(This article is the fourth in a series on Liouville and contact geoemtry. The first was on Liouville (exact symplectic) geometry on surfaces. The second went from them to convex surfaces in 3-dimensional contact geometry. The third went back, from convex surfaces in 3-dimensional contact geometry, to 2-dimensional Liouville geometry, and showed how convex surfaces can be regarded as two Liouville structures, pieced together along a dividing set.)

We’ve seen that there are excellent things called *convex surfaces* in 3-dimensional contact geometry, closely related to Liouville geometry. Indeed, on convex surfaces we have wondeful *foliations*. So when you slice a contact 3-manifold along a convex surface, you get wonderful foliations. We’re now going to consider the relationship between these foliations on surfaces, and contact structures

Recall a convex surface is a surface \(S\) in a 3-dimensional contact manifold \((M, \xi)\) which have a transverse contact vector field \(X\). We’ve seen how you can use the vector field \(X\) to define a transverse coordinate \(t\) and hence describe a neighbourhood of \(S\) as \(S \times [-1,1]\), where \(S = S \times {0}\), with \(t\) giving the latter coordinate. With respect to these coordinates, \(\xi\) has a contact form \(\alpha\) of the form

\[

\alpha = \beta + u \; dt,

\]

where \(\beta\) is a 1-form on \(S\), and \(u\) is a real-valued function on \(S\). (We’re assuming everything is smooth here.)

The neighbourhood \(S \times [-1,1]\) of \(S\) comes as a family of surfaces \(S_t = S \times {t}\), with \(S = S_0\). The contact vector field \(X\) flows \(S_0\) to each \(S_t\), and hence each surface \(S_t\) in this family looks the same with respect to contact geometry. The contact planes are preserved by the flow of \(X\).

Thus, on each surface \(S_t\), we have the *same* characteristic foliation — where by “same”, we mean the surfaces and their foliations are related by the flow of \(X\). When you map one surface \(S_t\) in this family to another by a flow of \(X\), you’ll also map the characteristic foliation on one surface to the characteristic foliation on the other. Thinking of \(X\) as being “vertical” and \(S\) as “horizontal”, it means that the contact structure is “vertically invariant” — it doesn’t change as you move in the vertical direction, from one surface \(S_t\) of the family to another.

This is a really nice structure to have. Rather than having to think about all the surfaces \(S_t\) near \(S\), you really only have to think about one, because they are all the “same”, in this sense.

The most amazing thing about convex surfaces is Giroux’s proof that *almost any* embedded surface in a contact 3-manifold is convex. So, for “almost any” embedded surface \(S\), there is a transverse contact vector field, and then you know that you can take \(S\) as part of a family of surfaces \(S_t\) foliating a neighbourhood of \(S\), which are “all the same” in the above sense, and hence you only need to think about one of these surfaces!

What does “almost any” surface mean here? Giroux describes it in terms of a property of the characteristic foliation: if the characteristic foliation is “almost Morse-Smale”, then the surface is convex. And “almost Morse-Smale” is a generic property. In particular, given *any* embedded surface, after a \(C^\infty\) small perturbation, the surface becomes convex.

(For surfaces with boundary, we often want to preserve a little more structure, and so some further details are required. But we will not pursue them here.)

However, let’s now imagine we took a surface \(S\), and we didn’t know about any transverse contact vector field — i.e. we didn’t know if \(S\) was convex or not. Then, we could still take a neighbourhood of \(S\), of the form \(S \times [-1,1]\), and define \(t\) as the coordinate on the latter factor. Then we would again obtain a family of surfaces \(S_t = S \times {t}\) foliating a neighbourhood of \(S\), with \(S = S_0\). But now the surfaces \(S_t\) could all have *different* characteristic foliations \(\mathcal{F}_t\).

It would be a mess! That is why knowing \(S\) is convex, or equivalently, having a transverse contact vector field, really helps.

Thinking of \(t\) as time, you can think of the family of foliations \(\mathcal{F}_t\) as a “movie” of foliations. Or, since you are probing the contact structure by considering how it cuts the slices \(S_t\), you can think of it as a form of “tomography” — and this is what Giroux calls it.

In a 2000 paper, Giroux studied these sorts of “movies” or “tomography”. The paper is called “Structures de contact en dimension trois et bifurcations des feuilletages de surfaces”, which translates as “Contact structures in dimenson three and bifurcations of foliations of surfaces”. The French word “feuilletage” is much nicer than the English word “foliation”.

Given a surface \(S\) in a contact manifold \((M, \xi)\) with a product neighbourhood \(S \times [-1,1]\), you obtain a movie of foliations \(\mathcal{F}_t\) on the surfaces \(S_t\). If \(S\) is convex then, by choosing the product neighbourhood right (as above), all \(\mathcal{F}_t\) are the “same” (as discussed above) — the “movie” is just one frame, repeated!

But if all you have are the foliations \(\mathcal{F}_t\), you can’t say much at all. In fact, given a family of foliations \(\mathcal{F}_t\) on \(S_t\), it’s not even clear that they come from a contact structure at all!

So a first question is: which families of foliations arise from contact structures? Equivalently: what movies of foliations describe contact structures. Or: what tomography can you get from slicing a contact structure. This question is not easy.

Giroux however gives some answers. I want to consider one of his results, which he calls the “realisation lemma”.

Here is one possible line of reasoning. We have seen that certain characteristic foliations arise from convex surfaces. A characteristic foliation \(\mathcal{F}\) on a convex surface has a dividing set \(\Gamma\), splitting \(S\) into two pieces \(S_+\) and \(S_-\), and on each piece \(\mathcal{F}\) can be directed by a vector field which expands an area form. We’ll say that such a foliation *divides* \(S\). It would be very pleasing to see this lovely Liouville geometry on each slice. The nicest case would be if we had the *same* Liouville geometry on each slice.

Does it follow, in these circumstances, that the foliations come from a contact structure?

In other words, suppose we have \(S \times [-1,1]\), and let \(\mathcal{F}\) be a foliation which divides \(S\). Suppose we have the foliation \(\mathcal{F}_t = \mathcal{F}\) on each slice \(S_t = S \times {t}\). Is this the movie of foliations of a contact structure on \(S \times [-1,1]\)?

Giroux proved that the answer is yes. (This is proposition I.3.4 of his 1991 paper “ConvexitÃ© en topologie de contact”.) In fact we have already seen the basic idea of the proof. Since \(S_+\) and \(S_-\) have Liouville structures, we may take a 1-form \(\beta\) on \(S_+ \cup S_-\), such that \(d\beta\) is an area form on \(S_+ \cup S_-\), and the dual vector field \(X\) with respect to \(\beta\) directs \(\mathcal{F}\) on \(S_+ \cup S_-\). As we saw previously, \(\beta\) being Liouville is equivalent to \(\beta + dt\) being a contact form on \((S_+ \cup S_-) \times [-1,1]\). So we have a contact structure on \((S_+ \cup S_-) \times [-1,1]\). It remains to extend this over \(\Gamma \times [-1,1]\). And Giroux shows that this can be done. He shows that you can patch them together, to obtain a contact form on \(S \times [-1,1]\) which takes the form \(\beta + u \; dt\), where \(\beta\) is a 1-form on \(S\), and \(u\) is a real-valued function on \(S\).

Indeed, having a contact form of this type, the contact structure obtained on \(S \times [-1,1]\) is not just any old contact structure: it’s invariant in the \([-1,1]\) direction. It’s “vertically invariant”, and so we have a contact vector field in the \([-1,1]\) direction. This direction of course is transverse to \(S\), so we have \(S\) is convex.

Great. So a “constant” movie of foliations, where the foliation \(\mathcal{F}\) divides each slice (i.e. has a dividing set which cuts each slice into pieces on which there is a Liouville structure) is always the movie of a contact structure — and indeed a vertically invariant contact structure exhibiting \(S\) as convex.

But let’s suppose we have a slightly worse situation. Suppose we have different foliations \(\mathcal{F}_t\) appearing on the slices \(S_t\), but each individual foliation \(\mathcal{F}_t\) still divides \(S_t\). In other words, each foliation \(\mathcal{F}_t\) has a dividing set \(\Gamma\) which splits \(S\) into an \(S+\) and \(S_-\), and \(\mathcal{F}_t\) can be directed by a vector field which expands an area form on \(S+\) and \(S_-\). Here \(\Gamma\), \(S_+\) and \(S_-\) might all vary with \(t\), so we should really write something like \(\Gamma_t\), \(S_{t,+}\) and \(S_{t,-}\) to indicate the dependence on \(t\).

In these circumstances, are the foliations \(\mathcal{F}_t\) the movie of a contact structure?

It’s not quite so clear. When you have a foliation which can vary with \(t\), the contact condition becomes more complicated.

When you just have a single foliation, with a dividing set and Liouville structures on either side, then you get a contact form of the type \(\alpha = \beta + u \; dt\), where \(\beta\) is a 1-form and \(u\) a real-valued function on \(S\). The condition for a 1-form to be a contact form is that \(\alpha \wedge d\alpha\) be a volume form, i.e. a non-degenerate 3-form. When \(\alpha = \beta + u \; dt\) we have

\[

\alpha \wedge d\alpha

= (\beta + u \; dt) \wedge (d\beta + du \wedge dt)

= (u \; d\beta + \beta \wedge du ) \wedge dt .

\]

So given that \(\beta\) and \(u\) are purely *on S*, i.e. have no \(t\)-dependence, the condition for \(\alpha = \beta + u \; dt\) to be a contact form is precisely that \(u \; d\beta + \beta \wedge du\) be an area form on \(S\).

But when you have a family of foliations, even if they each divide \(S\), you would be looking for a contact form of the type \(\alpha = \beta + u \; dt\) again — but now \(\beta\) and \(u\) can depend on \(t\). Perhaps it’s better to write, as Giroux does, \(\beta_t\) and \(u_t\), to indicate the dependence on \(t\). You can think of \(\beta_t\) as a 1-form on \(S_t\), and \(u_t\) as a real-valued function on \(S_t\). The contact condition then becomes more complicated, because then \(d\beta = d\beta_t + dt \wedge \frac{\partial \beta_t}{\partial t}\). Here we write \(d\beta_t\) for the 2-form on \(S_t\) which arises by taking the differential of a 1-form on \(S_t\); but \(\beta\) also has a \(t\)-dependence, and so we also obtain a derivative with respect to \(t\). Let us write \(\dot{\beta}_t\) for \(\frac{\partial \beta_t}{\partial t}\). Then the contact condition is

\[

\alpha \wedge d\alpha

= ( u_t \; d\beta_t + \beta_t \wedge ( du_t – \dot{\beta}_t ) \wedge dt .

\]

So in this more general case, with \(\beta\) and \(u\) depending on \(t\), the condition for \(\alpha\) to be a contact form is that \(u_t \; d\beta_t + \beta_t \wedge (du_t – \dot{\beta}_t )\) be an area form on \(S\).

So the answer to the question may not be clear. When \(\alpha\) takes the form \(\beta_t + u_t \; dt\), even if each surface \(S_t\) has a dividing set, with Liouville structures on either side, this only means that on each slice we have the first condition above, that \(u_t \; d\beta_t + \beta_t \wedge du_t\) is an area form on \(S_t\). To show that we have a contact form, we need to show the latter condition, that \(u_t \; d\beta_t + \beta_t \wedge (du_t – \dot{\beta}_t)\). The term with a \(\dot{\beta}_t\), taking a derivative in the \(t\)-direction, makes a difference.

But in any case, Giroux shows the answer is yes. This is his “realisation lemma”, which is lemma 2.4 of the 2000 paper. And the proof is not too difficult. Let’s state the result and prove it.

REALISATION LEMMA (Giroux): Let \(\beta_t\) be a family of 1-forms on \(S\), and \(v_t\) a family of functions \(S \rightarrow \mathbb{R}\), such that for all \(t\),

\[

v_t \; d\beta_t + \beta_t \wedge dv_t

\]

is an area form on \(S\). Then there exists a contact structure on \(S \times [-1,1]\) with a contact form of the form \(\beta_t + u_t \; dt\), where each \(u_t\) is a real-valued function on \(S\).

In other words, if each \(S_t\) is divided by the foliation \(\mathcal{F}_t\), then there exists a contact structure on \(S \times [-1,1]\) with \(\mathcal{F}_t\) as its movie of characteristic foliations.

PROOF:

Choose an orientation on \(S\) which agrees with \(v_t \; d\beta_t + d\beta_t \wedge dv_t\), so that we can write

\[

v_t \; d\beta_t + d\beta_t \wedge dv_t > 0.

\]

Now we need to find functions \(u_t\) such that

\[

u_t \; d\beta_t + \beta_t \wedge ( du_t – \dot{\beta}_t ) > 0.

\]

Clearly, the only difference between these two inequalities is the term \(\beta_t \wedge \dot{\beta}_t\). But the \(\beta_t\) are fixed — it’s the \(u_t\) we get to choose. And \(S \times [-1,1]\) is a compact set. So \(\beta_t \wedge \dot{\beta}_t\) only gets so large.

Similarly, by compactness, \(v_t \; d\beta_t + d \beta_t \wedge dv_t\), as a positive area form, only gets so small. If we multiply \(v_t\) by a large constant \(K\), then \(v_t \; d\beta_t + d \beta_t \wedge dv_t\) is also gets multiplied by \(K\) — and thus can be guaranteed to be arbitrarily large everywhere. Indeed, we can make it so large that it overwhelms the term \(\beta_t \wedge \dot{\beta}_t\).

And that is what we do. Let \(u_t = K v_t\) for sufficiently large \(K>0\). This is all we need to do. QED.

(This proof assumes the \(v_t\) vary smoothly in \(t\). But even if the \(v_t\) don’t vary smoothly, or even continuously, in \(t\), one can use a partition of unity to construct the desired \(u_t\).)