# Fillings of Contact Manifolds

There are various notions of fillability of a contact 3-manifold, $(Y,\xi)$: weak symplectic filliability, strong symplectic fillability, and Stein fillability. When these notions were initially studied, it was not yet known whether they all coincided. It is fairly straightforward to show that a Stein filling is a strong symplectic filling, and a strong symplectic filling is a weak symplectic filling so $\{\text{Stein fillable}\} \subseteq \{\text{Strongly sympl. fill.}\} \subseteq \{\text{Weakly sympl. fill.}\}$
However, it took more time to find examples to show that each of these inclusions is strict. Now it seems useful to mention the examples used to prove this in one place. While looking up these examples I noticed there are a few equivalent definitions of each of these notions, which are interchangeably used throughout the literature, so I’ll start by discussing those equivalences. That much seems to fill up an entire post so I’ll keep this post about definitions and equivalences (which make the above inclusions easy to conclude), and next post I’ll summarize known examples that show the inclusions are strict. So this is just an intro post.

A weak symplectic filling of $(Y,\xi)$ is a symplectic manifold $(W,\omega)$ with boundary $\partial W=Y$ such that $\omega|_{\xi}>0$. This definition is only considered for fillings of contact 3-manifolds, since that is when the symplectic 2-form can be positive on the 2-dimensional contact planes. However, the other notions of fillability hold in higher dimensions.

A strong symplectic filling of $(Y,\xi)$ is a symplectic manifold $(W,\omega)$ satisfying certain conditions that can be defined in a few equivalent ways. The first way is to require that there exists a Liouville vector field V defined near the boundary of $W$ ( $\mathcal{L}_V\omega = \omega$) which is transverse to $\partial W=Y$ and a contact form for $\xi$ is given by $\alpha = i^*(\iota_V\omega)$.

One can verify any such an $\alpha$ is a contact form (i.e. $\alpha\wedge d\alpha >0$) since $\alpha \wedge d\alpha = i^*(\iota_V\omega)\wedge d(i^*(\iota_V\omega)) = i^*(\iota_V\omega) \wedge i^*(\mathcal{L}_V\omega) = i^*(\iota_V\omega \wedge \omega)$

( $i$ denotes the inclusion map of $\partial W$ into $W$ and $\iota_V$ denotes contraction by the vector field $V$. We have used $\mathcal{L}_V = \iota_V\circ d +d\circ \iota_V$ and $d\omega=0$.) The right most expression is a volume form by nondegeneracy of $\omega$ and the fact that $V$ is everywhere transverse to $\partial W$.

Another way to define a strong filling independently of the Liouville vector field, is to ask that $\omega$ be exact in a neighborhood of $\partial W$ and it has a primitive $\eta$ defined on this neighborhood of the boundary such that $\xi = \ker(i^*\eta)$ and $d\eta|_{\xi}>0$ (i.e. $i^*\eta$ is a contact form for $\xi$).

The first definition of a strong filling implies the second since $\eta=\iota_V\omega$ is a primitive for $\omega$ near the boundary, and the other conditions are satsified since $i^*(\iota_V\omega)$ is a contact form for $\xi$. Conversely, since $\omega$ is nondegenerate, contraction into $\omega$ gives an isomorphism between 1-forms and vector fields so there is a unique vector field $V$ such that $\iota_V\omega = \eta$. Therefore $\mathcal{L}_V\omega = d(\iota_V\omega) = d\eta = \omega$ so $V$ is a Liouville vector field which induces the correct contact form on the boundary because of its relation to $\eta$. If it were not transverse to $\partial W$ at some point then $i^*(\eta\wedge d\eta) = i^*(\iota_V\omega \wedge \omega)$ could not be positive at that point, so $i^*\eta$ would not be a contact form.

Note there is a shorter way of stating the second definition which is again equivalent. We must only require that $\omega$ restricts to an exact form on $\partial W$ such that $i^*\omega = d\alpha$ where $\alpha$ is a contact form for $\xi$. This is because $\alpha$ can be extended to a primitive for $\omega$ in a small neighborhood of $\partial W$ (note such a primitive exists because if $\omega$ restricts to something trivial in cohomology in $\partial W$, then it will represent something trivial in cohomology when restricted to a tubular neighborhood diffeomorphic to $\partial W\times [0,\varepsilon)$).

A Stein filling of a contact manifold, is a complex manifold $(X,J)$ for which there is a strictly plurisubharmonic function $\phi: X\to [0,c]$, meaning the form $\omega_{\phi}:= -dd^{\mathbb{C}}\phi$ is a symplectic form compatible with $J$ (where $d^{\mathbb{C}}\phi = d\phi \circ J$) or equivalently $\omega_\phi(v,Jv)>0$ for all $v\neq 0$. Cieliebak and Eliashberg call these functions $J$-convex in their new book and give a thorough discussion of various aspects of Stein and Weinstein structures. The contact structure induced on the boundary of a Stein filling is the hyperplane field of complex tangencies to the boundary, namely $T(\partial X)\cap J(T(\partial X))$.

A Weinstein filling of a contact manifold is an exact symplectic manifold $(W,\omega)$ such that $\omega=d\eta$ and there is a Liouville vector field $V$ on all of $W$ for which $\iota_V\omega = \eta$ and $V$ is gradient-like for some Morse function on $W$ for which $\partial W$ is the maximal level set. It is clear from this definition that Weinstein fillable implies strongly symplectically fillable. It takes a lot more work to show that Weinstein fillability is equivalent to Stein fillability (see Cieliebak and Eliashberg’s book for a thorough explanation).

There are a few other characterizations of Stein fillings which are often more useful for constructions. Eliashberg proved that in dimensions greater than four, a $2n$-manifold can be given a Stein structure if and only if it has a Morse function with critical points of index at most $n$. Attaching the index $n$ handles is the trickiest part and in dimension four, additional hypotheses are needed. Specifically the 2-handles must be attached along Legendrian knots in the boundary of the 0- and 1-handles with framings on the attaching circles exactly one less than the contact framing. A standard way to keep track of the contact framing in $\partial(\natural_k S^1\times B^3) = \#_k S^1\times S^2$ in a Kirby diagram is given by Gompf here .

Finally if one thinks of contact structures in terms of open book decompositions, a contact manifold is Stein fillable if and only if it is supported by an open book decomposition whose monodromy can be factored into positive Dehn twists. A Stein filling is given by the corresponding Lefschetz fibration over a disk. This correspondence is in this paper by Loi and Piergallini and this paper by Akbulut and Ozbagci.

Discussion of how these notions differ coming soon.

1. Paolo