# Contact Structures on Folded Sums of Contact Mapping Tori and Surface
Bundles

Mehmet Firat Arikan  Dept. of Mathematics, Middle East Technical University,
Ankara, TURKEY<EMAIL_ADDRESS>
###### Abstract.

We introduce a “folded sum operation” which glues two compact mapping tori
along their common (diffeomorphic) boundaries. We call the resulting closed
manifold a “folded sum of mapping tori”, and it naturally fibers over the
circle. We show that any folded sum $M$ of two “contact” mapping tori (their
fibers are compact exact symplectic manifolds with contactomorphic convex
boundaries) admits a cooriented contact structure which is “compatible” (in a
certain sense) with the resulting fibration map from $M$ onto the circle.
Moreover, we study the tightness of the resulting contact structure on $M$ for
certain cases. This gives a new method of constructing closed contact
manifolds in any odd dimension. In particular, it recovers some earlier
constructions and results. More interestingly, in dimension three, it provides
the existence of tight contact structures on a large class of (hyperbolic)
surface bundles over the circle.

###### Key words and phrases:

Contact form, contact structure, folded sum, mapping torus, exact symplectic,
tight, hyperbolic, surface bundle

###### 2010 Mathematics Subject Classification:

58D27, 58A05, 57R65

##  1\. Introduction

For any smooth manifold $\Sigma$ and a chosen self-diffeomorphism
$\phi\in\textrm{Diff}(\Sigma)$, the mapping torus $\mathcal{M}(\Sigma,\phi)$
is the quotient manifold of $\Sigma\times[0,1]$ obtained by identifying each
point $x\in\Sigma\times\\{0\\}$ with its image
$\phi(x)\in\Sigma\times\\{1\\}$. (If $\Sigma$ is orientable, so is
$\mathcal{M}(\Sigma,\phi)$ provided $\phi\in\textrm{Diff}\,^{+}(\Sigma)$.) By
its definition, $\mathcal{M}(\Sigma,\phi)$ fibers over $S^{1}$ and $\phi$ is
the monodromy. Equivalently, in bundle theoretical words, there is a (locally
trivial) fiber bundle $\pi:\mathcal{M}(\Sigma,\phi)\longrightarrow S^{1}$ with
fibers $\Sigma$ and the structure group $\textrm{Diff}(\Sigma)$.

Throughout this note, we will restrict our attention to the case where
$\Sigma$ is a compact exact symplectic manifold with convex boundary
$\partial\Sigma$ and $\phi$ is an exact symplectomorphism which is identity
near $\partial\Sigma$ (for precise definitions see Section 2). Such a mapping
torus will be called a contact mapping torus.

If we require $\phi\in\textrm{Diff}\,^{+}(\Sigma)$ to be identity near
$\partial\Sigma$ (i.e., $\phi\in\textrm{Diff}\,^{+}(\Sigma,\partial\Sigma)$),
then one can glue $\mathcal{M}(\Sigma,\phi)$ and $\partial\Sigma\times D^{2}$
along their boundaries (so that the angular coordinate on $D^{2}$ agrees with
the map $\pi$) to obtain a closed orientable manifold

$Y(\Sigma,\phi)=\mathcal{M}(\Sigma,\phi)\cup_{\partial}(\partial\Sigma\times
D^{2}).$

Such a decomposition is called an (abstract) open book of the resulting
manifold $Y(\Sigma,\phi)$ and will be denoted by $\mathcal{OB}(\Sigma,\phi)$.

If $\mathcal{M}(\Sigma,\phi)$ is a contact mapping torus, then
$\mathcal{OB}(\Sigma,\phi)$ is a contact open book. That is, one can construct
a contact form (and so structure) on $\mathcal{M}(\Sigma,\phi)$ and glue with
the natural one on $\partial\Sigma\times D^{2}$ to obtain a smooth contact
form on $Y(\Sigma,\phi)$. This construction is due to Thurston and
Winkelnkemper [11] for dimension three, and to Giroux [7, 8] for any higher
odd dimensions. Their construction of the contact form on
$\mathcal{M}(\Sigma,\phi)$ is based on the fact that the monodromy is an exact
symplectomorphism of the fiber (which is identity near the boundary), and also
that the identification map building $\mathcal{M}(\Sigma,\phi)$ can be chosen
in such a way that it pulls back the product contact form on
$\Sigma\times[0,1]$ to itself.

In this note, we will construct contact forms on contact mapping tori by
making use of their (exact) symplectic bundle structures. As pointed out in
Remark 3.5, the resulting contact structures will contactomorphic to the ones
constructed in [11, 7, 8]. Contact mapping tori are the simplest compact
contact symplectic fibrations which are studied in [1] where a natural bundle
theoretical method constructing contact forms on them is given. This bundle
theoretical construction has more advantages in constructing contact forms on
folded pair of contact mapping tori. More precisely, it is easier to adapt it
to the technique given by Geiges and Stipsicz in [6] where they construct
contact forms on products of closed $4$-manifolds with $S^{1}$.

The paper is organized as follows: In Section 2, we will briefly recall the
related contact and symplectic background. A bundle theoretical construction
of contact structures on contact mapping tori will be given in Section 3. By
combining techniques given in [1] and [6], we will construct contact forms on
folded sums of contact mapping tori in Section 4. We will discuss the
tightness of such contact structures for certain cases in Section 5 where some
basic examples are also given. We will close the paper by presenting an
application of folded sum operation in Section 6 where the existence of tight
contact structures on a large class of (hyperbolic) surface bundles over
$S^{1}$ will be shown.

##  2\. Preliminaries

We start with by recalling some necessary background. For more details in
contact and symplectic topology, we refer the reader to [5] and [9].

A smooth global $1$-form $\alpha$ on a smooth oriented $(2n+1)$-dimensional
manifold $Y$ is a _positive contact form_ if $\alpha\wedge(d\alpha)^{n}>0$.
The kernel $\xi=\textrm{Ker}(\alpha)$ of such $\alpha$ is called a _positive
(co-oriented) contact structure_ on $Y$, and the pair $(Y,\xi)$ is called a
_contact manifold_. Two contact manifolds $(Y,\xi=\textrm{Ker}(\alpha))$ and
$(Y^{\prime},\xi^{\prime}=\textrm{Ker}(\alpha^{\prime}))$ are
_contactomorphic_ if there exists a diffeomorphism $f:Y\longrightarrow
Y^{\prime}$ such that $f_{\ast}(\xi)=\xi^{\prime}$ (equivalently,
$f^{*}(\alpha^{\prime})=\lambda\alpha$ for some nonzero smooth map
$\lambda:Y\to\mathbb{R}$). Such a map $f$ is called a contactomorphism. Two
contact structures $\xi_{0},\xi_{1}$ on a $Y$ are said to be _isotopic_ if
there exists a $1$-parameter family $\xi_{t}$ ($0\leq t\leq 1$) of contact
structures joining them.

A positive symplectic form on a smooth oriented $2n$-manifold $\Sigma$ is a
smooth $2$-form $\omega$ which is closed (i.e., $d\omega=0$) and nondegenerate
(i.e., $\omega^{n}>0$). The pair $(\Sigma,\omega)$ is called a symplectic
manifold. Two symplectic manifolds $(\Sigma_{1},\omega_{1})$ and
$(\Sigma_{2},\omega_{2})$ are _symplectomorphic_ if there exists a
diffeomorphism $f:\Sigma_{1}\longrightarrow\Sigma_{2}$ such that
$f^{*}(\omega_{2})=\omega_{1}$.

A compact _exact symplectic manifold_ (necessarily with boundary) is a
symplectic manifold $(\Sigma,\omega=d\beta)$ where the $\omega$-dual Liouville
vector field $X$ of $\beta$ (defined by $\iota_{X}\omega=\beta$) should point
strictly outwards along $\partial\Sigma$. In particular,
$\beta|_{T\partial\Sigma}$ is a contact form on $\partial\Sigma$, and the pair
$(\partial\Sigma,\textrm{Ker}(\beta|_{T\partial\Sigma}))$ is called the convex
boundary of $(\Sigma,d\beta)$. An _exact symplectomorphism_
$\phi:(\Sigma_{1},d\beta_{1})\longrightarrow(\Sigma_{2},d\beta_{2})$ between
two exact symplectic manifolds is a diffeomorphism such that
$\phi^{*}(\beta_{2})-\beta_{1}$ is exact.

An exact symplectic filling of a closed contact manifold
$(Y,\xi=\textrm{Ker}(\alpha))$ is a compact exact symplectic manifold
$(\Sigma,d\beta)$ such that $Y=\partial\Sigma$ (as oriented manifolds) and
$\alpha=\beta|_{TY}$. In this case, using the flow of $X$, a collar
neighborhood of $Y$ in $\Sigma$ can be identified with the part
$(\partial\Sigma\times(-\epsilon,0],d(e^{t}\alpha))$ of the symplectization
$(\partial\Sigma\times\mathbb{R},d(e^{t}\alpha))$ of $(Y,\xi)$ for some small
$\epsilon>0$. Thus, on such a collar neighborhood, $\beta=e^{t}\alpha$, and
$X=\partial_{t}$. Moreover, if $(\Sigma_{1},d\beta_{1})$,
$(\Sigma_{2},d\beta_{2})$ are two exact symplectic fillings of the same
contact manifold $(Y,\xi)$, then by multiplying with suitable smooth functions
we may assume that $\beta_{1}$ and $\beta_{2}$ agrees along their common
boundary, i.e., $\beta_{1}|_{TY}=\alpha=\beta_{2}|_{TY}$ and
$\xi=\textrm{Ker}(\alpha)$. Without loss of generality, we will assume this
throughout the work presented here.

Finally, if $(\Sigma,d\beta)$ is a compact exact symplectic manifold, then we
consider the group

$\textrm{Exact}(\Sigma,d\beta)=\\{\phi\in\textrm{Diff}(\Sigma)\,|\,\phi^{*}(\beta)-\beta\;\textrm{
is exact}\\}.$

of exact symplectomorphisms which lies in the symplectic group

$\textrm{Symp}(\Sigma,d\beta)=\\{\phi\in\textrm{Diff}(\Sigma)\,|\,\phi^{*}(d\beta)=d\beta\\}.$

Inside $\textrm{Exact}(\Sigma,d\beta)$, we have the subgroup
$\textrm{Exact}(\Sigma,\partial\Sigma,d\beta)$ of exact symplectomorphisms
which are identity near $\partial\Sigma$. That is,

$\textrm{Exact}(\Sigma,\partial\Sigma,d\beta)=\\{\phi\in\textrm{Exact}(\Sigma,d\beta)\,|\,\phi\,\textrm{
is identity near }\,\partial\Sigma\\}.$

It will be the structure group for our mapping tori considered as fiber
bundles over $S^{1}$.

##  3\. Contact Structures on Contact Mapping Tori

Let us first define our main objects of interest as fiber bundles over the
circle:

######  Definition 3.1.

Let $\Sigma$ be an oriented compact manifold with nonempty boundary and
consider $\phi\in\textrm{Diff}\,^{+}(\Sigma,\partial\Sigma)$. The mapping
torus (associated to $\Sigma$ and $\phi$) is a smooth fiber bundle
$\pi:\mathcal{M}(\Sigma,\phi)\longrightarrow S^{1}$ with fibers $\Sigma$ and
the structure group $\textrm{Diff}\,^{+}(\Sigma,\partial\Sigma)$ such that its
total space is the smooth manifold with the identification

$\mathcal{M}(\Sigma,\phi)\cong\Sigma\times[0,1]/\sim\quad\textrm{where}\;\;(x,0)\sim(\phi(x),1).$

In the case $\Sigma$ is a compact exact symplectic manifold and the monodromy
$\phi$ is an exact symplectomorphism which is identity near $\partial\Sigma$,
the mapping torus $\mathcal{M}(\Sigma,\phi)$ defined in Definition 3.1 will be
called a contact mapping torus and it admits a structure of compact contact
symplectic fibration over $S^{1}$ [1]. One can construct “compatible” contact
structures on such fibrations using their bundle structures. In precise words,
we have:

######  Definition 3.2 ([1]).

Let $\pi:\mathcal{M}(\Sigma,\phi)\longrightarrow S^{1}$ be any mapping torus
such that $\phi\in\textrm{Diff}\,^{+}(\Sigma,\partial\Sigma)$ and
$\mathcal{M}(\Sigma,\phi)$ admits a co-oriented contact structure $\eta$. Then
$\eta$ is said to be _compatible with_ $\pi$ if there exists a contact form
$\sigma$ for $\eta$ such that $d\sigma$ restricts to an exact symplectic form
on every fiber of $\pi$.

######  Theorem 3.3.

Let $\pi:\mathcal{M}(\Sigma,\phi)\longrightarrow S^{1}$ be any contact mapping
torus with the structure group $\emph{Exact}(\Sigma,\partial\Sigma,d\beta)$.
Then the total space $\mathcal{M}(\Sigma,\phi)$ admits a (co-oriented) contact
structure $\eta=\emph{Ker}(\sigma)$ which is compatible with $\pi$, that is,
$d\sigma$ restricts to an exact symplectic form on every fiber. Moreover, the
contact form $\sigma$ is equal to a product form on some collar neighborhood
of the boundary $\partial\mathcal{M}(\Sigma,\phi)\cong\partial\Sigma\times
S^{1}$.

###### Proof.

Let $(\Sigma,d\beta)$ be any fixed fiber of $\pi$. Then the transition maps
are contained in the structure group
$\textrm{Exact}(\Sigma,\partial\Sigma,d\beta)$. Since
$\pi:\mathcal{M}(\Sigma,\phi)\longrightarrow S^{1}$ is locally trivial fiber
bundle, $\mathcal{M}(\Sigma,\phi)$ is constructed by patching the trivial
$\Sigma$-bundles over open arcs in $S^{1}$. Therefore, there is an open cover
$\\{U_{\rho}\\}_{\rho}$ of $S^{1}$ and diffeomorphisms
$\varphi_{\rho}:\pi^{-1}(U_{\rho})\to U_{\rho}\times\Sigma$ satisfying

$\pi|_{\pi^{-1}(U_{\rho})}=p_{1}\circ\varphi_{\rho}$

(where $p_{1}:U_{\rho}\times\Sigma\longrightarrow U_{\rho}$ is the projection
onto the first-factor) such that the collection
$\\{U_{\rho},\varphi_{\rho}\\}_{\rho}$ trivializes $\pi$. For each $\rho$ by
restricting $\varphi_{\rho}$ to each fiber $\Sigma_{\theta}:=\pi^{-1}(\theta)$
over $U_{\rho}$ and then projecting onto $\Sigma$-factor, we obtain a smooth
map $\varphi_{\rho}(\theta):\Sigma_{\theta}\to\Sigma$, and as a result, a
pullback form

$\lambda_{\theta}:=(\varphi_{\rho}(\theta))^{*}(\beta).$

Note that $(\Sigma_{\theta},d\lambda_{\theta})$ is a compact exact symplectic
manifold for each $\theta\in U_{\rho}\subset S^{1}$. Also using the projection
$p_{2}:U_{\rho}\times\Sigma\longrightarrow\Sigma$ onto the second factor, we
have the pullback form

$\lambda_{\rho}:=p_{2}^{*}(\beta)\in\Omega^{1}(U_{\rho}\times\Sigma).$

Hence, we obtain a $1$-form
$\phi_{\rho}^{*}(\lambda_{\rho})\in\Omega^{1}(\pi^{-1}(U_{\rho}))$. By
construction,
$\phi_{\rho}^{*}(\lambda_{\rho})|_{\Sigma_{\theta}}=\lambda_{\theta}$ for each
$\theta\in U_{\rho}$. Let $f_{\rho}:S^{1}\to[0,1]$ be a partition of unity
subordinating to the open cover $\\{U_{\rho}\\}_{\rho}$ of $S^{1}$. We define

$\lambda_{\beta}:=\sum_{\rho}(f_{\rho}\circ\pi)\varphi_{\rho}^{*}(\lambda_{\rho})\;\in\Omega^{1}(\mathcal{M}(\Sigma,\phi)).$

If $\iota_{\theta}:\Sigma_{\theta}\longrightarrow\mathcal{M}(\Sigma,\phi)$ is
the inclusion map of the fiber over $\theta$, then we compute

$\iota_{\theta}^{*}(\lambda_{\beta})=\iota_{\theta}^{*}\left(\sum_{\rho}(f_{\rho}\circ\pi)\varphi_{\rho}^{*}(\lambda_{\rho})\right)=\sum_{\rho}(f_{\rho}\circ\pi)\iota_{\theta}^{*}(f_{\rho}^{*}(\lambda_{\rho}))=\lambda_{\theta},$

that is, $\lambda_{\beta}$ restricts to $\lambda_{\theta}$ on each fiber
$\Sigma_{\theta}$, and so $d\lambda_{\beta}=d\lambda_{\theta}$ is
nondegenerate on each $\Sigma_{\theta}$. In fact,
$d\lambda_{\beta}\in\Omega^{2}(\mathcal{M}(\Sigma,\phi))$ is nondegenerate on
all vertical subspaces

$\textrm{Vert}_{x}:=\textrm{Ker}(D\pi(x))=T_{x}\Sigma_{\theta}\subset
T_{x}\mathcal{M}(\Sigma,\phi),\quad\textrm{for }\theta\in\Sigma,\textrm{ and
for }x\in\pi^{-1}(\theta)$

where $D\pi(x)$ denotes the derivative map of $\pi$ at $x$. Therefore, at each
$x\in\mathcal{M}(\Sigma,\phi)$, we have the horizontal space (the symplectic
orthogonal complement of $\textrm{Vert}_{x}$ in
$T_{x}\mathcal{M}(\Sigma,\phi)$ with respect to $d\lambda_{\beta}$) defined by

$\textrm{Hor}_{x}:=\\{u\in
T_{x}\mathcal{M}(\Sigma,\phi)\,:\,d\lambda_{\beta}(u,v)=0,\;\forall
v\in\textrm{Vert}_{x}\\}\subset T_{x}\mathcal{M}(\Sigma,\phi).$

At each $x\in\Sigma_{\theta}$, since the restriction
$d\lambda_{\theta}=d\lambda_{\beta}|_{\Sigma_{\theta}}$ is of full rank and
the derivative map
$D\pi(x):T_{x}\mathcal{M}(\Sigma,\phi)=\textrm{Vert}_{x}\oplus\textrm{Hor}_{x}\to
T_{\theta}S^{1}$ vanishes on $\textrm{Vert}_{x}$, we conclude that
$D\pi(x)|_{\textrm{Hor}_{x}}:\textrm{Hor}_{x}\to T_{\theta}S^{1}$ is an
isomorphism. Now for any $K>0$, consider the $1$-form
$\sigma\in\Omega^{1}(\mathcal{M}(\Sigma,\phi))$ defined by

$\sigma:=\lambda_{\beta}+K\pi^{*}(d\theta).$

Let $\textrm{dim}(\Sigma)=2n$. Then we compute

$\displaystyle\sigma\wedge(d\sigma)^{n}$ $\displaystyle=$
$\displaystyle(\lambda_{\beta}+K\pi^{*}(d\theta))\wedge(d\lambda_{\beta}+K\pi^{*}(d^{2}\theta))^{n}=(\lambda_{\beta}+K\pi^{*}(d\theta))\wedge(d\lambda_{\beta})^{n}$
$\displaystyle=$
$\displaystyle\lambda_{\beta}\wedge(d\lambda_{\beta})^{n}+K\pi^{*}(d\theta)\wedge(d\lambda_{\beta})^{n}.$

Observe that at each $x\in\mathcal{M}(\Sigma,\phi)$, the form
$\sigma\wedge(d\sigma)^{n}|_{x}$ is a positive volume form on the tangent
space $T_{x}\mathcal{M}(\Sigma,\phi)$ provided $K=K_{x}>0$ is chosen large
enough. Since $\mathcal{M}(\Sigma,\phi)$ is compact, there exists a
sufficiently large $K>0$ such that $\sigma\wedge(d\sigma)^{n}|_{x}$ is a
positive volume form on $T_{x}\mathcal{M}(\Sigma,\phi)$ for all
$x\in\mathcal{M}(\Sigma,\phi)$. Equivalently, for any large enough $K>0$,
$\sigma$ is a global contact form and defines a (co-oriented) contact
structure $\eta=\textrm{Ker}(\sigma)$ on $\mathcal{M}(\Sigma,\phi)$.

Moreover, for each $\theta\in S^{1}$ we have

$\iota_{\theta}^{*}(\sigma)=\iota_{\theta}^{*}(\lambda_{\beta}+K\pi^{*}(d\theta))=\iota_{\theta}^{*}(\lambda_{\beta})+K(\pi\circ\iota_{\theta})^{*}(d\theta)=\iota_{\theta}^{*}(\lambda_{\beta})+0=\lambda_{\theta}$

which means, by definition, that $\eta$ is compatible with $\pi$ as required.

Finally, observe that since the structure group of
$\pi:\mathcal{M}(\Sigma,\phi)\to S^{1}$ is
$\textrm{Exact}(\Sigma,\partial\Sigma,d\beta)$ (which is a subgroup of
$\textrm{Diff}\,^{+}(\Sigma,\partial\Sigma)$) and $S^{1}$ is compact, there
exists $\epsilon>0$ such that a collar neighborhood $N$ of
$\partial\mathcal{M}(\Sigma,\phi)\cong\partial\Sigma\times S^{1}$ has an
identification $N=(-\epsilon,0]\times\partial\Sigma\times S^{1}$. Moreover,
the $1$-form $\lambda_{\beta}$ is constructed in such a way that it restricts
to $\lambda_{\theta}$ on each fiber $\Sigma_{\theta}$. But all these
$\lambda_{\theta}$’s are obtained by pulling back the same $\beta$ fixed at
the beginning. Hence, on each collar neighborhood $N\cap\Sigma_{\theta}$,
$\lambda_{\theta}$ (and so $\lambda_{\beta}$) restricts to $\beta$.
Equivalently, on the collar neighborhood $N$, $\sigma$ is the product form
$\beta+Kd\theta$ as claimed.

∎

######  Definition 3.4 ([1]).

The contact form $\sigma$ on $\mathcal{M}(\Sigma,\phi)$ constructed in the
above proof will be called a bundle contact form, and its kernel $\eta$ will
be called a bundle contact structure.

######  Remark 3.5.

One can see that the mapping torus
$\pi:\mathcal{M}(\Sigma,\phi)\longrightarrow S^{1}$ together with the bundle
contact structure $\eta=\textrm{Ker}(\sigma)$ is, in fact, contactomorphic to
the contact manifold
$(\mathcal{M}^{\prime}(\Sigma,\phi),\eta^{\prime}=\textrm{Ker}(\sigma^{\prime}))$
constructed in [11, 7, 8] where

$\mathcal{M}^{\prime}(\Sigma,\phi)=\Sigma\times[0,1]/\sim\;,\qquad(x,0)\sim(\phi(x),1)$

and the identification map of “$\sim$” is directly used in the construction of
the contact form $\sigma^{\prime}$. To verify this claim, first close up the
contact manifolds
$(\mathcal{M}(\Sigma,\phi),\eta),(\mathcal{M}^{\prime}(\Sigma,\phi),\eta^{\prime})$
by gluing two copies of $\partial\Sigma\times D^{2}$ equipped with suitable
contact structures which can be smoothly glued to $\eta$ and $\eta^{\prime}$,
respectively. This is just the standard procedure in obtaining contact open
books from contact mapping tori used in [11, 7, 8]. Therefore, we obtain two
isomorphic contact open books $\mathcal{OB}:=\mathcal{OB}(\Sigma,\phi)$,
$\mathcal{OB}^{\prime}:=\mathcal{OB}^{\prime}(\Sigma,\phi)$ on the closed
contact manifolds, say

$(M,\xi=\textrm{Ker}(\alpha)),\;\;(M^{\prime},\xi^{\prime}=\textrm{Ker}(\alpha^{\prime})),$

respectively, where $\alpha$ (resp. $\alpha^{\prime}$) is obtained from
$\sigma$ (resp. $\sigma^{\prime}$) by (smoothly) gluing a suitable contact
form on $\partial\Sigma\times D^{2}$. Note that the open books
$\mathcal{OB},\mathcal{OB}^{\prime}$ support $\xi,\xi^{\prime}$, respectively,
and since they are isomorphic, there exists a fiber preserving diffeomorphism
$f:M\to M^{\prime}$, that is, taking the fibers of $\mathcal{OB}$ onto those
of $\mathcal{OB}^{\prime}$ and mapping any (trivial) neighborhood of the
binding of $\mathcal{OB}$ to that of $\mathcal{OB}^{\prime}$. Observe that the
pullback form $\alpha^{*}:=f^{*}(\alpha^{\prime})$ defines a contact structure
$\xi^{*}=\textrm{Ker}(\alpha^{*})$ which is also supported by $\mathcal{OB}$.
Therefore, by Giroux correspondence, there exists an isotopy
$\\{\xi_{t}|t\in[0,1]\\}$ of contact structures on $M$ between $\xi_{0}=\xi$
and $\xi_{1}=\xi^{*}$. In particular, $(M,\xi)$ and $(M,\xi^{*})$ are
contactomorphic. Moreover, by the construction of $\xi^{*}$, the map $f$ is a
contactomorphism between $(M,\xi^{*})$ and $(M^{\prime},\xi^{\prime})$. Hence,
taking their composition defines a contactomorphism
$g:(M,\xi)\to(M^{\prime},\xi^{\prime})$. Finally, considering $g$ between the
complements of the suitable neighborhoods of the bindings of $\mathcal{OB}$
and $\mathcal{OB}^{\prime}$ gives the desired contactomorphism between
$(\mathcal{M}(\Sigma,\phi),\eta)$ and
$(\mathcal{M}^{\prime}(\Sigma,\phi),\eta^{\prime})$.

##  4\. Contact Structures on Folded Sums of Mapping Tori

For a given mapping torus $\pi:\mathcal{M}(\Sigma,\phi)\longrightarrow S^{1}$
with $\phi\in\textrm{Diff}\,^{+}(\Sigma,\partial\Sigma)$, the boundary

$\partial\mathcal{M}(\Sigma,\phi)=\bigcup_{\theta\in
S^{1}}\partial\pi^{-1}(\theta)\cong\partial\Sigma\times S^{1}$

has a trivial neighborhood which is diffeomorphic to a product
$(-\epsilon,0]\times\partial\Sigma\times S^{1}.$ This makes the following
construction possible where $\overline{\Sigma}$ denotes $\Sigma$ with opposite
orientation.

######  Definition 4.1.

Let $\pi_{i}:\mathcal{M}(\Sigma_{i},\phi_{i})\longrightarrow S^{1}$ (for
$i=1,2$) be two mapping tori with
$\phi_{i}\in\textrm{Diff}\,^{+}(\Sigma_{i},\partial\Sigma_{i})$ such that
$\partial\Sigma_{1}=Y=\partial\Sigma_{2}$. Then the folded sum of $\pi_{1}$
and $\pi_{2}$ is a smooth fiber bundle
$\pi:\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2})\longrightarrow
S^{1}$ constructed as follows:

  * (i)

The total space of $\pi$ is a smooth manifold with the identification

$\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2})=(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2})\times[0,1]/\sim$

where $(x,0)\sim(\tilde{\phi}_{1}(x),1)$ for any $x\in\Sigma_{1}$ and
$(y,0)\sim(\tilde{\phi}_{2}(y),1)$ for any $y\in\Sigma_{2}$. Here
$\tilde{\phi}_{1}$ (resp. $\tilde{\phi}_{2}$) is the extension of $\phi_{1}$
(resp. $\phi_{2}$) via identity over $\Sigma_{2}$ (resp. $\Sigma_{1}$).

  * (ii)

The fiber of $\pi$ over $\theta\in S^{1}$ is the boundary sum of the fiber of
$\pi_{1}$ over $\theta$ with the corresponding one of $\pi_{2}$ with the
orientation reversed, that is,

$\pi^{-1}(\theta)=\pi_{1}^{-1}(\theta)\cup_{\partial}\overline{\pi_{2}^{-1}(\theta)}\cong\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2}.$

  * (iii)

The monodromy of $\pi$ is the composition
$\tilde{\phi}_{1}\circ\tilde{\phi}_{2}$ which lies in the group

$\textrm{Diff}\,^{+}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},Y):=\\{\phi\in\textrm{Diff}\,^{+}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2})\;|\;\phi\,\textrm{
is identity near }\,Y\\}\\}$

which is, indeed, the folded structure group of $\pi$.

If we further assume that $\Sigma_{1},\Sigma_{2}$ are compact exact symplectic
manifolds with coinciding convex boundaries, and if each $\phi_{i}$ is chosen
to be exact symplectomorphism of $\Sigma_{i}$ (which is identity near
boundary), then one can construct (see Theorem 4.3) a contact structure $\eta$
on the folded sum (defined as in Definition 4.1). The contact form, say
$\sigma$, on the folded sum (defining $\eta$) is obtained by gluing the ones
on the contact mapping tori $\mathcal{M}(\Sigma_{i},\phi_{i})$ constructed in
the proof of Theorem 3.3. The resulting contact structure is “compatible” with
the fiber bundle structure on the folded sum in the sense that $d\sigma$
defines a folded symplectic structure when restricted to each fiber
$\Sigma_{1}\cup_{\partial}\overline{\Sigma}_{2}$. More precisely,

######  Definition 4.2.

Let
$\pi:\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2})\longrightarrow
S^{1}$ be any folded sum of two mapping tori (as in Definition 4.1) such that
$\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2})$
admits a co-oriented contact structure $\eta$. Then $\eta$ is said to be
_compatible with_ $\pi$ if there exists a contact form $\sigma$ for $\eta$
such that $d\sigma$ restricts to a folded exact symplectic form on every fiber
of $\pi$.

######  Theorem 4.3.

Let
$\pi:\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2})\longrightarrow
S^{1}$ be any folded sum of two contact mapping tori with the structure groups
$\emph{Exact}(\Sigma_{i},\partial\Sigma_{i},d\beta_{i})$ such that
$(\Sigma_{1},d\beta_{1})$ and $(\Sigma_{2},d\beta_{2})$ have the same convex
boundary. Then the total space
$\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2})$
admits a (co-oriented) contact structure $\eta=\emph{Ker}(\sigma)$ which is
compatible with $\pi$, that is, $d\sigma$ restricts to a folded exact
symplectic form on every fiber.

###### Proof.

Let $\pi_{i}:\mathcal{M}(\Sigma_{i},\phi_{i})\longrightarrow S^{1}$ (for
$i=1,2$) be the contact mapping tori which construct $\pi$. Fix a fiber
$(\Sigma_{i},d\beta_{i})$ of $\pi_{i}$ for each $i$. By assumption
$(\Sigma_{1},d\beta_{1})$ and $(\Sigma_{2},d\beta_{2})$ are both exact
symplectic fillings of their common convex boundary, say $(Y,\xi)$. Without
loss of generality, assume $\xi=\textrm{Ker}(\alpha)$ where
$\beta_{1}|_{TY}=\alpha=\beta_{2}|_{TY}$, and so for each $i$ we have
$\beta_{i}=e^{t}\alpha$ on some collar neighborhood $(-\epsilon,0]\times Y$ of
$Y=\partial\Sigma_{i}$ in $\Sigma_{i}$.

By Theorem 3.3, there exists a bundle contact structure
$\eta_{i}=\textrm{Ker}(\sigma_{i})$ on each total space
$\mathcal{M}(\Sigma_{i},\phi_{i})$ compatible with $\pi_{i}$. The bundle
contact form defining $\eta_{i}$ is explicitly given by

$\sigma_{i}=\lambda_{\beta_{i}}+K_{i}\pi_{i}^{*}(d\theta)\in\Omega^{1}(\mathcal{M}(\Sigma_{i},\phi_{i}))\quad\textrm{for
}\,K_{i}>0\,\textrm{ large enough},$

and it is a product form on a sufficiently small collar neighborhood of the
boundary $Y\times S^{1}$ of $\mathcal{M}(\Sigma_{i},\phi_{i})$. More
precisely, there exists $\epsilon_{i}>0$ such that

$\sigma_{i}=\beta_{i}+K_{i}\,d\theta\quad\textrm{on
}\,(-1-\epsilon_{i},-1]\times Y\times S^{1}.$

(Here we intentionally make a $1$ unit shift in the Liouville direction so
that $\partial\Sigma_{i}=\\{-1\\}\times Y$ for each $i$. This will provide an
easier formulation of the gluing arguments below.) Set

$K=\textrm{max}(K_{1},K_{2})$ and
$\epsilon=\textrm{min}(\epsilon_{1},\epsilon_{2})$.

Then for each $i$, $\sigma_{i}=\lambda_{\beta_{i}}+K\pi_{i}^{*}(d\theta)$ is a
bundle contact form on $\mathcal{M}(\Sigma_{i},\phi_{i})$ and equal to the
product form $\sigma_{i}=\beta_{i}+Kd\theta$ on $(-1-\epsilon,-1]\times
Y\times S^{1}$. Also note that replacing $\alpha$ with $(1/K)\alpha$ (they
define the same contact structure on $Y$), and after shifting along the
$t$-direction, we may take

$\beta_{i}=Ke^{t+1}\alpha$ on $(-1-\epsilon,-1]\times Y$ for each $i$.

Now, up to diffomorphism, we can consider the total space of $\pi$ as follows:

$\displaystyle\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2})=\mathcal{M}(\Sigma_{1},\phi_{1})\bigcup_{\\{-1\\}\times
Y\times S^{1}}[-1,1]\times Y\times S^{1}\bigcup_{\\{1\\}\times Y\times
S^{1}}\overline{\mathcal{M}(\Sigma_{2},\phi_{2})}$.

Choose two smooth functions $f,g:(-1-\epsilon,1+\epsilon)\to\mathbb{R}$
satisfying:

  * (1)

$f$ is an even function such that $f(t)=Ke^{t+1}$ near $(-1-\epsilon,-1]$,

  * (2)

$g$ is an odd function such that $g(t)=K$ near $(-1-\epsilon,-1]$,

  * (3)

$f^{\prime}g-fg^{\prime}>0$ everywhere on $(-1-\epsilon,1+\epsilon)$,

  * (4)

$f^{\prime}(t)=0$ if and only if $t=0$.

(See Figure 1 for the graphs of such $f$ and $g$.)

Figure 1. Smooth functions on the interval $(-1-\epsilon,1+\epsilon)$.

Consider the $1$-form $\sigma$ on
$\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2})$
defined by the following rules:

$\displaystyle{\sigma=\begin{cases}\lambda_{\beta_{1}}+K\pi_{1}^{*}(d\theta)&\textnormal{on
\qquad\;\;\;}\textrm{int}(\mathcal{M}(\Sigma_{1},\phi_{1}))\\\
\;\;\;f\alpha+g\,d\theta&\textnormal{on \;\;\;}(-1-\epsilon,1+\epsilon)\times
Y\times S^{1}\\\ \lambda_{\beta_{2}}-K\pi_{2}^{*}(d\theta)&\textnormal{on
\qquad\;\;\;}\textrm{int}(\overline{\mathcal{M}(\Sigma_{2},\phi_{2})})\end{cases}}$

Observe the conditions (1) and (2) imply that $\sigma$ is a smooth $1$-form.
Since $\sigma_{i}$ is a positive contact form on each
$\mathcal{M}(\Sigma_{i},\phi_{i})$, checking that $\sigma$ is a positive
contact form on the interiors
$\textrm{int}(\mathcal{M}(\Sigma_{1},\phi_{1}))$,
$\textrm{int}(\overline{\mathcal{M}(\Sigma_{2},\phi_{2})})$ is immediate.
Moreover, if $\textrm{dim}(\Sigma_{i})=2n$, a standard computation gives that
on a slightly bigger middle piece $(-1-\epsilon,1+\epsilon)\times Y\times
S^{1}$ we have

$\sigma\wedge(d\sigma)^{n}=nf^{n-1}(f^{\prime}g-fg^{\prime})\,dt\wedge\alpha\wedge(d\alpha)^{n-1}\wedge
d\theta$

which implies, by the condition (3), that $\sigma$ is a positive contact form
on the middle piece as well. Therefore, we obtain a co-oriented contact
structure $\eta=\textrm{Ker}(\sigma)$ on the folded sum
$\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2})$.

For the last claim, first for any $\theta\in S^{1}$ the $2$-form $d\sigma$
restricts to exact symplectic structures on the fibers $\pi_{1}^{-1}(\theta)$
and $\overline{\pi_{2}^{-1}(\theta)}$. (This is because $d\sigma$ restricts to
$d\lambda_{\beta_{i}}$ on each $\mathcal{M}(\Sigma_{i},\phi_{i})$ and so
Theorem 3.3 applies.) Secondly, observe that for any $\theta\in S^{1}$ the
restriction of $d(f\alpha+gd\theta)$ to the tangent spaces of the region
$A_{\theta}:=(-1-\epsilon,1+\epsilon)\times Y\times\\{\theta\\}$ is
nondegenerate everywhere except when $t=0$. To see this, we compute

$d(f\alpha+gd\theta)|_{TA_{\theta}}=(f^{\prime}dt\wedge\alpha+fd\alpha+g^{\prime}dt\wedge
d\theta)|_{TA_{\theta}}=f^{\prime}dt\wedge\alpha+fd\alpha$

from which we have

$[d(f\alpha+gd\theta)|_{TA_{\theta}}]^{n}=[f^{\prime}dt\wedge\alpha+fd\alpha]^{n}=nf^{\prime}f^{n-1}dt\wedge\alpha\wedge(d\alpha)^{n-1}+f^{n}(d\alpha)^{n}.$

When we consider this as a top form on $TA_{\theta}$, the last term vanishes
since $\alpha\in\Omega^{1}(Y)$ and $\textrm{dim}(Y)=2n-1$ (or, alternatively,
$d\alpha$ has the full rank equal to $2n-2$ on $\xi\subset TY$). As a result,
we conclude that on the tangent spaces of $A_{\theta}$, we have

$[d(f\alpha+gd\theta)|_{TA_{\theta}}]^{n}=nf^{\prime}f^{n-1}dt\wedge\alpha\wedge(d\alpha)^{n-1}$

which is a positive volume form everywhere except when $f^{\prime}=0$
($\Leftrightarrow t=0$ by the condition (4)). Hence, except the points on the
“fold” $\\{0\\}\times Y\times\\{\theta\\}$, the form $d\sigma$ is
nondegenerate on each fiber $\pi^{-1}(\theta)$, that is, the contact structure
$\eta$ is compatible with the fibration $\pi$.

∎

######  Definition 4.4.

The contact form $\sigma$ on the folded sum
$\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2})$
constructed in the above proof will be called a folded contact form, and its
kernel $\eta$ will be called a folded contact structure.

######  Remark 4.5.

The construction of the folded contact form $\sigma$ (end so the structure
$\eta$) on the folded sum
$\pi:\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2})\longrightarrow
S^{1}$ depends on the choice of a pair $(f,g)$ of smooth functions satisfying
the conditions $(1)-(4)$. So one should write $\sigma(f,g)$ for $\sigma$ and
$\eta(f,g)$ for $\eta$ and understand the effect of replacing $(f,g)$ with any
other pair of smooth functions satisfying the same conditions. If we alter
$(f_{0},g_{0})=(f,g)$ to another pair $(f_{1},g_{1})$ through the family
$\\{(f_{s},g_{s})\,|\,s\in[0,1]\\}$ for which the conditions $(1)-(4)$ are
satisfied (for all $s$), then (by following the steps in the above proof) one
easily checks that $\\{\sigma(f_{s},g_{s})\,|\,s\in[0,1]\\}$ is a family of
contact $1$-forms on
$\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2})$,
and so the family $\\{\eta(f_{s},g_{s})\,|\,s\in[0,1]\\}$ of their kernels
defines an isotopy of contact structures compatible with the fibration map
$\pi$. Hence, we may drop the pair $(f,g)$ from the notation if it not
necessary to emphases which functions are used in defining $\sigma$ (and so
$\eta$).

######  Remark 4.6.

By the construction in the above proof, the contact mapping tori

$(\mathcal{M}(\Sigma_{1},\phi_{1}),\textrm{Ker}(\lambda_{\beta_{1}}+K\pi_{1}^{*}(d\theta))\;\textrm{
and
}\;(\overline{\mathcal{M}(\Sigma_{2},\phi_{2})},\textrm{Ker}(\lambda_{\beta_{2}}-K\pi_{2}^{*}(d\theta))$

are both codimension zero contact submanifolds of
$(\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2}),\textrm{Ker}(\sigma))$.
In fact, the same is true for the codimension zero submanifolds

$\mathcal{M}(\Sigma_{1},\phi_{1})\cup([-1,0]\times Y\times S^{1})\;\textrm{
and }\;\overline{\mathcal{M}(\Sigma_{2},\phi_{2})}\cup([0,1]\times Y\times
S^{1})$

with their inherited contact structures coming from $\sigma$. For the
subsequent results, one needs to show the submanifolds in the above pairs are
contactomorphic, respectively. This is in the content of the next theorem.

######  Theorem 4.7.

Let
$\pi:\mathcal{M}:=\mathcal{M}(\Sigma_{1}\cup_{Y}\overline{\Sigma}_{2},\tilde{\phi}_{1}\circ\tilde{\phi}_{2})\longrightarrow
S^{1}$ be any folded sum of two contact mapping tori
$\pi_{i}:\mathcal{M}_{i}:=\mathcal{M}(\Sigma_{i},\phi_{i})\longrightarrow
S^{1}$ with the structure groups
$\emph{Exact}(\Sigma_{i},\partial\Sigma_{i},d\beta_{i})$ such that
$(\Sigma_{1},d\beta_{1})$ and $(\Sigma_{2},d\beta_{2})$ have the same convex
boundary $(Y,\emph{Ker}(\alpha))$. Then for any folded contact structure
$\eta(f,g)=\emph{Ker}(\sigma(f,g))$ on $\mathcal{M}$, there are
contactomorphisms

$(\mathcal{M}_{1},\emph{Ker}(\lambda_{\beta_{1}}+K\pi_{1}^{*}(d\theta)))\cong\left(\mathcal{M}_{1}\bigcup_{\\{-1\\}\times
Y\times S^{1}}([-1,0]\times Y\times S^{1}),\emph{Ker}(\sigma|_{1})\right),$

and

$(\overline{\mathcal{M}_{2}},\emph{Ker}(\lambda_{\beta_{2}}-K\pi_{2}^{*}(d\theta)))\cong\left(\overline{\mathcal{M}_{2}}\bigcup_{\\{1\\}\times
Y\times S^{1}}([0,1]\times Y\times S^{1}),\emph{Ker}(\sigma|_{2})\right).$

where $\sigma|_{1}:=\sigma(f,g)|_{T(\mathcal{M}_{1}\cup([-1,0]\times Y\times
S^{1}))}$ and
$\sigma|_{2}:=\sigma(f,g)|_{T(\overline{\mathcal{M}_{2}}\cup([0,1]\times
Y\times S^{1}))}$.

###### Proof.

Since we will use the second contactomorphism in the next section, we will
only give the proof of the second one. The proof of the first one is similar.

First observe that showing the existence of the second contactomorphism is
equivalent to finding a contactomorphism $\Phi$:

$\left(\overline{\mathcal{M}_{2}}\bigcup_{\\{1\\}\times Y\times
S^{1}}([0,1]\times Y\times
S^{1}),\textrm{Ker}(\lambda)\right)\stackrel{{\scriptstyle\cong}}{{\Phi}}\left(\overline{\mathcal{M}_{2}}\bigcup_{\\{1\\}\times
Y\times S^{1}}([0,1]\times Y\times S^{1}),\textrm{Ker}(\sigma|_{2})\right)$

where
$\displaystyle{\lambda=\begin{cases}\lambda_{\beta_{2}}-K\pi_{2}^{*}(d\theta)&\textnormal{on
\quad\;\;\;\;\qquad}\overline{\mathcal{M}_{2}}\\\
\;\;\;Ke^{-t+1}\alpha&\textnormal{on \;\;\;\;\;}[0,1]\times Y\times
S^{1}\end{cases}}$ because the inclusion map

$(\overline{\mathcal{M}_{2}},\textrm{Ker}(\lambda_{\beta_{2}}-K\pi_{2}^{*}(d\theta)))\hookrightarrow\left(\overline{\mathcal{M}_{2}}\bigcup_{\\{1\\}\times
Y\times S^{1}}([0,1]\times Y\times S^{1}),\textrm{Ker}(\lambda)\right)$

has a homotopy inverse which retracts (respecting the product contact
structures) the collar neighborhood of $[0,1+\epsilon)\times Y\times S^{1}$ of
$\\{0\\}\times Y\times S^{1}$ into the collar neighborhood
$[1,1+\epsilon)\times Y\times S^{1}$ of $\\{1\\}\times Y\times S^{1}$.

To find $\Phi$, consider two smooth $1$-parameter families
$f_{s},g_{s}:[0,1+\epsilon)\rightarrow\mathbb{R}$ defined by

$f_{s}(t)=(1-s)Ke^{-t+1}+s\,f(t)\textrm{ \;\;and\;\;
}g_{s}(t)=(1-s)(-K)+s\,g(t)\quad(\textrm{where}\,s\in[0,1\,]).$

Let $\\{\lambda_{s}\,|\,s\in[0,1]\\}$ be the family of $1$-forms on
$\overline{\mathcal{M}_{2}}\bigcup_{\\{1\\}\times Y\times S^{1}}([0,1]\times
Y\times S^{1})$ given by

$\displaystyle{\lambda_{s}=\begin{cases}\lambda_{\beta_{2}}-K\pi_{2}^{*}(d\theta)&\textnormal{on
\quad\;\;\;\;\qquad}\overline{\mathcal{M}_{2}}\\\
\;\;\;f_{s}\alpha+g_{s}\,d\theta&\textnormal{on \;\;\;\;\;}[0,1]\times Y\times
S^{1}\end{cases}}$ .

Then clearly $\lambda_{s}$ is a contact form on $\overline{\mathcal{M}_{2}}$
for each $s$. Moreover, a similar computation used in the proof of Theorem 4.3
gives that on $[0,1]\times Y\times S^{1}$ we have

$\lambda_{s}\wedge(d\lambda_{s})^{n}=nf_{s}^{n-1}(f_{s}^{\prime}g_{s}-f_{s}g_{s}^{\prime})\,dt\wedge\alpha\wedge(d\alpha)^{n-1}\wedge
d\theta$

where we assume $\textrm{dim}(\Sigma_{i})=2n$. Also for each $s$ we compute

$\displaystyle f_{s}^{\prime}g_{s}-f_{s}g_{s}^{\prime}$ $\displaystyle=$
$\displaystyle[-(1-s)Ke^{-t+1}+sf^{\prime}][(1-s)(-K)+sg]-[(1-s)Ke^{-t+1}+sf][sg^{\prime}]$
$\displaystyle=$
$\displaystyle[(1-s)^{2}K^{2}e^{-t+1}]+[-(1-s)sKe^{-t+1}g]+[-(1-s)sKf^{\prime}]$
$\displaystyle+$
$\displaystyle[-(1-s)^{2}Ke^{-t+1}g^{\prime}]+[s^{2}(f^{\prime}g-fg^{\prime})].$

Observe that each term is non-negative and the sum is strictly positive for
each $s\in[0,1]$ and for all $t\in[0,1+\epsilon)$. (For the last term recall
$f^{\prime}g-fg^{\prime}>0$.) Therefore, $\\{\lambda_{s}\\}$ is a family of
contact forms on $\overline{\mathcal{M}_{2}}\bigcup_{\\{1\\}\times Y\times
S^{1}}([0,1]\times Y\times S^{1})$ joining $\lambda_{0}=\lambda$ to
$\lambda_{1}=\sigma|_{2}$. Hence, their kernels
$\textrm{Ker}(\lambda),\textrm{Ker}(\sigma|_{2})$ are isotopic contact
structures. In particular, we obtain a contactomorhism $\Phi$ as claimed. ∎

##  5\. Tight Contact Structures on Folded Sums of Mapping Tori

An _overtwisted disk_ in a contact $3$-manifold $(M,\xi)$ is an $n$-disk $D$
such that the tangent plane $T_{p}\partial D$ at any point $p\in\partial D$
coincides with the contact plane $\xi_{p}$ at $p$ [4]. As a generalization of
this definition, overtwisted disks are also defined for $(2n+1)$-dimensional
contact manifolds $(n\geq 2)$ as embedded $2n$-disks with a certain germ of a
contact structure [2]. In all odd dimensions, a contact structure $\xi$ is
called _overtwisted_ if there is an embedded overtwisted disk in $(M,\xi)$,
otherwise it is called _tight_. Constructing (or finding) tight structures is
more challenging than finding overtwisted ones. In particular, one should be
careful in constructing a new contact manifold out of two tight pieces if one
wishes to get a tight structure on the resulting manifold. Here we will show
(see Theorem 5.2) that the folded sum operation “respects” the tightness in
certain cases.

In the proof of Theorem 5.2, we will make use of contact vector fields. A
vector field $Z$ on a contact manifold $(M,\xi=\textrm{Ker}(\alpha))$ is said
to be _contact_ if its flow preserves the contact distribution $\xi$,
equivalently, if $\mathcal{L}_{Z}\alpha=f\alpha$ for some smooth function
$f:M\to\mathbb{R}$. The following fundamental lemma in contact geometry
characterizes contact vector fields on a given contact manifold. See [5] for
more details.

######  Lemma 5.1.

Let $(M,\emph{Ker}(\alpha))$ be any contact manifold and $R_{\alpha}$ denote
the “Reeb vector field” of $\alpha$ (i.e.,
$\alpha(R_{\alpha})=1,\;\iota_{R_{\alpha}}d\alpha=0$). Then there is a one-to-
one correspondence

$\\{Z\in\Gamma(M)\,|\;Z\;\emph{ is
contact}\\}\longleftrightarrow\\{H:M\to\mathbb{R}\,|\;H\;\emph{ is smooth}\\}$

between the set of all contact vector fields on $M$ and the set of all smooth
functions on $M$. The correspondence is given by $Z\to H_{Z}:=\alpha(Z)$
($H_{Z}$ is called the “contact Hamiltonian” of the contact vector field $Z$),
and $H\to Z_{H}$ where $Z_{H}$ is the contact vector field uniquely determined
by the equations $\alpha(Z_{H})=H$ and
$\iota_{Z_{H}}d\alpha=dH(R_{\alpha})\alpha-dH$. ∎

In what follows, we will make use of the trivial (or standard) contact mapping
torus

$\pi_{st}:\mathcal{M}_{st}:=\mathcal{M}(D^{2n},id_{D^{2n}})=D^{2n}\times
S^{1}\longrightarrow S^{1}\quad(\textrm{the projection onto}\;\;S^{1})$

with the trivial structure group $\textrm{Exact}(D^{2n},\partial
D^{2n},d\lambda_{st})=\\{Id_{D^{2n}}\\}$. The fiber $(D^{2n},d\lambda_{st})$
of $\pi_{st}$ is the standard exact symplectic (indeed Stein) filling of the
standard (tight) contact sphere
$(S^{2n-1},\xi_{st}=\textrm{Ker}(\lambda_{st}))$ where $\lambda_{st}$ is the
standard Liouville form which is written as
$\lambda_{st}=\sum_{i}(x_{i}dy_{i}-y_{i}dx_{i})$ in the standard coordinates
$x_{1},y_{1},...,x_{n},y_{n}$ on $D^{2n}\subset\mathbb{C}^{n}$. It is a well-
known fact that (or by Theorem 3.3) we have the standard (product) bundle
contact form
$\sigma_{st}=\lambda_{st}+\pi_{st}^{*}(d\theta)=\lambda_{st}+d\theta$ on
$\mathcal{M}_{st}$ which defines the standard bundle contact structure
$\eta_{st}=\textrm{Ker}(\sigma_{st})$ which is, clearly, tight by Remark 3.5.

######  Theorem 5.2.

For $n\geq 1$, let
$\pi:\mathcal{M}:=\mathcal{M}(\,D^{2n}\,\cup_{S^{2n-1}}\overline{\Sigma},\widetilde{Id}_{D^{2n}}\circ\tilde{\phi})\longrightarrow
S^{1}$ be the folded sum of the contact mapping tori
$\pi_{st}:\mathcal{M}_{st}\longrightarrow S^{1}$ and
$\pi_{1}:\mathcal{M}_{1}:=\mathcal{M}(\Sigma,\phi)\longrightarrow S^{1}$ where
$\pi_{1}$ has the structure group $\emph{Exact}(\Sigma,\partial\Sigma,d\beta)$
and $(\Sigma,d\beta)$ has the convex boundary $(S^{2n-1},\xi_{st})$. Also let
$\eta_{1}=\emph{Ker}(\sigma_{1})$ be a bundle contact structure on
$\mathcal{M}_{1}$, and consider the folded contact structure
$\eta(f,g)=\emph{Ker}(\sigma(f,g))$ on $\mathcal{M}$ obtained from $\eta_{st}$
and $\eta_{1}$. Then $\eta(f,g)$ is tight if $\eta_{1}$ is tight.

###### Proof.

As before, without loss of generality, we may assume that

$\xi_{st}=\textrm{Ker}(\alpha)\quad\textrm{ where
}\quad\lambda_{st}|_{TS^{2n-1}}=\alpha=\beta|_{TS^{2n-1}}.$

Suppose $\sigma_{1}=\lambda_{\beta}+K\pi_{1}^{*}(d\theta)$ is the bundle
contact form on $\mathcal{M}_{1}$ defining $\eta_{1}$ where $K>0$ is a
sufficiently large fixed real constant. Then by modifying the contact form
$\sigma_{st}$ on $\mathcal{M}_{st}$ as
$\sigma_{st}=\lambda_{st}+K\pi_{st}^{*}(d\theta)$ (still denoted by
$\sigma_{st}$ by abuse of notation), and replacing $\alpha$ with $(1/K)\alpha$
(and modifying $\lambda_{st}$ and $\beta$ accordingly near their common
boundary $S^{2n-1}$), one obtains the following description of the folded
contact form $\sigma(f,g)$ on $\mathcal{M}$:

$\displaystyle{\sigma(f,g)=\begin{cases}\lambda_{st}+K\pi_{st}^{*}(d\theta)&\textnormal{on
\qquad\;\quad\quad}\textrm{int}(\mathcal{M}_{st})\\\
\;\;\;f\alpha+g\,d\theta&\textnormal{on \;\;\;}(-1-\epsilon,1+\epsilon)\times
S^{2n-1}\times S^{1}\\\ \lambda_{\beta}-K\pi_{1}^{*}(d\theta)&\textnormal{on
\qquad\;\quad\quad}\textrm{int}(\overline{\mathcal{M}_{1}})\end{cases}}$

where, as in the proof of Theorem 4.3, we choose $\epsilon>0$ sufficiently
small so that

$\lambda_{st}=Ke^{t+1}\alpha=\beta_{1}$ on $(-1-\epsilon,-1]\times S^{2n-1}$.

For simplicity, we will write $\sigma:=\sigma(f,g)$. The Reeb vector field of
$\sigma$ is given by

$\displaystyle{R_{\sigma}=\begin{cases}\qquad\qquad(1/K)\partial_{\theta}&\textnormal{on
\qquad\;\quad\quad}\textrm{int}(\mathcal{M}_{st})\\\
\dfrac{-g^{\prime}}{f^{\prime}g-fg^{\prime}}R_{\alpha}+\dfrac{f^{\prime}}{f^{\prime}g-fg^{\prime}}\partial_{\theta}&\textnormal{on
\;\;\;}(-1-\epsilon,1+\epsilon)\times S^{2n-1}\times S^{1}\\\
\;\quad\qquad-(1/K)\partial_{\theta}&\textnormal{on
\qquad\;\quad\quad}\textrm{int}(\overline{\mathcal{M}_{1}})\end{cases}}$

where $R_{\alpha}$ is the Reeb vector field of the contact form $\alpha$ on
$S^{2n-1}\;(=\partial D^{2n}=\partial\overline{\Sigma})$.

Now, on the contrary, suppose that $(\mathcal{M},\eta(f,g))$ is overtwisted.
Let $\mathcal{D}_{\textrm{ot}}\approx D^{2n}$ be an embedded overtwisted
$2n$-disk in $(\mathcal{M},\eta(f,g))$. Then, by assumption, one must have

$\mathcal{D}:=\mathcal{D}_{\textrm{ot}}\cap\left[\mathcal{M}_{st}\bigcup_{\\{-1\\}\times
S^{2n-1}\times S^{1}}([-1,0)\times S^{2n-1}\times S^{1})\right]\neq\emptyset.$

Otherwise $\eta_{1}$ would be overtwisted. To see this, if $\mathcal{D}$ were
an empty set, then from Lemma 4.7 (and by denoting the restriction
$\sigma|_{T(\overline{\mathcal{M}_{1}}\cup([0,1]\times S^{2n-1}\times
S^{1}))}$ by $\sigma|_{1}$), we would have

$\mathcal{D}_{\textrm{ot}}\subset\left(\underbrace{\overline{\mathcal{M}_{1}}\bigcup_{\\{1\\}\times
S^{2n-1}\times S^{1}}([0,1]\times S^{2n-1}\times S^{1})}_{\textrm{call
}A},\textrm{Ker}(\sigma|_{1})\right)\cong(\overline{\mathcal{M}_{1}},\textrm{Ker}(\lambda_{\beta}-K\pi_{1}^{*}(d\theta)))$

which would imply that there is an overtwisted disk in
$(\mathcal{M}_{1},\eta_{1}=\textrm{Ker}(\lambda_{\beta}+K\pi_{1}^{*}(d\theta)))$.
However, $\eta_{1}$ is assumed to be tight. Indeed, to prove the theorem, we
will get a contradiction by showing that one can isotope the overtwisted disk
$\mathcal{D}_{\textrm{ot}}$ using the flows of suitable contact vector fields
until its final image is contained in

$A:=\overline{\mathcal{M}_{1}}\bigcup_{\\{1\\}\times S^{2n-1}\times
S^{1}}([0,1]\times S^{2n-1}\times S^{1}).$

First, consider the contact Hamiltonian function
$H_{\sigma}:\mathcal{M}_{st}\to\mathbb{R}$ of $R_{\sigma}$ given by

$\displaystyle{H_{\sigma}:=\sigma(R_{\sigma})=\begin{cases}\;\;\;\quad
1&\textnormal{on \qquad\;\quad\quad}\textrm{int}(\mathcal{M}_{st})\\\
\dfrac{f^{\prime}}{f^{\prime}g-fg^{\prime}}&\textnormal{on
\;\;\;}(-1-\epsilon,1+\epsilon)\times S^{2n-1}\times S^{1}\\\ \;\;\;\quad
1&\textnormal{on
\qquad\;\quad\quad}\textrm{int}(\overline{\mathcal{M}_{1}})\end{cases}}$

By multiplying $H_{\sigma}$ with suitable smooth function
$\mu_{0}:\mathcal{M}_{st}\to\mathbb{R}$ and then pushing
$\mathcal{D}_{\textrm{ot}}$ using the flow of the contact vector field
$Z_{\mu_{0}H_{\sigma}}$, we can make $\mathcal{D}_{\textrm{ot}}$ disjoint from
a fiber of $\pi_{st}$. This can be done because $\mathcal{D}_{\textrm{ot}}$ is
contractible, and so it can not contain any circle representing the pullback
$\pi_{st}^{*}([S^{1}])$ (which is nonzero in $\pi_{1}(\mathcal{M}_{st},*)$) of
the generator of $\pi_{1}(S^{1},*)\cong\mathbb{Z}$. Assuming such an isotopy
has been already applied, suppose that $\mathcal{D}_{\textrm{ot}}$ misses the
fiber $\pi_{st}^{-1}(\theta_{0})$ for some $\theta_{0}\in S^{1}$.

Let $D_{\theta}^{2n}:=\pi_{st}^{-1}(\theta)$ be any fiber of $\pi_{st}$ over
$\theta\in S^{1}$. The pair $(D_{\theta}^{2n},d\lambda_{st})$ is, indeed, a
compact Liouville domain. Any compact Liouville domain retracts onto its
compact _core_ under the backward flow of its Liouville vector field [3]. The
Liouville flow, say $X_{\theta}$, of $(D_{\theta}^{2n},d\lambda_{st})$
retracts $D_{\theta}^{2n}$ onto its origin (a point). Therefore,
$\textrm{Core}(D_{\theta}^{2n},d\lambda_{st})=\\{pt\\}$ and the union of these
cores gives the “core” of the trivial contact mapping torus
$\mathcal{M}_{st}=D^{2n}\times S^{1}$ which is just a copy of $S^{1}$. More
precisely,

$\textrm{Core}(\mathcal{M}_{st})=\bigcup_{\theta\in
S^{1}}\textrm{Core}(D_{\theta}^{2n},d\lambda_{st})=\bigcup_{\theta\in
S^{1}}\\{pt\\}=\\{pt\\}\times S^{1}\cong S^{1}.$

Since $\mathcal{D}_{\textrm{ot}}\cap\pi_{st}^{-1}(\theta_{0})=\emptyset$, the
intersection $\mathcal{D}\cap\textrm{Core}(\mathcal{M}_{st})$ lies in the
contact submanifold

$(\mathcal{M}_{st}\setminus\pi_{st}^{-1}(\theta_{0}),\textrm{Ker}(\lambda_{st}+Kd\theta))\cong(D^{2n}\times\mathbb{R},\textrm{Ker}(\lambda_{st}+Kdz))\cong(\mathbb{R}^{2n+1},\textrm{Ker}(\alpha_{0}))$

where $z$ is the coordinate on $\mathbb{R}$ and
$\alpha_{0}=\sum_{i=1}^{n}x_{i}dy_{i}+Kdz$. (The explicit rule of the last
contactomorphism can be found, for instance, in Example 2.9. in [5].) There
are $2n+1$ linearly independent (globally defined) contact vector fields on
$(\mathbb{R}^{2n+1},\textrm{Ker}(\alpha_{0}))$. In the coordinates
$x_{1},y_{1},...,x_{n},y_{n},z$, they are given as follows:

$Z_{1}=\partial_{y_{1}}\;,...,\;Z_{n}=\partial_{y_{n}},\quad
Z_{n+1}=\partial_{x_{1}}-y_{1}\partial_{z}\;,...,\;Z_{2n}=\partial_{x_{n}}-y_{n}\partial_{z}\;,\quad
Z_{2n+1}=\partial_{z}$

The corresponding contact Hamiltonian functions (as in Lemma 5.1),
respectively, are

$H_{1}=x_{1}\;,...,\;H_{n}=x_{n},\quad
H_{n+1}=-y_{1}\;,...,\;H_{2n}=-y_{n}\;,\quad H_{2n+1}=1.$

Considering the compact set $\mathcal{D}\cap\textrm{Core}(\mathcal{M}_{st})$
inside $(\mathbb{R}^{2n+1},\textrm{Ker}(\alpha_{0}))$, there exist smooth
functions $\mu_{1},\mu_{2},...,\mu_{2n+1}:\mathbb{R}^{2n+1}\to\mathbb{R}$
compactly supported in a neighborhood of
$\mathcal{D}\cap\textrm{Core}(\mathcal{M}_{st})$ in
$(\mathbb{R}^{2n+1},\textrm{Ker}(\alpha_{0}))$ such that by pushing
(isotoping) $\mathcal{D}_{\textrm{ot}}$ using the flow of the contact vector
field corresponding to the contact Hamiltonian
$\oplus_{i=1}^{2n+1}\mu_{i}H_{i}$ (as in Lemma 5.1), we can make
$\mathcal{D}_{\textrm{ot}}$ disjoint from $\textrm{Core}(\mathbb{R}^{2n+1})$.
As a result, by the last two contactomorphisms, assuming that
$\mathcal{D}_{\textrm{ot}}$ is already pushed, we may start with an
overtwisted disk (still denoted by $\mathcal{D}_{\textrm{ot}}$) which is
disjoint from $\textrm{Core}(\mathcal{M}_{st})$.

Next by pulling back the Liouville vector field $X_{\theta}$ of a fiber
$D_{\theta}^{2n}$ using the projection map $\mathcal{M}_{st}=D^{2n}\times
S^{1}\longrightarrow D_{\theta}^{2n}$, we obtain a smooth vector field $X$ on
$\mathcal{M}_{st}$. Note that the restriction of $X$ on each fiber of
$\mathcal{M}_{st}$ is the Liouville vector field of that fiber by
construction. In particular, for each point $p$ on a given fiber
$D_{\theta}^{2n}$ with
$p\in\mathcal{M}_{st}\setminus\textrm{Core}(\mathcal{M}_{st})$, there is a
unique flow line of $X$ connecting $p$ to a unique point $q$ in the convex
boundary of $D_{\theta}^{2n}$. Now consider the following smooth vector field
on the contact open subdomain $\mathcal{M}_{st}\bigcup_{\\{-1\\}\times
S^{2n-1}\times S^{1}}((-1-\epsilon,1+\epsilon)\times S^{2n-1}\times S^{1})$ of
the folded sum $\mathcal{M}$:

$\displaystyle{Z=\begin{cases}\qquad\;\;X+\theta\partial_{\theta}&\textnormal{on
\qquad\;\quad\quad}\textrm{int}(\mathcal{M}_{st})\\\
\dfrac{fg}{f^{\prime}g-fg^{\prime}}\partial_{t}+\theta\partial_{\theta}&\textnormal{on
\;\;\;}(-1-\epsilon,1+\epsilon)\times S^{2n-1}\times S^{1}\end{cases}}$

Note that the smoothness of $Z$ follows from the conditions $(1)-(4)$ on $f$
and $g$. Moreover, similar to the flow lines of $X$ (as discussed above),
there is unique flow line of $X+\theta\partial_{\theta}$ connecting each point
$p\in D_{\theta}^{2n}$ (with
$p\in\mathcal{M}_{st}\setminus\textrm{Core}(\mathcal{M}_{st})$) to a unique
point

$q^{\prime}\in\\{-1\\}\times
S^{2n-1}\times\\{\theta^{\prime}\\}\subset\\{-1\\}\times S^{2n-1}\times
S^{1}=\partial\mathcal{M}_{st}\quad(\textrm{for some }\theta^{\prime}\in
S^{1}),$

and this flow line (uniquely and smoothly) extends to a flow line of $Z$
connecting each such $p\in D_{\theta}^{2n}$ (with
$p\in\mathcal{M}_{st}\setminus\textrm{Core}(\mathcal{M}_{st})$) to a unique
point

$q^{\prime\prime}\in\\{-1\\}\times
S^{2n-1}\times\\{\theta^{\prime\prime}\\}\subset\\{0\\}\times S^{2n-1}\times
S^{1}\quad(\textrm{for some }\theta^{\prime\prime}\in S^{1}).$

Finally, by multiplying the contact Hamiltonian $H_{Z}=\theta$ of $Z$ (recall
Lemma 5.1) with a suitable smooth function
$\mu:\mathcal{M}_{st}\bigcup_{\\{-1\\}\times S^{2n-1}\times
S^{1}}((-1-\epsilon,1+\epsilon)\times S^{2n-1}\times
S^{1})\longrightarrow\mathbb{R}$ compactly supported on a suitable open
domain, and then pushing $\mathcal{D}_{\textrm{ot}}$ using the flow of the
contact vector field $Z_{\mu H_{Z}}$, we can isotope
$\mathcal{D}_{\textrm{ot}}$ so that its final image lies in $A$. More
precisely, $\mu$ is chosen in such a way that:

  * •

$\mu\equiv 1$ on $\displaystyle\mathcal{M}_{st}\bigcup_{\\{-1\\}\times
S^{2n-1}\times S^{1}}((-1-\epsilon,0]\times S^{2n-1}\times S^{1})$,

  * •

$\mu\equiv 0$ on $[1,1+\epsilon)\times S^{2n-1}\times S^{1}$,

  * •

$\mu(t,x,\theta)=h(t)$ on $[0,1]\times S^{2n-1}\times S^{1}$ where $h(t)$ is a
smooth function such that $h(0)=1$, $h(1)=0$, $h^{\prime}(t)=0$ near $t=0$ and
$t=1$.

However, as explained in an early part of the proof, an existence of an
embedded overtwisted disk in $A$ contradicts to the assumption that the
contact structure $\eta_{1}$ on $\mathcal{M}_{1}$ is tight.

∎

######  Example 5.3.

Taking the folded sum of two copies of the trivial contact mapping torus
$\pi_{st}:\mathcal{M}_{st}\longrightarrow S^{1}$ gives a fiber bundle

$\pi:\mathcal{M}:=\mathcal{M}(\,D^{2n}\,\cup_{S^{2n-1}}\overline{D^{2n}},\widetilde{Id}_{D^{2n}}\circ\widetilde{Id}_{D^{2n}})\longrightarrow
S^{1}$

whose total space

$\mathcal{M}=S^{2n}\times S^{1}=D^{2n}\times S^{1}\bigcup_{S^{2n-1}\times
S^{1}}D^{2n}\times S^{1}$

admits a folded contact structure $\eta=\textrm{Ker}(\sigma)$ constructed as
in the proof of Theorem 4.3. The folded contact form $\sigma$ is obtained from
two copies of standard (product) bundle contact form
$\sigma_{st}=\lambda_{st}+d\theta$ on $\mathcal{M}_{st}=D^{2n}\times S^{1}$
which defines the standard bundle contact structure
$\eta_{st}=\textrm{Ker}(\sigma_{st})$ on each copy of $\mathcal{M}_{st}$.
Since $\eta_{st}$ is tight, one concludes that $\eta$ is also tight by Theorem
5.2. We note that $\eta$ is, in fact, Stein fillable because $(S^{2n}\times
S^{1},\eta)$ is the convex boundary of the Stein domain $D^{2n+1}\times S^{1}$
obtained by attaching a Stein $1$-handle $D^{1}\times D^{2n+1}$ to the Stein
$0$-handle $D^{2n+2}$.

##  6\. Tight Contact Structures on (Hyperbolic) Surface Bundles

In this section we will apply the above results to show existence of tight
contact structures on a large class of closed hyperbolic $3$-manifolds.
Indeed, we will prove the existence of tight structures for even more general
family of closed $3$-manifolds, and the above result will be a corollary. We
prove:

######  Theorem 6.1.

Let $\pi:\mathcal{M}\longrightarrow S^{1}$ be any surface bundle such that its
fiber is an oriented closed surface $\Sigma$ and the monodromy
$\phi\in\emph{Diff}\,^{+}(\Sigma)$ is identity on an open disk neighborhood
$U_{p}$ of a point $p\in\Sigma$. Let $\Sigma_{1}=\overline{\Sigma\setminus
D_{p}}$ be an oriented surface with boundary obtained from $\Sigma$ by
removing an open disk $D_{p}\subset U_{p}$ and then reversing the orientation,
so that
$\phi_{1}:=\phi|_{\Sigma_{1}}\in\emph{Diff}\,^{+}(\Sigma_{1},\partial\Sigma_{1})$.
Then the total space $\mathcal{M}$ admits a structure of a folded sum of
contact mapping tori with the same $\pi$ and $\phi$, and so it admits a folded
contact structure $\eta$. Moreover, if the contact structure supported by
$\mathcal{OB}(\Sigma_{1},\phi_{1})$ is tight, then so is $\eta$.

###### Proof.

First note that an open disk $U_{p}$ (on which $\phi$ is identity) exists for
any such surface bundle over $S^{1}$ with the cost of altering the total space
$\mathcal{M}$ in its diffeomorphism type: To see this, any such bundle map
$\pi$ over $S^{1}$ admits a section $\mathcal{S}$. Pick a fiber $\Sigma$, and
say $\mathcal{S}\cap\Sigma=\\{p\\}$. Then replacing $p$ with a small closed
disk $D$ gives a surface $\tilde{\Sigma}\cong\Sigma$. Also extending the
monodromy $\phi$ over $D$ as identity we obtain
$\tilde{\phi}\in\textrm{Diff}\,^{+}(\tilde{\Sigma})$, and hence a new surface
bundle $\tilde{\pi}:\tilde{\mathcal{M}}\longrightarrow S^{1}$ with fiber
$\tilde{\Sigma}$ and the monodromy $\tilde{\phi}$. (Here $\tilde{\pi}$ is the
obvious extension of $\pi$.) Clearly, $\tilde{\mathcal{M}}\cong\mathcal{M}$
and $U_{p}:=\textrm{int}(D)$ is the required open disk neighborhood of the
point at its center (which we still denote by $p$). Hence, we may assume the
existence of $U_{p}$ as in the statement of the theorem.

Consider the mapping torus
$\pi_{1}:\mathcal{M}_{1}:=\mathcal{M}(\Sigma_{1},\phi_{1})\longrightarrow
S^{1}$ whose fiber is $\Sigma_{1}=\overline{\Sigma\setminus D_{p}}$ and the
monodromy is the restriction
$\phi_{1}:=\phi|_{\Sigma_{1}}\in\textrm{Diff}\,^{+}(\Sigma_{1},\partial\Sigma_{1})$.
Note that $\mathcal{M}_{1}\subset\mathcal{M}$ and $\pi_{1}$ is just the
resriction $\pi|_{\mathcal{M}_{1}}$. By a sequence of results (see Lemma
4.4.6, Lemma 7.3.1, and Lemma 7.3.2 in [5], for instance), we may assume
$\phi_{1}\in\textrm{Exact}(\Sigma_{1},\partial\Sigma_{1},d\beta)$ for some
exact symplectic form $d\beta$ on the fiber $\Sigma_{1}$ of $\pi_{1}$. That
is, $\pi_{1}:\mathcal{M}_{1}\longrightarrow S^{1}$ is a contact mapping torus
with the structure group
$\textrm{Exact}(\Sigma_{1},\partial\Sigma_{1},d\beta)$. In particular, there
exists a bundle contact structure $\eta_{1}$ (which is compatible with
$\pi_{1}$) on $\mathcal{M}_{1}$ by Theorem 3.3.

By construction, the complement
$\mathcal{M}\setminus\mathcal{M}_{1}=\mathcal{M}_{st}$ is the total space of
the trivial contact mapping torus
$\pi_{st}:\mathcal{M}_{st}=\mathcal{M}(D_{p},id_{D_{p}})=D_{p}\times
S^{1}=D^{2}\times S^{1}\longrightarrow S^{1}$ with the trivial structure group
$\textrm{Exact}(D^{2},\partial D^{2},d\lambda_{st})=\\{Id_{D^{2}}\\}$. (Here
we identify $D_{p}$ with $D^{2}$.) Note that we have
$\pi_{st}=\pi|_{\mathcal{M}_{st}}$. Moreover, the fibers (over the same point)
$(\Sigma_{1},d\beta)$, $(D^{2},d\lambda_{st})$ meet along their common
boundary which is just a circle $\partial\Sigma_{1}=S^{1}$, and so, they can
be assumed to have a common convex boundary (because any contact (volume) form
$\alpha$ on $S^{1}$ defines a trivial contact structure
$\textrm{Ker}(\alpha)=\emptyset$). Hence, the total space $\mathcal{M}$ of
$\pi$ admits a structure of a folded sum of contact mapping tori:

$\pi:\mathcal{M}=\mathcal{M}(\,D_{p}\,\cup_{S^{1}}\overline{\Sigma}_{1},\widetilde{Id}_{D_{p}}\circ\tilde{\phi_{1}})\longrightarrow
S^{1}$

with the projection map agreeing with $\pi$ and the monodromy
$\widetilde{Id}_{D_{p}}\circ\tilde{\phi_{1}}$ agreeing with $\phi$. In
particular, there exists a folded contact structure (which is compatible with
$\pi$) on $\mathcal{M}$ by Theorem 4.3. Thus, the first claim has been proved.

Now, assume that $\mathcal{OB}(\Sigma_{1},\phi_{1})$ supports a tight contact
structure $\xi$ on some contact $3$-manifold. Then the complement of an open
neighborhood of the binding can be considered as a contact mapping torus
$\mathcal{M}^{\prime}(\Sigma_{1},\phi_{1})$ equipped with the tight contact
structure $\eta_{1}^{\prime}$ (the restriction of $\xi$). By Remark 3.5, the
contact manifold $(\mathcal{M}_{1}=\mathcal{M}(\Sigma_{1},\phi_{1}),\eta_{1})$
is contactomorphic to
$(\mathcal{M}^{\prime}(\Sigma_{1},\phi_{1}),\eta_{1}^{\prime})$, and
therefore, it is tight. As a result, the folded contact structure $\eta$ on
the folded sum $\mathcal{M}$ is also tight by Theorem 5.2.

∎

As an immediate consequence, we have

######  Corollary 6.2.

Let $\pi:\mathcal{M}\longrightarrow S^{1}$ be any surface bundle such that its
fiber is an oriented closed surface $\Sigma$ (of genus $\geq 2$) and the
monodromy $\phi\in\emph{Diff}\,^{+}(\Sigma)$ is pseudo-Anosov, and hence
$\mathcal{M}$ admits an complete hyperbolic structure of finite volume. Let
$\Sigma_{1}$ and $\phi_{1}$ be as in the Theorem 6.1. Then if the contact
structure supported by the open book $\mathcal{OB}(\Sigma_{1},\phi_{1})$ is
tight, then $\mathcal{M}$ admits a tight contact structure.

###### Proof.

By the theorem of Thurston [10], if a closed $3$-manifold $M$ admits a surface
bundle structure $\Sigma_{g}\hookrightarrow M\to S^{1}$ such that the fiber
$\Sigma_{g}$ is a closed surface of genus $g$, then $M$ admits a complete
hyperbolic metric of finite volume provided that $g\geq 2$ and the monodromy
is a pseudo-Anosov homeomorphism of $\Sigma_{g}$. So, $\mathcal{M}$ is a
closed hyperbolic $3$-manifold by [10]. Moreover, a direct application of
Theorem 6.1 shows that $\mathcal{M}$ admits a tight contact structure.

∎

Acknowledgments. The author would like to thank Mustafa Korkmaz and Yıldıray
Ozan for helpful conversations.

## References

  * [1] F. Arikan, _Contact symplectic fibrations and bundle contact structures_ , Mediterr. J. Math. (2023) 20:332, https://doi.org/10.1007/s00009-023-02540-w.
  * [2] M. Borman, Y. Eliashberg, E. Murphy, _Existence and classification of overtwisted contact structures in all dimensions_ , Acta Math. (2015) 215(2): 281–361. DOI: 10.1007/s11511-016-0134-4
  * [3] K. Cieliebak and Y. Eliashberg, From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds, American Mathematical Society Colloquium Publications, 59. American Mathematical Society, Providence, RI, (2012).
  * [4] Y. Eliashberg, _Classification of overtwisted contact structures on 3-manifolds_ , Invent. math. (1989) 98, 623–637.
  * [5] H. Geiges, An Introduction to Contact Topology, Cambridge University Press, (2008).
  * [6] H. Geiges, A. Stipsicz, Contact structures on product five-manifolds and fibre sums along circles, Mathematische Annalen. (2010) 348, 195–210.
  * [7] E. Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing), Higher Ed. Press. (2002), 405–414. MR 2004c:53144
  * [8] E. Giroux, Contact structures and symplectic fibrations over the circle, trans-parencies of a seminar talk.
  * [9] D. McDuff, D. Salamon, Introduction to Symplectic Topology, Oxford University Press, (1995).
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  * [11] W. Thurston, H. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. (1975) 52, 345–347.


On the moments of Cox rate-and-state models

Z.Baki$\mbox{}^{a}$ and M.N.M. van Lieshout$\mbox{}^{b,a}$

$\mbox{}^{a}$ Department of Applied Mathematics, University of Twente

P.O. Box 217, NL-7500 AE, Enschede, The Netherlands.

$\mbox{}^{b}$ CWI, P.O. Box 94079, NL-1090 GB Amsterdam, The Netherlands

ABSTRACT: Rate-and-state models are widely used physical models for the
relation between changes in pore pressure due to fluid injection or gas
extraction and the induced seismic hazard in a field. We consider the
modification where the pore pressure measurements are affected by noise and
provide explicit expressions for the first and second moments of the state
variable. We show that when the pressure increases, there is positive
correlation. In the case of decreasing pressure, both positive and negative
correlation is possible. Using the delta method, approximate first and second
moments of the rate variable are derived and compared to empirical moments.

Keywords: Cox process, induced seismicity, pore pressure, rate-and-state
model.

2020 MSC Classification: 60G55.

##  1 Introduction

The study of induced earthquakes caused by extraction or injection of fluids
or gases is an important research topic. In the Netherlands, the Groningen gas
field discovered in the late 1950s was viewed in a mostly positive light for
decades until the occurrence of earthquakes from the 1990s onward caused a
shift in public opinion that eventually led to the closing of the field. Thus,
for planning purposes, it is important to be able to predict seismic hazard
based on field measurements, for instance of pore pressure or, equivalently,
Coulomb stress. One of the most widely used methodologies to do so is the
rate-and-state model [1, 4, 7] which is now considered to be the state-of-the-
art technique [5].

In the rate-and-state model, the earthquake intensity $\lambda(t)$, the rate
at time $t$, is assumed to be inversely proportional to a state variable
$\Gamma(t)$, that is,

$\lambda(t)=\frac{1}{\Gamma(t)},\quad t\geq 0.$

The state variable $\Gamma(t)$ is defined by the ordinary differential
equation

$d\Gamma(t)=\alpha\left[dt+\Gamma(t)dX(t)\right],$

where $X(t)\geq 0$ is the pore pressure at time $t$ and $\alpha>0$ is a
parameter. Multiplying both sides by $\exp(-\alpha X(t))$ and discretising in
time steps of length $\Delta>0$, we obtain the corresponding Euler difference
equation

$\Gamma((k+1)\Delta)=\left(\Gamma(k\Delta)+\Delta\alpha\right)\exp\left[\alpha\left\\{X((k+1)\Delta)-X(k\Delta)\right\\}\right].$
(1)

Additionally, a spatial location may be added to the model. For a fuller
discussion, we refer to [5].

In this paper, we consider stochastic rate-and-state models in which noise is
added to the pore pressure $X$. After the formal definition of this model in
Section 2, we give explicit expressions of the first and second moments of the
random state variable. We then apply the delta method to approximate the first
and second moments of the random rate variable. The paper finishes with an
illustration of the accuracy of the approximations for various pore pressure
scenarios.

##  2 Cox rate-and-state models

Let $\Psi$ be a Cox process (cf. [3, Section 5.2]) on $k\Delta$,
$k=0,1,2,\dots$, with driving random measure

$\Lambda(k\Delta)=\frac{1}{\Gamma(k\Delta)},\quad k\in{\mathbb{N}}_{0}.$ (2)

In contrast to the classic deterministic case discussed in Section 1, we
assume that the state $\Gamma$ is a stochastic process defined by equation (1)
for a deterministic starting state $\gamma_{0}>0$ and random pore pressure $X$
of the form $X(k\Delta)=m(k\Delta)+E(k\Delta)$. Here, $m\geq 0$ is a
deterministic mean and $E$ is Gaussian white noise. In other words, each
$E(k\Delta)$ is normally distributed with mean zero and variance $\sigma^{2}$,
and for $k\neq k^{\prime}$, $E(k\Delta)$ and $E(k^{\prime}\Delta)$ are
mutually independent.

The Euler difference equation (1) can be solved explicitly. Indeed, for each
$k\in{\mathbb{N}}$,

$\Gamma(k\Delta)=\exp\left[\alpha
X(k\Delta)\right]\left\\{\alpha\Delta\sum_{i=0}^{k-1}\exp\left[-\alpha
X(i\Delta)\right]+\gamma_{0}\exp\left[-\alpha X(0)\right]\right\\}$ (3)

under the convention that $\Gamma(0)=\gamma_{0}$. The Cox rate-and-state model
description is complete upon requiring that $N(k\Delta)$, the number of
induced earthquakes during the time period $[k\Delta,(k+1)\Delta)$, is Poisson
distributed with rate parameter $\Lambda(k\Delta)\Delta$, $k=0,1,\dots$.

##  3 Moments of the state variable

Since the randomness in the driving random measure (2) of our Cox process is
induced by the state process $\Gamma$, we investigate its first and second-
moment properties first.

######  Theorem 1.

Let $\Gamma$ be defined by (3). Set $c=e^{\alpha^{2}\sigma^{2}}$ and define,
for $i,j\in{\mathbb{N}}_{0}$,

$f_{ij}=\exp\left[\alpha\left\\{m(i\Delta)-m(j\Delta)\right\\}\right].$

Then, for $k\in{\mathbb{N}}$,

$\displaystyle{\mathbb{E}}\Gamma(k\Delta)$ $\displaystyle=$ $\displaystyle
c\left(\alpha\Delta\sum_{i=0}^{k-1}f_{ki}+\gamma_{0}f_{k0}\right),$
$\displaystyle{\rm{Var}}\Gamma(k\Delta)$ $\displaystyle=$
$\displaystyle\alpha^{2}\Delta^{2}c^{2}(c^{2}-1)\sum_{i=0}^{k-1}f_{ki}^{2}+\alpha^{2}\Delta^{2}c^{2}(c-1)\sum_{i=0}^{k-1}\sum_{i\neq
j=0}^{k-1}f_{ki}f_{kj}$ $\displaystyle+$ $\displaystyle
2\alpha\Delta\gamma_{0}c^{2}f_{k0}^{2}\left(c^{2}-1+(c-1)\sum_{i=1}^{k-1}f_{0i}\right)+\gamma_{0}^{2}f_{k0}^{2}c^{2}(c^{2}-1)$

and, for $0<k<l$,

$\displaystyle{\rm{Cov}}(\Gamma(k\Delta),\Gamma(l\Delta))$ $\displaystyle=$
$\displaystyle\alpha^{2}\Delta^{2}c^{2}\sum_{i=0}^{k-1}\left[f_{ki}f_{li}(c-1)-f_{li}\left(1-\frac{1}{c}\right)\right]$
(4) $\displaystyle+$
$\displaystyle(2\alpha\Delta\gamma_{0}+\gamma_{0}^{2})c^{2}f_{k0}f_{l0}(c-1)-\alpha\Delta\gamma_{0}c^{2}f_{l0}\left(1-\frac{1}{c}\right).$

###### Proof.

Recall the moment-generating function of a normal distribution to conclude
that

${\mathbb{E}}e^{tZ}=\exp\left(\mu t+\sigma^{2}t^{2}/2\right)$

when $Z$ is normally distributed with mean $\mu$ and variance $\sigma^{2}$.
The formula for the mean follows immediately. Moreover, for $0<k\leq l$,

$\displaystyle{\rm{Cov}}(\Gamma(k\Delta),\Gamma(l\Delta))$ $\displaystyle=$
$\displaystyle\alpha^{2}\Delta^{2}\sum_{i=0}^{k-1}\sum_{j=0}^{l-1}{\rm{Cov}}\left(e^{\alpha\left[X(k\Delta)-X(i\Delta)\right]},e^{\alpha\left[X(l\Delta)-X(j\Delta)\right]}\right)$
(5) $\displaystyle+$
$\displaystyle\alpha\Delta\gamma_{0}\sum_{i=0}^{k-1}{\rm{Cov}}\left(e^{\alpha\left[X(k\Delta)-X(i\Delta)\right]},e^{\alpha\left[X(l\Delta)-X(0)\right]}\right)$
$\displaystyle+$
$\displaystyle\alpha\Delta\gamma_{0}\sum_{j=0}^{l-1}{\rm{Cov}}\left(e^{\alpha\left[X(k\Delta)-X(0)\right]},e^{\alpha\left[X(l\Delta)-X(j\Delta)\right]}\right)$
$\displaystyle+$
$\displaystyle\gamma_{0}^{2}{\rm{Cov}}\left(e^{\alpha\left[X(k\Delta)-X(0)\right]},e^{\alpha\left[X(l\Delta)-X(0)\right]}\right).$

Let us work out the terms in the expression on the right-hand side of equation
(5) one by one. First, consider the double sum over $i$ and $j$. Because of
the independence of the components of the random vector $E$, summands for
which $k$, $l$, $i$ and $j$ are all different do not contribute. Therefore,
for $0<k=l$, the total contribution of the double sum is

$\alpha^{2}\Delta^{2}c^{2}\sum_{i=0}^{k-1}f_{ki}^{2}\left(c^{2}-1\right)+\alpha^{2}\Delta^{2}c^{2}\sum_{i=0}^{k-1}f_{ki}\sum_{j\neq
i;j=0}^{k-1}f_{kj}\left(c-1\right).$

For $0<k<l$, the double sum contributes non-zero entries for $i=j$ (which
cannot be equal to $k$ or $l$) and for $i\neq j=k$. Their total contribution
is

$\alpha^{2}\Delta^{2}c^{2}\sum_{i=0}^{k-1}f_{ki}f_{li}\left(c-1\right)-\alpha^{2}\Delta^{2}c^{2}\sum_{i=0}^{k-1}f_{li}\left(1-\frac{1}{c}\right).$

Next consider the two single sums on the right-hand side of (5). If $0<k=l$,
they are identical and each one is equal to

$\alpha\Delta\gamma_{0}c^{2}\left[f_{k0}^{2}(c^{2}-1)+(c-1)\sum_{i=1}^{k-1}f_{ki}f_{k0}\right].$

In the case that $0<k<l$, the sum over $i$ has a non-zero contribution only
for $i=0$, and the sum over $j$ has non-vanishing contributions for $j=0$ and
$j=k$. Adding these up, we obtain

$\alpha\Delta\gamma_{0}c^{2}f_{l0}\left[2f_{k0}(c-1)-\left(1-\frac{1}{c}\right)\right].$

Finally, the last term in the expression on the right-hand side of equation
(5) reads $\gamma_{0}^{2}f_{k0}^{2}c^{2}(c^{2}-1)$ for $0<k=l$ and
$\gamma_{0}^{2}f_{k0}f_{l0}c^{2}(c-1)$ when $0<k<l$. Expression (4) now
follows from tallying up the various contributions. ∎

When fluid is injected into a field, the pore pressure typically increases. In
this case, the state variables are positively correlated.

######  Corollary 1.

Let $\Gamma$ be as in Theorem 1. Then, if the function $m(k\Delta)$ is
increasing in $k\in{\mathbb{N}}_{0}$,
${\rm{Cov}}(\Gamma(k\Delta),\Gamma(l\Delta))\geq 0$ for all
$k,l\in{\mathbb{N}}_{0}$.

###### Proof.

Obviously, the claim holds for $k=l$. For $0\leq i<k<l$,
$\alpha^{2}\Delta^{2}f_{li}\\{f_{ki}c^{2}(c-1)-c(c-1)\\}\geq 0$ if
$\sigma^{2}=0$ or $cf_{ki}\geq 1$. The latter condition is equivalent to

$\alpha^{2}\sigma^{2}+\alpha\\{m(k\Delta)-m(i\Delta)\\}\geq 0$

and is implied by the assumption that $m$ is increasing. By the same argument,
for increasing pore pressure,
$\alpha\Delta\gamma_{0}f_{l0}\\{f_{k0}c^{2}(c-1)-c(c-1)\\}\geq 0$ and an
appeal to (4) completes the proof. ∎

When the pore pressure decreases, for example due to gas extraction, the
picture is more varied.

######  Corollary 2.

Let $\Gamma$ be as in Theorem 1 and assume that the function $m(k\Delta)$ is
decreasing in $k\in{\mathbb{N}}_{0}$.

(i)

If $\alpha\sigma^{2}>m(0)$, then
${\rm{Cov}}(\Gamma(k\Delta),\Gamma(l\Delta))\geq 0$ for all
$k,l\in{\mathbb{N}}_{0}$.

(ii)

If

$\alpha\sigma^{2}<\min_{i\in{\mathbb{N}}_{0}}\left\\{m(i\Delta)-m((i+1)\Delta)\right\\},$

the minimal drop in pressure in between observation epochs,

${\rm{Cov}}(\Gamma(k\Delta),\Gamma(l\Delta))-(\alpha\Delta\gamma_{0}+\gamma_{0}^{2})c^{2}(c-1)f_{k0}f_{l0}\leq
0$

for all $0\leq k<l$.

###### Proof.

For $0\leq i<k<l$, consider $\alpha^{2}\Delta^{2}f_{li}c(c-1)\\{cf_{ki}-1\\}$.
The term in between curly brackets is negative if and only if

$\alpha\left\\{m(k\Delta)-m(i\Delta)\right\\}+\alpha^{2}\sigma^{2}<0.$

Since $m$ is decreasing, under assumption (ii),

$\alpha^{2}\sigma^{2}<m(i\Delta)-m((i+1)\Delta)\leq m(i\Delta)-m(k\Delta).$

We conclude that $\alpha^{2}\Delta^{2}f_{li}c(c-1)\\{cf_{ki}-1\\}\leq 0$.
Since the same argument can be used to show that
$\alpha\Delta\gamma_{0}f_{l0}c(c-1)\\{cf_{k0}-1\\}\leq 0$, (ii) is seen to
hold. Under assumption (i),

$\alpha^{2}\sigma^{2}>m(0)\geq m(i\Delta)\geq m(i\Delta)-m(k\Delta)$

and therefore $\alpha^{2}\Delta^{2}f_{li}c(c-1)\\{cf_{ki}-1\\}\geq 0$.
Similarly, $\alpha\Delta\gamma_{0}f_{l0}c(c-1)\\{cf_{k0}-1\\}\geq 0.$ The
observation that $\Gamma(0)=\gamma_{0}$ is deterministic completes the proof
of (i). ∎

Years |  |  |  |  |  |  |
---|---|---|---|---|---|---|---
1995–2001 | 179.81 | 177.39 | 174.86 | 172.20 | 169.42 | 166.50 | 163.48
2002–2008 | 160.32 | 157.05 | 153.65 | 150.13 | 146.49 | 142.72 | 138.82
2009–2015 | 134.81 | 130.68 | 126.43 | 122.04 | 117.53 | 112.91 | 108.16
2016–2021 | 103.28 | 98.29 | 93.17 | 87.94 | 82.56 | 77.08 |
Table 1: Estimated pore pressure in bara on January 1st in the years 1995–2021
near the town of Slochteren, The Netherlands.

######  Example 1.

Table 1 lists estimated pore pressure values near the town of Slochteren in
the Groningen gas field in The Netherlands for January 1st, 1995–2021 [6]. The
estimated standard deviation is $\sigma=7.17$.

Note that the pore pressure values are decreasing due to gas extraction. Since
the intensity of induced earthquakes was very low in 1995, $\gamma_{0}$ can be
considered infinite. Equation (4) then implies that the covariance matrix of
the random vector $\Gamma(k\Delta)_{k}$ has positive entries only.

######  Example 2.

Next, let us suppose that – in contrast to the previous example – the initial
seismicity is very high, i.e. $\gamma_{0}=0$. Assume a linearly decreasing
sequence of pore pressures $m(0)=3$, $m(1)=2$ and $m(2)=1$ and set $\alpha=1$.
Then the covariance matrix of the random vector $(\Gamma(1),\Gamma(2))$ is
readily calculated. Indeed,
${\rm{Var}}\Gamma(1)=e^{-2}(e^{4\sigma^{2}}-e^{2\sigma^{2}})$ and
${\rm{Var}}\Gamma(2)=(e^{-2}+e^{-4})(e^{4\sigma^{2}}-e^{2\sigma^{2}})+2e^{-3}(e^{3\sigma^{2}}-e^{2\sigma^{2}})$.
As for the off-diagonal entry,

$\displaystyle{\rm{Cov}}(\Gamma(1),\Gamma(2))$ $\displaystyle=$
$\displaystyle{\rm{Cov}}\left(e^{-1}e^{E(1)-E(0)},e^{-2}e^{E(2)-E(0)}+e^{-1}e^{E(2)-E(1)}\right)$
$\displaystyle=$ $\displaystyle
e^{2\sigma^{2}}\left\\{e^{-3}(e^{\sigma^{2}}-1)-e^{-2}(1-e^{-\sigma^{2}})\right\\}$

is negative for $\sigma^{2}<1$ and positive for $\sigma^{2}>1$.

##  4 Approximate moments of the rate variable

Recall that in the rate-and-state model defined in Section 2, the rate of
induced earthquakes is inversely proportional to the state. However, due to
its form (3), the moments of ${1}/{\Gamma(k\Delta)}$ are intractable. Thus, in
this section, we use the delta method to approximate them in terms of the
tractable moments of $\Gamma(k\Delta)$ derived in the previous section.

######  Theorem 2.

Let $\Gamma$ be defined by (3). Then, for $k\in{\mathbb{N}}_{0}$,

${\mathbb{E}}\left[\frac{1}{\Gamma(k\Delta)}\right]\approx\frac{1}{{\mathbb{E}}\Gamma(k\Delta)}+\frac{{\rm{Var}}\Gamma(k\Delta)}{({\mathbb{E}}\Gamma(k\Delta))^{3}}.$

Note that the approximation of the expectation of $\Gamma(k\Delta)^{-1}$ is at
least as large as its ‘plug-in estimator’ $1/{\mathbb{E}}\Gamma(k\Delta)$.

###### Proof.

Apply the delta method based on the Taylor expansion

$\frac{1}{x_{0}+h}\approx\frac{1}{x_{0}}-\frac{h}{x_{0}^{2}}+\frac{1}{2!}\frac{2h^{2}}{x_{0}^{3}}$

around $x_{0}$ equal to the expectation of $\Gamma(k\Delta)$. Upon taking the
expectation, one obtains that

${\mathbb{E}}\left[\frac{1}{\Gamma(k\Delta)}\right]\approx\frac{1}{{\mathbb{E}}\Gamma(k\Delta)}-{\mathbb{E}}\left[\frac{\Gamma(k\Delta)-{\mathbb{E}}\Gamma(k\Delta)}{({\mathbb{E}}\Gamma(k\Delta))^{2}}\right]+{\mathbb{E}}\left[\frac{(\Gamma(k\Delta)-{\mathbb{E}}\Gamma(k\Delta))^{2}}{({\mathbb{E}}\Gamma(k\Delta))^{3}}\right].$

The middle term on the right-hand side is zero, and the claim follows. ∎

######  Theorem 3.

Let $\Gamma$ be defined by (3). Then, for $k,l\in{\mathbb{N}}_{0}$,

$\displaystyle{\mathbb{E}}\left[\frac{1}{\Gamma(k\Delta)\Gamma(l\Delta)}\right]$
$\displaystyle\approx$
$\displaystyle\frac{1}{{\mathbb{E}}\left[\Gamma(k\Delta)\right]\,{\mathbb{E}}\left[\Gamma(l\Delta)\right]}+\frac{{\rm{Cov}}(\Gamma(k\Delta),\Gamma(l\Delta))}{({\mathbb{E}}\Gamma(k\Delta))^{2}({\mathbb{E}}\Gamma(l\Delta))^{2}}$
$\displaystyle+$
$\displaystyle\frac{{\rm{Var}}\Gamma(k\Delta)}{{\mathbb{E}}\Gamma(l\Delta)({\mathbb{E}}\Gamma(k\Delta))^{3}}+\frac{{\rm{Var}}\Gamma(l\Delta)}{{\mathbb{E}}\Gamma(k\Delta)({\mathbb{E}}\Gamma(l\Delta))^{3}}.$

###### Proof.

The Taylor expansion of the function $(x,y)\mapsto 1/(xy)$ around the point
$\left({\mathbb{E}}\Gamma(k\Delta),{\mathbb{E}}\Gamma(l\Delta)\right)$ yields
the approximation

$\displaystyle\frac{1}{\Gamma(k\Delta)}\frac{1}{\Gamma(l\Delta)}$
$\displaystyle\approx$
$\displaystyle\frac{1}{{\mathbb{E}}\Gamma(k\Delta)}\frac{1}{{\mathbb{E}}\Gamma(l\Delta)}+\frac{\left(\Gamma(k\Delta)-{\mathbb{E}}\Gamma(k\Delta)\right)\,\left(\Gamma(l\Delta)-{\mathbb{E}}\Gamma(l\Delta)\right)}{({\mathbb{E}}\Gamma(k\Delta))^{2}({\mathbb{E}}\Gamma(l\Delta))^{2}}$
$\displaystyle-$
$\displaystyle\frac{\Gamma(k\Delta)-{\mathbb{E}}\Gamma(k\Delta)}{({\mathbb{E}}\Gamma(k\Delta))^{2}{\mathbb{E}}\Gamma(l\Delta)}-\frac{\Gamma(l\Delta)-{\mathbb{E}}\Gamma(l\Delta)}{({\mathbb{E}}\Gamma(l\Delta))^{2}{\mathbb{E}}\Gamma(k\Delta)}$
$\displaystyle+$
$\displaystyle\frac{\left(\Gamma(k\Delta)-{\mathbb{E}}\Gamma(k\Delta)\right)^{2}}{({\mathbb{E}}\Gamma(k\Delta))^{3}{\mathbb{E}}\Gamma(l\Delta)}+\frac{\left(\Gamma(l\Delta)-{\mathbb{E}}\Gamma(l\Delta)\right)^{2}}{{\mathbb{E}}\Gamma(k\Delta)({\mathbb{E}}\Gamma(l\Delta))^{3}}$

up to second-order moments. Taking expectations proves the claim. ∎

Combining Theorems 2 and 3, we obtain the following corollary.

######  Corollary 3.

Let $\Gamma$ be defined by (3). Then, for $k,l\in{\mathbb{N}}_{0}$,

${\rm{Cov}}\left(\frac{1}{\Gamma(k\Delta)},\frac{1}{\Gamma(l\Delta)}\right)\approx\frac{{\rm{Cov}}(\Gamma(k\Delta),\Gamma(l\Delta))}{({\mathbb{E}}\Gamma(k\Delta))^{2}({\mathbb{E}}\Gamma(l\Delta))^{2}}.$

##  5 Simulation examples

Figure 1: $95\%$ pointwise confidence intervals for mean (left) and variance
(right) of $\Gamma(k\Delta)^{-1}$ as a function of $k$ when $m(\Delta
k)=6-{1}/{(0.5k+1)}$, $\alpha=0.01$, $\Delta=0.1$, $\gamma_{0}=0.2$ and
$\sigma^{2}=2.0$. The dots correspond to the approximation values of Theorem 2
and Corollary 3.

Figure 2: $95\%$ pointwise confidence intervals for the mean (left) and
variance (right) of $\Gamma(k\Delta)^{-1}$ as a function of $k$ when
$m(k\Delta)=5+{10}/{(2k+1)}$, $\alpha=0.01$, $\Delta=0.1$, $\gamma_{0}=0.2$
and $\sigma^{2}=2.0$. The dots correspond to the approximation values of
Theorem 2 and Corollary 3.

To investigate the accuracy of the approximations in Section 4, we compare the
formulae in Theorem 2 and Corollary 3 with population estimates. Recall that
for an i.i.d. sample $X_{1},\dots,X_{n}$ of some random variable $X_{1}$ with
mean $\mu$ and variance $\sigma^{2}$, approximate confidence intervals for
$\mu$ and $\sigma^{2}$ take the form

$\left(\bar{X}_{n}-\frac{S_{n}}{\sqrt{n}}\xi_{1-\alpha/2},\quad\bar{X}_{n}+\frac{S_{n}}{\sqrt{n}}\xi_{1-\alpha/2}\right)$

for $\mu$, and

$\left(\frac{S_{n}^{2}}{1+\xi_{1-\alpha/2}\sqrt{(}\frac{2\zeta}{n-1})},\quad\frac{S_{n}^{2}}{1-\xi_{1-\alpha/2}\sqrt{(}\frac{2\zeta}{n-1})}\right)$

for $\sigma^{2}$. Here $\xi_{1-\alpha/2}$ is the $(1-\alpha/2)$-quantile of
the standard normal distribution. Furthermore,
$\zeta=\frac{1}{2}(\gamma_{4}-1)$ for $\gamma_{4}$ the ratio of the fourth
central moment and the squared variance. Since $\gamma_{4}$ is unknown, we
estimate it by
$\widehat{\gamma_{4}}={n\sum(X_{i}-m)^{4}}/{((n-1)^{2}S_{n}^{4})}$ where $m$
is the sample median [2]. In our case, $X_{i}=1/\Gamma_{i}(k\Delta)$. A
similar approach may be taken for the covariance, although we do not pursue it
here.

We consider two cases: increasing and decreasing pore pressure. For Figure 1,
define the pore pressure by the increasing function

$m(\Delta k)=6-\frac{1}{1+k/2}.$

With parameter values $\alpha=0.01$, $\Delta=0.1$, $\gamma_{0}=0.2$ and
$\sigma^{2}=2.0$, the $95\%$ pointwise confidence intervals for the mean and
variance are given in the left- and right-most panels. In both cases, the
sample size was $n=500$. It can be seen that the approximations are quite
adequate.

For Figure 2, take the decreasing function

$m(\Delta k)=5+\frac{10}{2k+1}.$

With the same parameter values and sample size, the $95\%$ pointwise
confidence intervals for the mean and variance are given in the left- and
right-most panels. Again the approximations are satisfactory.

##  6 Conclusion

In this paper, we modified the rate-and-state model proposed in the earth
sciences to describe the relation between changes in pore pressure and induced
seismic hazard [1, 4, 7] in such a way that measurement error in the pore
pressure can be taken into account. For the time series of state variables
$\Gamma$ in the resulting Cox process [3], closed-form expressions for the
first and second moments were derived. Additionally, we provided sufficient
conditions for positive and negative correlation. On top of that, we gave
approximate expressions for the first two moments of the rate variable and
verified in simulated examples that the approximations are satisfactory.

## Acknowledgements

This research was funded by the Dutch Research Council NWO through their
DEEP.NL programme (grant number DEEP.NL.2018.033).

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