun showed in
the same paper that the two cases can be distinguished by asking when the
Yamabe invariant is realised.

######  Theorem 3.2.

Let $M$ be a connected compact complex surface which admits a Kähler metric
with $Y(M)=0$, i.e. $\kappa(M)=0$ or $1$. Then $Y(M)$ is realised if and only
if $\kappa(M)=0$ and $M$ is minimal.

In addition, LeBrun computed the value of Yamabe invariant precisely in the
general type case. If $M$ is a general type surface with minimal model $X$,
then $Y(M)=Y(X)=-4\pi\sqrt{2c_{1}(X)^{2}}$. The Yamabe invariant in the
positive case is harder to compute in general. The only known value with
$\kappa(M)=-\infty$ is $Y(\mathbb{CP}^{2})=12\sqrt{2}\pi$, see [32]. Also note
that while the value of the Yamabe invariant is unchanged after blowing up a
Kähler surface of Kodaira dimension 0, 1, or 2 (i.e. it is a bimeromorphic
invariant in these cases) it is unknown whether this is true of Kähler
surfaces of Kodaira dimension $-\infty$.

There is no analogue of Theorem 3.1 in higher dimensions, which is why we have
set our attention to complex dimension two. To see this, let $m\geq 3$, and
consider a smooth degree $m+3$ hypersurface $M$ of $\mathbb{CP}^{m+1}$; note
that $M$ is of general type. As $M$ is a simply connected non-spin closed
smooth manifold of real dimension $2m\geq 6$, it admits a metric of positive
scalar curvature by [12, Corollary C], and hence $Y(M)>0$ in contrast with the
surface case. More generally, Petean [41] has shown that any simply connected
manifold of real dimension at least five has non-negative Yamabe invariant.

In the non-Kähler setting, much less is known about the values of the Yamabe
invariant. All of our knowledge is restricted to Hopf surfaces and their
blowups. In [44, Section 2], Schoen showed that $Y(S^{1}\times
S^{n-1})=Y(S^{n})$ which can be explicitly calculated as the latter is
realised by the round metric. In particular, primary Hopf surfaces are all
diffeomorphic to $S^{1}\times S^{3}$ and we have $Y(S^{1}\times
S^{3})=Y(S^{4})=8\sqrt{6}\pi$. It follows from a result of Gursky and LeBrun
[15, Theorem B] that the Yamabe invariant of a primary Hopf surface blownup at
$k$ points, which is diffeomorphic to $M_{k}:=(S^{1}\times
S^{3})\\#k\overline{\mathbb{CP}^{2}}$, satisfies $12\sqrt{2}\pi\leq
Y(M_{k})\leq 4\pi\sqrt{2k+16}$ for $k=1,2,3$; it is also true for $k>3$, but
the upper bound is no better than the upper bound $Y(S^{4})$. In particular,
we see that $Y(M_{k})\neq Y(S^{1}\times S^{3})$ for $k=1,2,3$, so the Yamabe
invariant is not a bimeromorphic invariant for class $\mathrm{VII}$ surfaces.
The upper bound also applies to secondary Hopf surfaces blownup at $k>0$
points. However, while those secondary Hopf surfaces in class (1) of Theorem
2.2 and their blowups admit metrics of positive scalar curvature, and hence
have positive Yamabe invariant, this is not clear for those secondary Hopf
surfaces in class (2).

Note, all of the known results in the previous paragraph are consistent with
what one would expect the non-Kähler analogue of LeBrun’s Theorem would
predict: surfaces with Kodaira dimension $-\infty$ have positive Yamabe
invariant. Modulo the status of the global spherical shell conjecture, the
only non-Kähler surfaces which could possibly violate this prediction are
secondary Hopf surfaces in class (2), Inoue surfaces, and their blowups.

While complex surfaces are the focus of this article, the Yamabe invariant has
been computed for many four-manifolds which do not admit a complex structure,
see [19], [20], [40], [42], [50], and [51].

##  4\. Inoue Surfaces, Kodaira Surfaces, and their blowups

In this section we prove the main result: the Yamabe invariants of Inoue
surfaces and their blowups are zero. We begin by first noting that work of
Paternian and Petean implies that the Yamabe invariant of these manifolds is
non-negative.

Recall that a $\mathcal{T}$-structure on a closed smooth manifold $M$ is a
finite open cover $\\{U_{1},\dots,U_{N}\\}$ and a non-trivial torus action on
each $U_{i}$ such that each intersection $U_{i_{1}}\cap\dots\cap U_{i_{k}}$ is
invariant under the torus actions on $U_{i_{1}},\dots,U_{i_{k}}$ and the torus
actions commute. This notion was introduced by Cheeger and Gromov in [7] as a
special case of a more general notion called an $\mathcal{F}$-structure,
although they stated the definition using different terminology.

A result of Paternain and Petean [38, Theorem 7.2] asserts that, provided
$\dim M>2$, the existence of a $\mathcal{T}$-structure implies $Y(M)\geq 0$.
They also showed that all Inoue surfaces admit a $\mathcal{T}$-structure [39,
Theorem B]. The simplest case is those Inoue surfaces of type $S_{M}^{+}$ (or
$S_{M}^{-}$ as they are diffeomorphic). Any such surface is the mapping torus
of the self-diffeomorphism of $T^{3}$ induced by $M^{T}$ . Let $p\colon
S_{M}^{+}\to S^{1}$ denote the projection, then let
$U_{1}=p^{-1}(S^{1}\setminus\\{1\\})$ and
$U_{2}=p^{-1}(S^{1}\setminus\\{-1\\})$. Note that $U_{1}$ and $U_{2}$ are both
diffeomorphic to $(0,1)\times T^{3}$ so they admit effective torus actions
acting by translations on the second factor. Moreover, the intersection
$U_{1}\cap U_{2}$ is invariant under the torus actions, and as the
diffeomorphism $T^{3}\to T^{3}$ is linear, they commute and hence $X$ has a
$\mathcal{T}$-structure.

As for the blowups of Inoue surfaces, note that
$Y(\overline{\mathbb{CP}^{2}})=12\sqrt{2}\pi>0$, so by a theorem of Kobayashi
on the Yamabe invariant of a connected sum [25, Theorem 2 (a)], it follows
that the blowups also have non-negative Yamabe invariant. Alternatively, as a
non-trivial $S^{1}$-action is a $\mathcal{T}$-structure, we see that
$\overline{\mathbb{CP}^{2}}$ admits a $\mathcal{T}$-structure, and hence so do
the blowups of Inoue surfaces by [38, Theorem 5.9]. In summary, we have the
following:

######  Proposition 4.1.

Let $M$ be an Inoue surface or a blownup Inoue surface. Then $Y(M)\geq 0$.

So all that remains to be shown is that these surfaces do not admit metrics of
positive scalar curvature. As all Inoue surfaces are mapping tori by
Proposition 2.3, we first note the following elementary fact:

######  Proposition 4.2.

Let $F$ be a closed orientable $n$-dimensional manifold, and let $p\colon
M_{f}\to S^{1}$ be the mapping torus of some orientation-preserving
homeomorphism $f\colon F\to F$. The inclusion $F\hookrightarrow M_{f}$ induces
an injection $H_{n}(F;\mathbb{Z})\to H_{n}(M_{f};\mathbb{Z})$. Moreover, the
image of $[F]$ is primitive.

###### Proof.

Letting $U=p^{-1}(S^{1}\setminus\\{i\\})$ and
$V=p^{-1}(S^{1}\setminus\\{-i\\})$, we have inclusion maps $k^{U}\colon U\cap
V\to U$, $k^{V}\colon U\cap V\to V$, $\ell^{U}\colon U\to M_{f}$, and
$\ell^{V}\colon V\to M_{f}$. Identifying $F$ with $p^{-1}(1)$, we also have
inclusion maps $j:F\to U$ and $\iota:F\to M_{f}$. Now consider the following
diagram (all homology groups are with $\mathbb{Z}$ coefficients)

${H_{n+1}(M_{f})}$${H_{n}(U\cap V)}$${H_{n}(U)\oplus
H_{n}(V)}$${H_{n}(M_{f})}$${H_{n-1}(U\cap
V)}$${H_{n}(F)}$$\scriptstyle{(k^{U}_{*},k^{V}_{*})}$$\scriptstyle{\ell^{U}_{*}-\ell^{V}_{*}}$$\scriptstyle{(j_{*},0)}$$\scriptstyle{\iota_{*}}$

The first row is exact by Mayer-Vietoris, and as
$(\ell^{U}_{*}-\ell^{V}_{*})\circ(j_{*},0)=\ell^{U}_{*}\circ
j_{*}-\ell^{V}_{*}\circ 0=(\ell^{U}\circ j)_{*}=\iota_{*}$, the triangle
commutes.

Note that $H_{n+1}(M_{f})\cong\mathbb{Z}$, $H_{n}(U\cap V)\cong H_{n}(F\sqcup
F)\cong\mathbb{Z}^{2}$, and $H_{n}(U)\cong H_{n}(V)\cong
H_{n}(F)\cong\mathbb{Z}$. As each connected component of $U\cap V$ includes
into both $U$ and $V$, the map $\mathbb{Z}^{2}\to\mathbb{Z}^{2}$ is
multiplication by $\begin{bmatrix}1&1\\\ 1&1\end{bmatrix}$. The kernel of this
map is the span of $\begin{bmatrix}1\\\ -1\end{bmatrix}$, so by exactness, the
map $\mathbb{Z}\to\mathbb{Z}^{2}$ is multiplication by $\begin{bmatrix}1\\\
-1\end{bmatrix}$. Furthermore, $H_{n}(F)\cong\mathbb{Z}$ and the map
$(j_{*},0)$ corresponds to multiplication by $\begin{bmatrix}1\\\
0\end{bmatrix}$. As $\begin{bmatrix}1\\\ 0\end{bmatrix}$ is not in the image
of the map $\mathbb{Z}^{2}\to\mathbb{Z}^{2}$, it follows that $\iota_{*}$ is
injective and has the same image as $\ell^{U}_{*}-\ell^{V}_{*}$.

Suppose now that $\iota_{*}[F]=a\gamma$ for some $\gamma\in H_{n}(M_{f})$.
Then the image of $\gamma$ under $H_{n}(M_{f})\to H_{n-1}(U\cap V)$ is
$a$-torsion, but $H_{n-1}(U\cap V)\cong H_{n-1}(F\sqcup F)$ is torsion-free,
so the image of $\gamma$ is zero. By exactness, $\gamma$ is in the image of
$\ell_{U}^{*}-\ell_{V}^{*}$ and hence in the image of $\iota_{*}$. As $[F]$
generates $H_{n}(F)$, we see that $\gamma=\iota_{*}(b[F])$ for some
$b\in\mathbb{Z}$, but then $\iota_{*}[F]=a\gamma=\iota_{*}(ab[F])$. It follows
from the injectivity of $\iota_{*}$ that $a=\pm 1$, and hence $\iota_{*}[F]$
is primitive. ∎

In the case that $b_{n}(M_{f})=1$, as is the case for Inoue surfaces, we
obtain the following corollary.

######  Corollary 4.3.

Let $F$ be a closed orientable $n$-dimensional manifold, and let $p\colon
M_{f}\to S^{1}$ be the mapping torus of some orientation-preserving
homeomorphism $f\colon F\to F$. Suppose that $b_{n}(M_{f})=1$. Then
$H_{n}(M_{f};\mathbb{Z})$ is generated by the image of $[F]$ under the map
$H_{n}(F;\mathbb{Z})\to H_{n}(M_{f};\mathbb{Z})$ induced by the inclusion
$F\hookrightarrow M_{f}$.

We will need the following lemma:

######  Lemma 4.4.

Let $M$ be a closed connected smooth manifold with a closed connected smooth
hypersurface $\Sigma$. If $\Sigma$ is non-orientable, or $[\Sigma]\in
H_{n-1}(M;\mathbb{Z})$ is non-zero, then $M\setminus\Sigma$ is connected.

###### Proof.

Let $U$ be a tubular neighbourhood of $\Sigma$ in $M$, and let
$V=M\setminus\Sigma$. Then by Mayer-Vietoris, we have

$\dots\to H_{0}(U\cap V)\to H_{0}(U)\oplus H_{0}(V)\to H_{0}(M)\to 0.$

Note that $H_{0}(U)\cong H_{0}(M)\cong\mathbb{Z}$ as $\Sigma$ and $M$ are
connected, and $H_{0}(V)\cong\mathbb{Z}^{k}$ where $k$ is the number of
connected components of $V$. Also note that $U\cap V$ deformation retracts
onto the orientation double cover of $\Sigma$, so if $\Sigma$ is non-
orientable we have

$\dots\to\mathbb{Z}\to\mathbb{Z}^{k+1}\to\mathbb{Z}\to 0$

from which it immediately follows that $k=1$, i.e. $V=M\setminus\Sigma$ is
connected. If $\Sigma$ is orientable, we instead have

$\dots\to\mathbb{Z}^{2}\to\mathbb{Z}^{k+1}\to\mathbb{Z}\to 0$

which implies that $k\leq 2$. Suppose that $k=2$.

As $\Sigma$ and $M$ are orientable, the normal bundle is trivial, so there is
a diffeomorphism $\phi\colon U\to(-1,1)\times\Sigma$ such that
$\phi|_{\Sigma}$ is the inverse of the inclusion $i\colon\Sigma\to M$. Let
$M_{-}$ and $M_{+}$ be the connected components of $M\setminus\Sigma$ where
$\phi(M_{-}\cap U)=(-1,0)\times\Sigma$ and $\phi(M_{+}\cap
U)=(0,1)\times\Sigma$. Let $f\colon(-1,1)\to\mathbb{R}$ be a non-decreasing
smooth function such that $f|_{\left(-1,-\frac{1}{2}\right)}\equiv 0$ and
$f|_{\left(\frac{1}{2},1\right)}\equiv 1$. Let $\hat{f}\colon M\to\mathbb{R}$
be the function given by $\hat{f}(p)=f(\operatorname{pr}_{1}(\phi(p)))$ for
$p\in U$, and extended by $0$ on $M_{-}\setminus U$ and $1$ on $M_{+}\setminus
U$. Then $d\hat{f}$ is a smooth one-form such that $d\hat{f}|_{M\setminus
U}=0$ and $(\phi^{-1})^{*}(d\hat{f}|_{U})=g(t)dt$ where
$g\colon(-1,1)\to\mathbb{R}$ is a non-negative smooth function such that
$g|_{\left(-1,-\frac{1}{2}\right)\cup\left(\frac{1}{2},1\right)}\equiv 0$ and
$\int_{-1}^{1}g(t)dt=1$. It follows that for any $(n-1)$-form $\eta$ on $M$,
we have $\int_{M}\eta\wedge d\hat{f}=\int_{\Sigma}i^{*}\eta$, so $d\tilde{f}$
is a representative of the Poincaré dual of $[\Sigma]$ under the map
$\Phi\colon H^{1}(M;\mathbb{Z})\to H^{1}_{\text{dR}}(M)$ given by change of
coefficients followed by the de Rham isomorphism. As $H^{1}(M;\mathbb{Z})$ is
torsion-free, the map $\Phi$ is injective, so $\operatorname{PD}([\Sigma])=0$
as $[d\hat{f}]=0$ in $H^{1}_{\text{dR}}(M)$. Finally, as
$H_{n-1}(M;\mathbb{Z})$ is torsion-free, we see that $[\Sigma]=0$. ∎

We now briefly recall two methods which can be used to rule out the existence
of positive scalar curvature metrics on a closed orientable smooth manifold
$M$.

  1. (1)

We say that $(M,g)$ is enlargeable if for every $\varepsilon>0$, there is an
orientable Riemannian covering space which admits an $\varepsilon$-contracting
map onto $(S^{n},g_{\text{round}})$ which is constant at infinity and is of
non-zero degree. As $M$ is compact, any two metrics are Lipschitz equivalent,
so enlargeability is metric independent. The class of enlargeable manifolds is
quite large, it includes manifolds of non-positive sectional curvature,
nilmanifolds, and is closed under domination, i.e. if $M$ admits a map of non-
zero degree onto an enlargeable manifold, then $M$ is enlargeable. Gromov and
Lawson proved that if $M$ is spin and enlargeable, then $M$ does not admit
metrics of positive scalar curvature [13, Theorem A].

  2. (2)

Another approach, due to Schoen and Yau, is via stable minimal hypersurfaces.
In the proof of [45, Theorem 1], they showed that if $g$ is a positive scalar
curvature metric on $M$, and $\Sigma$ is a closed stable minimal hypersurface
of $(M,g)$, then $g|_{\Sigma}$ is conformal to a metric of positive scalar
curvature.

We will combine these two approaches to prove the following:

######  Theorem 4.5.

Inoue surfaces and their blowups do not admit positive scalar curvature
metrics.

###### Proof.

Let $M$ denote the possibly blownup Inoue surface, and let $X$ denote the
minimal model for $M$, so that $M$ is diffeomorphic to
$X\\#k\overline{\mathbb{CP}^{2}}$ for some $k\geq 0$. By Proposition 2.3, the
manifold $X$ is a mapping torus of a three-dimensional nilmanifold $F$ (either
a torus or a circle bundle over a two-dimensional torus).

Let $g$ be a positive scalar curvature metric on $M$. As $[F]\neq 0$ by
Proposition 4.2, there are closed oriented stable minimal hypersurface
$\Sigma_{1},\dots,\Sigma_{\ell}$ and integers $m_{1},\dots,m_{\ell}$ such that
$m_{1}[\Sigma_{1}]+\dots+m_{\ell}[\Sigma_{\ell}]=[F]$, see [30, Remark 3.4].
By construction, we have $[\Sigma_{j}]\neq 0$ for $j=1,\dots,\ell$. Letting
$\Sigma:=\Sigma_{1}$, and noting that $b_{3}(M)=b_{3}(X)=1$, we see that there
is a non-zero integer $n$ such that $[\Sigma]=n[F]$ by Corollary 4.3.

As $\Sigma$ and $M$ are orientable, the normal bundle is trivial, so there is
an embedding $\phi\colon(-1,1)\times\Sigma\to M$ such that $\phi(0,\cdot)$ is
the inclusion $i\colon\Sigma\hookrightarrow M$. Note that $M\setminus\Sigma$
is a connected non-compact manifold by Lemma 4.4. We can compactify
$M\setminus\Sigma$ to obtain a manifold $Y$ with boundary $\partial
Y=\Sigma\sqcup\Sigma$ and interior diffeomorphic to $M\setminus\Sigma$. To see
this, let $\phi_{-}\colon(-1,0)\times\Sigma\to M\setminus\Sigma$ and
$\phi_{+}\colon(0,1)\times\Sigma\to M\setminus\Sigma$ be the embeddings given
by restricting $\phi$ appropriately and set
$Y=(\phi([0,1)\times\Sigma)\sqcup(M\setminus\Sigma)\sqcup\phi((-1,0]\times\Sigma))/\sim$
where $\phi_{+}(t,p)\sim\phi(t,p)$ for $(t,p)\in(0,1)\times\Sigma$ and
$\phi_{-}(t,p)\sim\phi(t,p)$ for $(t,p)\in(-1,0)\times\Sigma$.

Fix a diffeomorphism $f\colon\Sigma\times\\{0,1\\}\to\partial Y$. Now let
$Z=(Y\times\mathbb{Z})/\sim$ where $(f(s,1),m)\sim(f(s,0),m+1)$ for all
$s\in\Sigma$ and $m\in\mathbb{Z}$. Note that the self-map of
$Y\times\mathbb{Z}$ given by $(y,m)\mapsto(y,m+1)$ descends to a fixed-point
free diffeomorphism of $Z$. The quotient of $Z$ by the $\mathbb{Z}$-action
generated by this diffeomorphism can be naturally identified with $M$, so we
have a regular covering $\rho\colon Z\to M$ with group of deck transformations
$\mathbb{Z}$. Hence, there is a short exact sequence

$0\to\pi_{1}(Z)\xrightarrow{\rho_{*}}\pi_{1}(M)\xrightarrow{\alpha}\mathbb{Z}\to
0.$

Using the fact that $X$ is a mapping torus, we can construct another covering.
First note that there is a regular covering $F\times\mathbb{R}\to X$ with deck
transformation group $\mathbb{Z}$ generated by $(x,t)\mapsto(f(x),t+1)$ where
$f$ is the diffeomorphism which gives rise to the mapping torus $X$. There is
a corresponding regular covering $\tau\colon M^{\prime}\to M$ where
$M^{\prime}$ is diffeomorphic to the connected sum of $F\times\mathbb{R}$ with
a copy of $k\overline{\mathbb{CP}^{2}}$ at $(p,m+\frac{1}{2})$ for some fixed
$p\in F$ and all $m\in\mathbb{Z}$. Hence, there is a short exact sequence

$0\to\pi_{1}(M^{\prime})\xrightarrow{\tau_{*}}\pi_{1}(M)\xrightarrow{\beta}\mathbb{Z}\to
0.$

As $b_{1}(M)=b_{1}(X)=1$, we see that
$\alpha,\beta\in\operatorname{Hom}(\pi_{1}(M),\mathbb{Z})\cong
H^{1}(M;\mathbb{Z})\cong\mathbb{Z}$, so there is $k\in\mathbb{Z}$ such that
$\alpha=k\beta$ or $\beta=k\alpha$. As $\alpha$ and $\beta$ are surjective, it
follows that $k=\pm 1$ and hence
$\pi_{1}(Z)\cong\ker\alpha=\ker\beta\cong\pi_{1}(M^{\prime})$. Therefore, the
two coverings are isomorphic; let $\Psi\colon Z\to M^{\prime}$ be an
isomorphism of covering spaces.

The inclusion of $\Sigma$ as one of the boundary components of $Y$ induces an
inclusion $\iota\colon\Sigma\hookrightarrow Z$ such that $\rho\circ\iota=i$,
so we have the following commutative diagram:

${\Sigma}$${Z}$${M^{\prime}}$${M}$$\scriptstyle{\iota}$$\scriptstyle{i}$$\scriptstyle{\rho}$$\scriptstyle{\Psi}$$\scriptstyle{\tau}$

As $i_{*}[\Sigma]=n[F]\neq 0$, we see that $(\Psi\circ\iota)_{*}[\Sigma]\neq
0$. In fact, as $\tau_{*}$ is an isomorphism on $H_{3}$, we see that
$(\Psi\circ\iota)_{*}[\Sigma]=n[F]$. Note that there is a map $g\colon
M^{\prime}\to F$ by first mapping to $F\times\mathbb{R}$ then projecting onto
$F$, and $g_{*}$ is an isomorphism on $H_{3}$. Therefore
$g\circ\Psi\circ\iota\colon\Sigma\to F$ is a map of three-manifolds with
$(g\circ\Psi\circ\iota)_{*}[\Sigma]=n[F]$; that is, the map has degree $n\neq
0$. As $F$ is a nilmanifold, it is enlargeable, and therefore $\Sigma$ is also
enlargeable. Moreover, as $\Sigma$ is a closed orientable three-manifold, it
is spin, and hence does not admit a metric of positive scalar curvature. This
is a contradiction as $\Sigma$ is a stable minimal hypersurface and $g$ is a
metric of positive scalar curvature. ∎

The previous argument proves the following statement immediately:

######  Theorem 4.6.

Let $X$ be the mapping torus of a diffeomorphism $f\colon F\to F$ where $F$ is
a closed enlargeable three-manifold. If $b_{1}(X)=1$, then $X\\#N$ does not
admit a positive scalar curvature metric for any closed orientable four-
manifold $N$ with $b_{1}(N)=0$.

######  Remark 4.1.

It follows from Proposition IV.6.10 and Corollary IV.6.12 of [31] that $X\\#N$
is weakly enlargeable, a concept which we will not define here. Every compact
enlargeable manifold is weakly enlargeable, but it is not known if the
converse is true. It is also not known if a non-spin weakly enlargeable
manifold can admit metrics of positive scalar curvature.

######  Theorem 4.7.

Let $M$ be an Inoue surface or a blownup Inoue surface. Then $Y(M)=0$, but it
is not realised.

###### Proof.

Combining Proposition 4.1 and Theorem 4.5 gives $Y(M)=0$. If the Yamabe
invariant were realised, then $M$ would admit a scalar flat metric, but any
such metric must be Ricci flat as $M$ does not admit positive scalar metrics
[24, Lemma 5.2]. By [34, Theorem A], non-Kähler surfaces cannot admit Ricci
flat metrics. ∎

So we see that the Kähler hypothesis of Theorem 3.1 is necessary.

We take this opportunity to record the following result, which was alluded to
by LeBrun in [33], and also follows from the extension of LeBrun’s Theorem to
the symplectic setting in [49]:

######  Theorem 4.8.

Let $M$ be a Kodaira surface or a blownup Kodaira surface. Then $Y(M)=0$ and
it is not realised.

###### Proof.

As in the Inoue surface case, Paternain and Petean showed that Kodaira
surfaces admit $\mathcal{T}$-structures [39, Theorem B], and hence Kodaira
surfaces and their blowups have non-negative Yamabe invariant.

Now note that by Proposition 2.4, primary Kodaira surfaces are symplectic. The
complex blowup of a complex manifold is diffeomorphic to the symplectic
blowup, so blownup primary Kodaira surfaces are symplectic. Such surfaces
therefore have a non-trivial Seiberg-Witten invariant [52], and as
$b^{+}=2h^{2,0}=2>1$, they cannot admit metrics of positive scalar curvature
[37, Corollary 2.3.8]. The same is true of their free quotients, namely
secondary Kodaira surfaces and their blowups.

Combining, we see that the Yamabe invariant of Inoue surfaces and their
blowups is zero. Again by [34, Theorem A], non-Kähler surfaces cannot admit
Ricci flat metrics and hence the Yamabe invariant is not realised. ∎

So we see that the Kähler hypothesis of Theorem 3.2 is also necessary.

Modulo the status of the global spherical shell conjecture, the only connected
compact non-Kähler surfaces for which the Yamabe invariant is unknown are the
secondary Hopf surfaces in class (2) of Theorem 2.2, properly elliptic non-
Kähler surfaces, and their blowups. The author plans to address these
remaining cases in a future paper.

## References

  * [1] Aubin, T., 1976. Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl.(9), 55, pp.269-296.
  * [2] Auslander, L., 1973. An exposition of the structure of solvmanifolds. Part I: Algebraic theory. Bulletin of the American Mathematical Society, 79(2), pp.227-261.
  * [3] Besse, A.L., 2007. Einstein manifolds. Springer Science & Business Media.
  * [4] Bogomolov, F., 1976. Classification of surfaces of class $\mathrm{VII}_{0}$ with $b_{2}=0$. Mathematics of the USSR-Izvestija, 10(2), pp.255-269.
  * [5] Bogomolov, F., 1982. Surfaces of class $\mathrm{VII}_{0}$ and affine geometry. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 46(4), pp.710-761.
  * [6] Buchdahl, N., 1999. On compact Kähler surfaces. In Annales de l’institut Fourier (Vol. 49, No. 1, pp. 287-302).
  * [7] Cheeger, J. and Gromov, M., 1985. Collapsing Riemannian manifolds while keeping their curvature bounded: I. Mathematical Sciences Research Institute.
  * [8] Cecchini, S. and Schick, T., 2018. Enlargeable metrics on nonspin manifolds. arXiv preprint arXiv:1810.02116.
  * [9] Dloussky, G., 2001. Complex surfaces with Betti numbers $b_{1}=1$, $b_{2}>0$ and finite quotients. Contemporary Mathematics, 288, pp.305-309.
  * [10] Dloussky, G., 2006. On surfaces of class $\mathrm{VII}^{+}_{0}$ with numerically anticanonical divisor. American journal of mathematics, 128(3), pp.639-670.
  * [11] Endo, H. and Pajitnov, A., 2019. On generalized Inoue manifolds. arXiv preprint arXiv:1903.08030.
  * [12] Gromov, M. and Lawson, H.B., 1980. The classification of simply connected manifolds of positive scalar curvature. Annals of Mathematics, pp.423-434.
  * [13] Gromov, M. and Lawson, H.B., 1980. Spin and scalar curvature in the presence of a fundamental group. I. Ann. of Math, 111(2), pp.209-230.
  * [14] Gromov, M. and Lawson, H.B., 1983. Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publications Mathématiques de l’IHÉS, 58, pp.83-196.
  * [15] Gursky, M. and LeBrun, C, 1998. Yamabe Invariants and Spinc Structures. Geometric & Functional Analysis GAFA, 8(6), pp. 965-977.
  * [16] Hasegawa, K., 2005. Complex and Kähler structures on compact solvmanifolds. Journal of Symplectic Geometry, 3(4), pp.749-767.
  * [17] Inoue, M., 1974. On surfaces of class VII0. Inventiones mathematicae, 24(4): pp.269-310.
  * [18] Inoue, M., 1975. An example of an analytic surface. Sugaku, 27, pp.358-364.
  * [19] Lebrun, C. and Isida, M., 2003. Curvature, connected sums, and Seiberg-Witten theory. Communications in analysis and geometry, 11(5), pp.809-836.
  * [20] Ishida, M., Matsuo, S. and Nakamura, N., 2016. Yamabe invariants and the $\mathrm{Pin}^{-}(2)$-monopole equations. Mathematical Research Letters, 23(4), pp.1049-1069.
  * [21] Kato, M., 1975. Topology of Hopf surfaces. Journal of the Mathematical Society of Japan, 27(2), pp.222-238.
  * [22] Kato, M., 1977. Compact complex manifolds containing “global” spherical shells. Proceedings of the Japan Academy, 53(1), pp.15-16.
  * [23] Kato, M., 1989. Erratum to “Topology of Hopf surfaces”. Journal of the Mathematical Society of Japan, 41(1), pp.173-174.
  * [24] Kazdan, J. and Warner, F., 1975. Prescribing curvatures. In Proceedings of Symposia in Pure Mathematics (Vol. 27, pp. 309-319).
  * [25] Kobayashi, O., 1987. Scalar curvature of a metric with unit volume. Mathematische Annalen, 279(2), pp.253-265.
  * [26] Kodaira, K., 1966. On the structure of compact complex analytic surfaces, II. American Journal of Mathematics, 88(3), pp.682-721.
  * [27] Lamari, A., 1999. Courants kählériens et surfaces compactes. In Annales de l’institut Fourier (Vol. 49, No. 1, pp. 263-285).
  * [28] Lattès, S., 1911. Sur les formes réduites des transformations ponctuelles dans le domaine d’un point double. Bulletin de la Société Mathématique de France, 39, pp.309-345.
  * [29] Laufer, H.B., 1971. Normal two-dimensional singularities. Princeton University Press.
  * [30] Lawson, H.B., 1974. Minimal varieties in real and complex geometry (Vol. 57). Les Presses de l’Université de Montréal.
  * [31] Lawson, H.B. and Michelsohn, M.L., 1989. Spin geometry (pms-38) (Vol. 38). Princeton university press.
  * [32] LeBrun, C., 1997. Yamabe constants and perturbed Seiberg-Witten equations. Comm. Anal. Geom., 5, pp.535-553.
  * [33] LeBrun, C., 1999. Kodaira dimension and the Yamabe problem. Communications in Analysis and Geometry, 7, pp.135-156.
  * [34] LeBrun, C., 2009, July. Einstein metrics, complex surfaces, and symplectic 4-manifolds. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 147, No. 1, pp. 1-8). Cambridge University Press.
  * [35] Li, J., Yau, S.T. and Zheng, F., 1994. On projectively flat Hermitian manifolds. Communications in Analysis and Geometry, 2(1), pp.103-109.
  * [36] Lu, P. and Tian, G., 1996. The complex structures on connected sums of $S^{3}\times S^{3}$, in ‘Manifolds and geometry’ (Pisa, 1993), 284–293. Sympos. Math., XXXVI, Cambridge Univ. Press, Cambridge.
  * [37] Nicolaescu, L.I., 2000. Notes on Seiberg-Witten theory (Vol. 28). American Mathematical Soc..
  * [38] Paternain, G.P. and Petean, J., 2003. Minimal entropy and collapsing with curvature bounded from below. Inventiones mathematicae, 151(2), pp.415-450.
  * [39] Paternain, G.P. and Petean, J., 2004. Entropy and collapsing of compact complex surfaces. Proceedings of the London Mathematical Society, 89(3), pp.763-786.
  * [40] Petean, J., 1998. Computations of the Yamabe invariant. Mathematical Research Letters, 5(6), pp.703-709.
  * [41] Petean, J, 2000. The Yamabe invariant of simply connected manifolds. J. Reine Angew. Math. 523, pp.225-231.
  * [42] Petean, J. and Ruiz, J.M., 2011. Isoperimetric profile comparisons and Yamabe constants. Annals of Global Analysis and Geometry, 40(2), pp.177-189.
  * [43] Schoen, R., 1984. Conformal deformation of a Riemannian metric to constant scalar curvature. Journal of Differential Geometry, 20(2), pp.479-495.
  * [44] Schoen, R.M., 1989. Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In Topics in calculus of variations (pp. 120-154). Springer, Berlin, Heidelberg.
  * [45] Schoen, R. and Yau, S.T., 1979. On the structure of manifolds with positive scalar curvature. Manuscripta mathematica, 28(1), pp.159-183.
  * [46] Schoen, R. and Yau, S.T., 2017. Positive scalar curvature and minimal hypersurface singularities. arXiv preprint arXiv:1704.05490.
  * [47] Siu, Y.T., 1983. Every $K3$ surface is Kähler. Inventiones mathematicae, 73(1), pp.139-150.
  * [48] Sternberg, S., 1957. Local contractions and a theorem of Poincaré. American Journal of Mathematics, 79(4), pp.809-824.
  * [49] Suárez-Serrato, P. and Torres, R., 2014. A note on collapse, entropy, and vanishing of the Yamabe invariant of symplectic $4$-manifolds. Journal of Geometry and Physics, 86, pp.383-391.
  * [50] Sung, C., 2009. Surgery, Yamabe invariant, and Seiberg–Witten theory. Journal of Geometry and Physics, 59(2), pp.246-255.
  * [51] Şuvaina, I., 2015. The Yamabe invariant of a class of symplectic $4$-manifolds. Journal of Geometry and Physics, 91, pp.54-65.
  * [52] Taubes, C.H., 1994. The Seiberg-Witten invariants and symplectic forms. Mathematical Research Letters, 1(6), pp.809-822.
  * [53] Teleman, A., 1994. Projectively flat surfaces and Bogomolov’s theorem on class $\mathrm{VII}_{0}$ surfaces. International Journal of Mathematics, 5(02), pp.253-264.
  * [54] Teleman, A., 2005. Donaldson theory on non-Kählerian surfaces and class $\mathrm{VII}$ surfaces with $b_{2}=1$. Inventiones mathematicae, 162(3), pp.493-521.
  * [55] Teleman, A., 2010. Instantons and curves on class $\mathrm{VII}$ surfaces. Annals of Mathematics, 172, pp.1749-1804.
  * [56] Teleman, A., 2018. Donaldson theory in non-Kählerian geometry, in “Modern Geometry: A Celebration of the Work of Simon Donaldson”, Proceedings of Symposia in Pure Mathematics, Vol. 99, pp.363-392.
  * [57] Trudinger, N.S., 1968. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 22(2), pp.265-274.
  * [58] Yamabe, H., 1960. On a deformation of Riemannian structures on compact manifolds. Osaka Mathematical Journal, 12(1): 21-37.