## Kylerec – On J-holomorphic curves, part 1

This to-be-2-part-because-this-got-long post is a continuation of the series on Kylerec 2017 starting with the previous post, and covers most of the talks from Days 2-3 of Kylerec, focusing on the use of J-holomorphic curves in the study of fillings. I should mention that two more sets of notes, by Orsola Capovilla-Searle and Cédric de Groote, have been uploaded to the website on this page. So if you wish to follow along, feel free to follow the notes there, and in particular, the relevant talks I’ll be discussing in this post are:

Part 1

• Day 1 Talk 1 – The introductory talk by (mostly) Roger Casals (with some words by Laura Starkston)
• Day 2 Talk 2 – Roberta Gaudagni’s talk introducing J-holomorphic curves
• Day 2 Talk 3 – Emily Maw’s talk on McDuff’s rational ruled classification

Part 2

It should be obvious in what follows which parts of the exposition correspond to which talks, although what follows is perhaps a pretty biased account, with some parts amplified or added, and others skimmed or skipped.

### J-holomorphic curves – basics

Gromov introduced the study of J-holomorphic curves into symplectic geometry in his famous 1985 paper, immediately revolutionizing the field. One might wonder why we care about these objects, and the rest of this post (along with part 2) should be a testament to some (but certainly not all) aspects of the power of the theory.

The “J” in “J-holomorphic” refers to some choice $J$ of almost complex structure on a manifold $M^{2n}$. Given an almost complex manifold, a J-holomorphic curve is a map $u : (\Sigma,j) \rightarrow (M^{2n},J)$ such that $(\Sigma,j)$ is a Riemann surface and $J \circ du = du \circ j$. In the case where $(M,J)$ is a complex manifold, we see this is precisely what it means to be holomorphic.

We are mostly concerned a choice of $J$ which is compatible with a symplectic manifold $(M,\omega)$. By this, we mean that the (0,2)-tensor $g(\cdot, \cdot) = \omega(\cdot,J\cdot)$ is a Riemannian metric. We say $J$ is tame if $\omega(v,Jv) > 0$ for each nonzero vector $v$ (note that $g$ as defined above is not necessarily symmetric in this case).

Proposition: The space of compatible almost complex structures on a symplectic manifold $(M,\omega)$ is non-empty and contractible. So is the space of tame almost complex structures.

This suggests either:

• Studying the space of J-holomorphic curves into $M$ for some particular choice of $J$.
• Study some invariant of spaces of J-holomorphic curves which does not depend on the choice of $J$ compatible (or tame) with respect to a given symplectic form $(M,\omega)$.

In walking down either of these paths, there are a large number of properties at our disposal. What is presented in this section is far from a conclusive list, and I have completely abandoned including proofs and motivation, so beware that there is a lot of subtlety involved in the analytic details. For many many many more details, consult this book of McDuff and Salamon.

Firstly, there is a dichotomy between somewhere injective curves and multiple covers. Some J-holomorphic curves will factor through branched covers, meaning that $u : \Sigma \rightarrow (M,J)$ factors as $(\Sigma,j) \rightarrow (\Sigma',j') \rightarrow (M,J)$ such that the first map is a branched cover of Riemann surfaces. J-holomorphic curves which are not multiply covered are called simple, and it turns out that simple curves are characterized by being somewhere injective, meaning there is some $z$ for which $u^{-1}u(z) = \{z\}$ and $du_z \neq 0$. Even better, somewhere injective means that $u$ is almost everywhere injective.

The main tool in the theory is the study of certain moduli spaces of J-holomorphic curves. There are many flavors of this, but we discuss a specific example to highlight the relevant aspects of the theory. The analytical details are typically easier for simple curves, so we denote by $\mathcal{M}^*(M,J)$ the moduli space of all simple $J$-holomorphic curves. It turns out to be fruitful to focus in on a specific piece of this space, so we often restrict to a given domain of definition, say some $\Sigma_g$, and also restrict the homology class $u_*[\Sigma]$ of the map $u : \Sigma_g \rightarrow (M,J)$ to some $A \in H_2(M)$. The main question is:

When is such a moduli space $\mathcal{M}_g^*(M,A,J)$ actually a smooth manifold?

This is certainly a subtle question, and it turns out that not every $J$ works. However, it is a theorem that for generic $J$, this moduli space is a smooth manifold of dimension $d = n(2-2g) + 2c_1(A)$, where $\dim M = 2n$.

Given our nice moduli space, we also might be interested in what happens as we change our choice of $J$, so that we go from one regular choice to another. A generic path of such almost complex structures will give a smooth cobordism between the moduli spaces, a property which allows us to cook up invariants which do not depend, for example, on choices of $J$ compatible with a given symplectic structure.

To note a few variants of the discussion so far, sometimes we will study J-holomorphic disks with certain boundary conditions, or J-holomorphic curves with punctures sent to a certain asymptotic limit. In all cases, the same analytic machinery already swept under the rug (Fredholm theory) will give that the moduli spaces in question are smooth for generic choices of almost complex structure, and the dimension of this moduli space is given by some purely topological quantity (by, for example, the Atiyah-Singer index theorem).

One common thing to do is to quotient out by the group action given by reparametrizing the domain of a given J-holomorphic curve. That is, we consider the equivalence relation $u \sim u \circ \phi$ where $\phi : (\Sigma,j) \rightarrow (\Sigma,j)$ is a biholomorphism. A more careful author would probably distinguish between the map $u$ as opposed to the corresponding equivalence class, which is really what one should mean when they say curve. Hence, one can quotient our moduli spaces $\mathcal{M}^*$ by reparametrization to obtain moduli spaces of curves. Usually, these are the main objects of interest.

So now we have our nice moduli space, in whatever situation we desire, and we can ask about studying limits of J-holomorphic curves in that moduli space. In general, no such curve might exist. The first reason for this is that any such curve $u : (\Sigma,j) \rightarrow (M,\omega,J)$ has an energy $E = \int_{\Sigma}u^*\omega$ attached to it (when $J$ is compatible with $\omega$). If this quantity diverges to $\infty$, then there can be no limiting curve. One can ask instead about what happens when the energy is bounded.

Consider the following sequence of holomorphic curves $u_n \colon \mathbb{C}P^1 \rightarrow \mathbb{C}P^1 \times \mathbb{C}P^1$ given by $z \mapsto (z, 1/(nz))$. We see that away from $z=0$, this is just converging to the curve $\mathbb{C}P^1 \times \{0\}$. But near $z = 0$, if we reparametrize the domain by $1/(nz)$, we see this converges to the sphere $\{0\} \times \mathbb{C}P^1$. In this case, our curve formed what is often called a bubble. More generally, a curve can split off many bubbles at a time. For an example of this, consider instead $u_n \colon \mathbb{C}P^1 \rightarrow \mathbb{C}P^1 \times \mathbb{C}P^1 \times \mathbb{C}P^1$ given by $z \mapsto (z,1/(nz),1/(n^2z))$, in which a new bubble forms at $\{0\} \times \{\infty\} \times \mathbb{C}P^1$ in addition to the one discussed above. More generally, a sequence of curves can limit to a curve with trees of bubbles sticking out.

Such bubble trees are called stable or nodal or cusp curves (or probably a lot of other things), depending upon how old your reference is and to whom you talk. The incredible theorem, which goes under the name of Gromov compactness, is that this is the only phenomenon which precludes a limit from existing. We state this vaguely as follows:

Theorem [Gromov ’85]: The moduli space of curves of energy bounded by some constant $E$ (modulo reparametrization of domain) can be compactified by adding in stable curves of total energy bounded by $E$.

Another generally important tool is that of the evaluation map. Suppose that we wish to study the moduli space $\mathcal{M}_g^*(M,A,J)$ of simple J-holomorphic maps $u : (\Sigma_g,j) \rightarrow (M,J)$ in the homology class $A \in H_2(M)$. Suppose $G = \text{Aut}(\Sigma_g,j)$ is the group of biholomorphisms of $(\Sigma_g,j)$. Then the group $G$ acts on $\mathcal{M}_g^*(M,A,J) \times \Sigma_g$ by $\phi \cdot (u,z) = (u \circ \phi^{-1},\phi(z))$. Notice then that the evaluation map $(u,z) \mapsto u(z)$ only depends on the orbit, and hence descends to a map $\text{ev} : \mathcal{M}_g^*(M,A,J) \times_G \Sigma_g \rightarrow M$. Proving enough properties of such an evaluation map sometimes allows us to compare the smooth topology of $\mathcal{M}_g^*(M,A,J)$ to that of $M$. There are other variants of this – sometimes we wish to evaluate at multiple points, or sometimes we consider J-holomorphic discs and want to evaluate along boundary points. And often the evaluation map extends to the compactified moduli spaces considered above.

Finally, we come to dimension 4, where curves might actually generically intersect each other. With respect to these intersections, there are two key results to highlight. The first is positivity of intersection (due to Gromov and McDuff), which states that if any two J-holomorphic curves intersect, then the algebraic intersection number at each intersection point is positive (and precisely equal to 1 at transverse intersections). This can be thought of as some sort of rudimentary version of a so-called adjunction inequality (due to McDuff), which states that if $u \colon (\Sigma,j) \rightarrow (M,J)$ is a simple J-holomorphic curve representing the class $A$ with geometric self-intersection number $\delta(u)$, then

$c_1(TV,J) \cdot A + 2\delta(u) \leq \chi(\Sigma) + A \cdot A$.

Further, when $u$ is immersed and with transverse self-intersections, this is an equality, yielding an adjunction formula.

### A first example – Fillable implies tight (in 3 dimensions)

On a first pass, I want to expand upon the example of fillability implying tightness in three dimensions which Roger Casals discussed in his introductory talk. Really, we prove the contrapositive – that an overtwisted contact manifold cannot be filled. For simplicity, we will consider exact fillings. This result is typically attributed to Gromov and Eliashberg, referencing Gromov’s ’85 paper as well as Eliashberg’s paper on filling by holomorphic discs from ’89. This is essentially the same proof in spirit, although we take a little bit of a cheat by considering exact fillings.

Firstly, recall that an overtwisted contact manifold $(M^3,\xi)$ is one such that there exists an embedding of a disk $\phi : D^2 \hookrightarrow M$, such that the so-called characteristic foliation $(d\phi)^{-1}\xi$ on $D^2$, which is actually a singular foliation, looks like the following image, with one singular point in the center and a closed leaf as boundary.

So now suppose $(M^3,\xi)$ has an exact filling $(W^4,\omega = d\lambda)$. We study the space of certain J-holomorphic disks with boundary on the overtwisted disk. The key is that a neighborhood of the overtwisted disk $D$ actually has a canonical neighborhood in $W$ up to symplectomorphism, and one can pick an almost complex structure $J$ to be in a standard form in this neighborhood. It turns out that with this standard choice, in a close enough neighborhood of the singular point $p$ in the interior of $D$, all somewhere injective J-holomorphic curves are precisely those living in a 1-parameter family, called the Bishop family, which radiate outwards from the singular point $p$.

Let us be a bit more precise, so that we can see this Bishop family explictly. Consider the standard 3-sphere $S^3 \subset \mathbb{C}^2$, with its standard contact structure given by the complex tangencies, i.e. $\xi = TS^3 \cap iTS^3$, with $i$ the standard complex structure on $\mathbb{C}^2$. Then consider the disk given by $z \mapsto (z, \sqrt{1-|z|^2})$. The characteristic foliation on this disk looks like the characteristic foliation near the center of the overtwisted disk, so a neighborhood of this disk in $D^4 \subset \mathbb{C}^2$ yields a model for a neighborhood of the center of the overtwisted disk. We may assume the almost complex structure in this neighborhood is just given by the standard one, $i$. Then the Bishop family is just the sequence of holomorphic disks given by $z \mapsto (sz,\sqrt{1-s^2})$ for $s$ a real constant near 0. That these are all of the somewhere injective disks is a relatively easy exercise in analysis. Namely, suppose we had such a disk of the form $z \mapsto (v_1(z),v_2(z))$. Then since boundary points are mapped to the overtwisted disk, $v_2(\partial D^2) \subset \mathbb{R}$. But each component of $v_2$ is harmonic, hence satisfies a maximum principle. Therefore, $v_2(D^2) \subset \mathbb{R}$. But by holomorphicity, $v_2$ cannot have real rank 1 and so must be constant. Hence, any disk in consideration must have $v_2$ is a real constant.

All of these disks live in $D^4 \subset \mathbb{C}^2$, but in particular in the slice where the second component $z_2$ is real, so we can draw this situation in $\mathbb{R}^3$ by forgetting the imaginary part of $z_2$. This is depicted in the following figure.

This Bishop family lives in some component of the moduli space of somewhere injective J-holomorphic disks with boundary on $D$. Perturbing $J$, one can assume this component is actually a smooth 1-dimensional manifold. We can compactify this moduli space by including stable maps, i.e. disks with bubbles, via Gromov compactness. On the Bishop family end, we see explicitly that the limit is just the constant disk at the point $p$. So there must be another stable curve at the other boundary of this moduli space. We prove no such other stable curve can exist.

Similar to how we proved that the only disks completely contained in a neighborhood of the singular point on the overtwisted disk must have been part of the Bishop family, one can use a maximum principle argument to conclude that every holomorphic disk entering this neighborhood must have been in the Bishop family. Alternatively, one can use a modified version of positivity of intersections to conclude that continuing the moduli space away from the Bishop family, these boundaries have to continue radiating outward. Either way, the moduli space has to stay away from the central singularity of the overtwisted disk $D$. But also, the boundary of a J-holomorphic disk cannot be tangent to $\xi$, and in particular cannot be tangent to $\partial D$. This is by a maximum principle which comes from analytic convexity properties of a filled contact manifold.

The only possible explanation is that this is a stable curve with some sphere bubble having formed in the interior of $(W,\lambda)$. But one checks that the relation $g(\cdot,\cdot) = \omega(\cdot,J\cdot)$ implies that for a $J$-holomorphic sphere $u : (S^2,j) \rightarrow (W,J)$, we have $\text{Area}_g(u) = \int_{S^2}u^*\omega$. This vanishes by Stokes’ Theorem since $\omega = d\lambda$ is exact, and so $u$ must be constant, and so there is no bubble. In other words, this cannot explain the other boundary point of the component of the moduli space containing the Bishop family, so this yields a contradiction.

### On McDuff’s The structure of ruled and rational symplectic 4-manifolds

Emily Maw’s talk from the workshop followed this paper by Dusa McDuff. In what follows, we shall consider triples $(V,C,\omega)$ such that $(V,\omega)$ is a smooth closed symplectic 4-manifold and $C$ is a rational curve, by which we mean a symplectically embedded $S^2$. We call a rational curve $C$ exceptional if $C \cdot C = -1$ with respect to the intersection product on $H_2(V)$ (with respect to its orientation coming from $\omega$). We say $(V,C,\omega)$ is minimal if $V \setminus C$ contains no exceptional curves. The main theorem is as follows:

Theorem [McDuff ’90]: If $(V,C,\omega)$ is minimal and $C \cdot C \geq 0$, then $(V,\omega)$ is symplectomorphic to either:

• $(\mathbb{C} P^2, \omega_{FS})$, in which case $C$ is either a complex line or a quadric (up to symplectomorphism).
• A symplectic $S^2$-bundle over a compact manifold $M$, in which case $C$ is either a fiber or a section (up to symplectomorphism).

Before describing the proof, which is the part involving J-holomorphic curve techniques, we apply this to weak fillings. We shall concern ourselves with fillings of the lens spaces $L(p,1)$ with their standard contact structures, where $p > 0$ is an integer. Let us first define this contact structure. Recall that the standard contact structure on $S^3$ is the one coming from complex tangencies by viewing $S^3 \subset \mathbb{C}^2$. Then the standard contact structure on $L(p,1)$ is the one given by the quotient $L(p,1) = S^3/(\mathbb{Z}/p\mathbb{Z})$ where the action of $1 \in \mathbb{Z}/p\mathbb{Z}$ given by $(z_1,z_2) \mapsto e^{2\pi i/p}(z_1,z_2)$ preserves the contact structure, so that it descends.

Theorem [McDuff ’90]: The lens spaces $L(p,1)$ all have minimal symplectic fillings $(Z,\omega)$, and when $p \neq 4$, these fillings are unique up to diffeomorphism, and further up to symplectomorphism upon fixing the cohomology class $[\omega]$. The space $L(4,1)$ has two nondiffeomorphic minimal fillings.

Proof (sketch): The complex line bundle $\mathcal{O}(p)$ over $S^2$ comes with a natural symplectic structure, and this forms a cap for $L(p,1)$. The zero section of $\mathcal{O}(p)$ is a rational curve of self intersection $p > 0$. McDuff’s explicit classification includes examples $(V,C)$ for any such given $p$, and $V \setminus C$ thus gives a minimal filling for $L(p,1)$. The remaining statements come from a more detailed analysis of the classification result.

Now, I will not go through all of the details of McDuff’s proof of the main theorem, but I will highlight where various J-holomorphic tools appear in the proof. Let me break up the proof into two big pieces.

Step 1: “Mega-Lemma” Consider $(V,C,\omega)$ minimal as above. There is a tame almost complex structure $J$ such that $[C]$ can be represented by a $J$-holomorphic stable curve of the form $S = S_1 \cup \cdots \cup S_m$, where:

• Each $A_i := [S_i]$ is $J$-indecomposable (meaning any stable curve representing $A_i$ must actually be a legitimate curve of one component)
• The almost complex structure $J$ is regular for all curves in the class $A_i$.
• The $S_i$ are distinct and embedded curves of self-intersection -1, 0, or 1, with at least one index for which $A_i \cdot A_i \geq 0$.

We didn’t prove this at the workshop, so I won’t discuss it in detail here. But this is a major reduction into cases. For example, if $m = 1$ and $S \cdot S = 1$, then it had already been shown that this implies that $V = \mathbb{C}P^2$. This bleeds into…

Step 2: Using the evaluation maps constructively

Let us discuss the proof of this last fact briefly. The idea is as follows. We consider the moduli space $\mathcal{M}^*(A,J)$ consisting of simple holomorphic spheres representing the class $A = [S]$. This comes with an evaluation map of the form

$\mathcal{M}^*(A,J) \times_{G} (S^2 \times S^2) \rightarrow V \times V$

where $G$ is the group of automorphisms of $S^2$. Both sides have dimension 8 and this evaluation map is injective away from the diagonal since $A \cdot A = 1$ and we have positivity of intersection. Therefore, this map has degree 1, and so any pair of distinct points on $V$ has a unique curve passing through it. This is enough to show $V = \mathbb{C}P^2$.

Let us do another case, but show that the adjunction formula also comes into play.

Proposition: Suppose $B$ is a simple homology class in $(V,\omega)$ (i.e. is not a multiple of another homology class) with $B \cdot B = 0$, and suppose $F$ is a rational embedded sphere representing $B$. Then there is a fibration $\pi \colon V \rightarrow M$ with symplectic fibers and such that $F$ is one of the fibers.

Proof (sketch): The idea is to consider the moduli space $\mathcal{M}^*_{0,1}(V,J,B)$ of rational embedded $J$-holomorphic curves with 1 marked point $p \in S^2$, and where $J$ is chosen to tame $\omega$ and such that $F$ is itself a $J$-holomorphic curve, and where we have quotiented by reparametrization of the domain. Then one can compute the dimension of this moduli space at a given curve $C$ in the appropriate way as

$d = \dim V + 2c_1(TV) \cdot [C] - 4$,

where the last -4 comes from quotienting by the subgroup of $PSL_2(\mathbb{C})$ fixing the marked point. Applying adjunction for the curve represented by $F$, so that $[C] \cdot [C] = 0$, yields $d = 4$. We also have an evaluation map

$\text{ev} : \mathcal{M}^*_{0,1}(V,J,B) \rightarrow V$

Since $B \cdot B = 0$, there is at most one $B$-curve through each point in $V$, so it follows that this evaluation map has degree at most 1, and hence equal to 1 by regularity. This yields the structure of a fibration $\pi : V \rightarrow M$ where the fibers are precisely the curves in our moduli space. Since the fibers are holomorphic, they are symplectic by the taming condition.

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## Kylerec Overview

Updates (June 11, 2017): Added link to other notes from Kylerec workshop, and fixed an error caught by Chris Wendl in the comments.

I’m very excited to be joining this blog!

This is the first of a series of posts about the content of the Kylerec workshop, held May 19-25 near Lake Tahoe, which focused on fillings of contact manifolds. Under the guidance of our mentors, Roger Casals, Steven Sivek, and Laura Starkston, we worked from the basic theory of fillings through some state-of-the-art results. Many of the basics have been discussed on this blog already in Laura Starkston’s posts from January 2013: Part 1 and Part 2 on Fillings of Contact Manifolds. For a more thorough introduction to types of filling and the differences between them, I suggest reading those posts (and the accompanying comments by Paolo Ghiginni and Chris Wendl). This post will remain self-contained anyway.

One can find notes that I took (except for three lectures, due to technical difficulties) at the Kylerec 2017 tab at this link. Other notes (with shorter load times, and including the ones I’m missing) will be posted on the Kylerec website soon are now posted on the Kylerec website here.

Comments and corrections are very welcome!

### Definitions

We quickly review the various notions of fillings of a contact manifold. We shall always assume that our manifolds are oriented and contact structures cooriented. As a starting point, one might be interested in smooth fillings of contact manifolds. It turns out that this problem is rather uninteresting. Every contact manifold of dimension $2n+1$ has a structure group which can be reduced to $U(n) \times 1$, but the complex bordism group is well known to satisfy $\Omega^U_{2n+1} = 0$. As a consequence, every contact manifold is smoothly fillable. We must therefore consider fillability questions which extend beyond the realm of complex bordism in order to discover interesting phenomena.

These notions are as follows, in (strictly!) increasing order of strength.

• We say a contact 3-manifold $(M^3,\xi)$ is weakly fillable if it is the smooth boundary of a symplectic manifold $(W^4,\omega)$ such that $\omega|_{\xi} > 0$. There is a generalization in higher dimensions due to Massot, Niederkrüger, and Wendl, but we omit it here. (Simply requiring that $\omega|_{\xi}$ is a positive symplectic form in the same conformal symplectic class as the natural one on $\xi$, i.e. is $d\alpha|_{\xi}$ up to scaling where $\alpha$ is a contact form for $\xi$, implies strong fillability in higher dimensions, by McDuff.)
• We say a contact manifold $(M^{2n-1},\xi)$ is strongly fillable if there is a weak filling $(W^{2n},\omega)$ such that one can find a Liouville vector field $V$ in a neighborhood of $M$, i.e. one such that $\mathcal{L}_V\omega = \omega$, such that $(\iota_V\omega)|_M$ gives a (properly cooriented) contact form for $\xi$.
• We say a contact manifold $(M^{2n-1},\xi)$ is exactly fillable if there is a strong filling such that the Liouville vector field $V$ can be extended to all of $(W,\omega)$. In other words, $M$ is the contact boundary of a Liouville domain $(W,\omega = d\alpha)$ where $\alpha = \iota_V\omega$.
• We say a contact manifold $(M^{2n-1},\xi)$ is Weinstein (or Stein) fillable if it is exactly fillable by some $(W,\omega = d\alpha)$, where $\alpha = \iota_V\omega$, such that there is also a Morse function $f$ on $W$ such that $V$ is gradient-like for $f$ and $M$ is a maximal regular level set. In other words, $M$ is the contact boundary of a Weinstein domain.

As a final remark, there is a notion of overtwistedness in contact manifolds. In 3-dimensions, this is characterized by the existence of an overtwisted disk. This was known to obstruct all types of fillings, due to Eliashberg and Gromov. In higher dimensions, overtwistedness was defined in a paper of Borman, Eliashberg, and Murphy, which was discussed on this blog by Laura Starkston and Roger Casals, starting with this post and concluding with this one. This definition implies the existence of a plastikstufe as defined by Niederkrüger, which had been already shown to obstruct fillings (strongly in the same paper, weakly in the paper by Massot, Niederkrüger, and Wendl). In other words, in any dimension, overtwistedness implies not fillable. A contact manifold which is not overtwisted is called tight, so equivalently, fillable implies tight, in all dimensions.

To summarize this section:

Tight < Weakly fillable < Strongly fillable < Exactly fillable < Weinstein fillable

where all of the inclusions turn out to be strict.

### Two Motivating Questions

Question 1: What tools do we have at each level of fillability?

The easiest type of filling to understand is that of the Weinstein filling, since Weinstein domains have an explicit surgery theory, which lends themselves to concrete geometric descriptions. Most notably, a Weinstein domain can be thought of as a symplectic Lefschetz fibration, which naturally has an open book decomposition on its boundary whose monodromy is a product of positive Dehn twists. Hence, Weinstein fillings and fillability can be studied through studying supporting open book decompositions for a contact manifold $(M,\xi)$.

Another rather powerful tool is the study of J-holomorphic curves. Let us provide a quick example: the proof that fillability of a contact 3-manifold implies tightness. One assumes by way of contradiction that an overtwisted contact 3-manifold has a filling. Then one considers a certain compact 1-dimensional moduli space of J-holomorphic curves with boundary on the overtwisted disk. One finds an explicit component of this moduli space which has one endpoint (a constant disk) but cannot have another endpoint, which contradicts the compactness of the moduli space. In higher dimensions, studying similar moduli spaces of J-holomorphic curves yields obstructions to fillings.

There are some other miscellaneous techniques. For example, Liouville domains have attached to them a symplectic homology, which provides another tool for the case of exact fillings. And in the case of 3-dimensional contact manifolds, one can also study the Seiberg-Witten invariants of a given filling.

Question 2: How can we study the topology of different fillings? Or tell when fillings are distinct even if they have the same homology?

J-holomorphic curves come with extra evaluation maps which allow one to study how the moduli space of curves compares to some underlying topology, e.g. of the filling or of the contact manifold. This is a technique which comes up many times in different contexts, and it sometimes allows us to produce maps between the filling or the contact manifold in question which do not exist for any other obvious reason.

Similarly, symplectic homology in its two flavors $SH$ and $SH^{+}$ fits into an exact triangle with Morse homology, and so one can understand the topology of a filling from its symplectic homology. One might be interested, for example, in studying fillings with $SH = 0$, in which case the homology of the filling is completely determined by $SH^{+}$. Alternatively, $SH$ can be used directly to distinguish fillings.

### Overview of Kylerec

More detailed posts about the contents of Kylerec will appear in future blog posts, but I will outline here precisely what was covered.

Day 1: After an overview talk, we spent the rest of the day studying the surgery theory of Weinstein manifolds, and began our study of the correspondence between Weinstein fillings, Lefschetz fibrations, and open book decompositions.

Day 2: We highlighted some results from this correspondence, and then turned towards an introduction to the theory of J-holomorphic curves, including applications of this theory to fillings via McDuff’s classification result as well as Wendl’s J-holomorphic foliations.

Day 3: On our short day, we first discussed some applications of J-holomorphic curves to high-dimensional fillings due to Barth, Geiges, and Zehmisch (for example reproving the result of Eliashberg, Floer, and McDuff that the standard sphere has a unique aspherical filling), and applied Wendl’s theorem (as discussed in Day 2) following a paper of Plamenevskaya and Van Horn-Morris to show that many contact structures on the lens spaces $L(p,1)$ have unique Weinstein fillings up to deformation equivalence.

Day 4: We discussed the Seiberg-Witten equations, how they appear in symplectic geometry, and how they are used by Lisca and Matic to distinguish contact structures on homology 3-spheres which are homotopic (through plane fields) but not isotopic (through contact structures). We also discussed how Calabi-Yau caps, as defined by Li, Mak, and Yasui, can be used to prove certain uniqueness results on fillings of unit cotangent bundles of surfaces, as in this paper by Sivek and Van Horn-Morris.

Day 5: On our last day, we focused mainly on symplectic homology (and its variants). In one talk, we performed computations which allowed us to distinguish contact structures on standard spheres (see Ustilovsky’s paper) and to compute the symplectic homology of fillings of certain Brieskorn spheres (see Uebele’s paper). We also discussed Lazarev’s generalization of M.-L.Yau’s theorem (that subcritical Weinstein fillings have isomorphic integral cohomology) to the flexible case.

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## Some open computational problems in link homology and contact geometry

I’m thrilled to join everyone at the best-named math blog.

I am just home from Combinatorial Link Homology Theories, Braids, and Contact Geometry at ICERM in Providence, Rhode Island.  The conference was aimed at students and non-experts with a focus on introducing open problems and computational techniques.  Videos of many of the talks are available at ICERM’s site.  (Look under “Programs and Workshops,” then “Summer 2014”.)

One of the highlights of the workshop was the ‘Computational Problem Session’ MC’d by John Baldwin with contributions from Rachel Roberts, Nathan Dunfield, Joanna Mangahas, John Etnyre, Sucharit Sarkar, and András Stipsicz.  Each spoke for a few minutes about open problems with a computational bent.

I’ve done my best to relate all the problems in order with references and some background.  Any errors are mine.  Corrections and additions are welcome!

### Rachel Roberts

Contact structures and foliations

Eliashberg and Thurston showed that a $C^2$ one-dimensional foliation of a three-manifold can be $C^0$-approximated by a contact structure (as long as it is not the product foliation on $S^1 \times S^2$).  Vogel showed that, with a few other restrictions, any two approximating contact structures lie in the same isotopy class.  In other words, there is a map $\Phi$ from $C^2$, taut, oriented foliations to contact structures modulo isotopy for any closed, oriented three-manifold.

Geography: What is the image of $\Phi$?

Botany: What do the fibers of $\Phi$ look like?

The image of $\Phi$ is known to be contained within the space of weakly symplectically fillable and universally tight contact structures.  Etnyre showed that if one removes “taut”, then $\Phi$ is surjective.  Etnyre and Baldwin showed that $\Phi$ doesn’t “see” universal tightness.

L-spaces and foliations

A priori the rank of the Heegaard Floer homology groups associated to a rational homology three-sphere Y are bounded by the first ordinary homology group: $\text{rank}(\hat{HF}(Y)) \geq |H_1(Y; \mathbb{Z})|$. An L-space is a rational homology three-sphere for which equality holds.

Conjecture: Y is an L-space if and only if it does not contain a taut, oriented, $C^2$ foliation.

Ozsváth and Szabó showed that L-spaces do not contain such foliations.  Kazez and Roberts proved that the theorem applies to a class of $C^0$ foliations and perhaps all $C^0$ foliations.  The classification of L-spaces is incomplete and we are led to the following:

Question: How can one prove the (non-)existence of such a foliation?

Existing methods are either ad hoc or difficult (e.g. show that the manifold does not act non-trivially on a simply-connected (but not necessarily Hausdorff!) one-manifold). Roberts suggested that Agol and Li’s algorithm for detecting “Reebless” foliations via laminar branched surfaces may be useful here, although the algorithm is currently impractical.

### Nathan Dunfield

What do random three-manifolds look like?

First of all, how does one pick a random three-manifold?  There are countably many compact three-manifolds (because there are countably many finite simplicial complexes, or because there are countably many rational surgeries on the countably many links in $S^3$, or because…) so there is no uniform probability distribution on the set of compact orientable three-manifolds.

To dodge this issue, we first consider random objects of bounded complexity, then study what happens as we relax the bound.  (A cute, more modest example: the probability that two random integers are relatively prime is $6/\pi^2$.1).  Fix a genus $g$ and write $G$ for the mapping class group of the oriented surface of genus $g$.  Pick some generators of $G$. Let $\phi$ be a random word of length $N$ in the chosen generators.   We can associate a unique closed, orientable three-manifold to $\phi$ by identifying the boundaries of two genus $g$ handlebodies via $\phi$.

Metaquestion: How is your favorite invariant distributed for random 3-manifolds of genus $g$?  How does it behave as $g \to \infty$?  Experiment! (Ditto for knots, links, and their invariants.)

Challenge: Show that your favorite conjecture about some class of three-manifolds or links holds with positive probability. For example:

Conjecture: a random three-manifold is not an $L$-space, has left-orderable fundamental group, admit a taut foliation, and admit a tight contact structure.

These methods can also be used to prove more traditional-sounding existence theorems. Perhaps you’d like to show that there is a three-manifold of every genus satisfying some condition. It suffices to show that a random three-manifold of fixed genus satisfies the condition with non-negative probability! For example,

Theorem: (Lubotzky-Maher-Wu, 2014): For any integers $k$ and $g$ with $g \geq 2$, there exist infinitely many closed hyperbolic three-manifolds which are integral homology spheres with Casson invariant $k$ and Heegaard genus $g$.

### Johanna Mangahas

What do generic mapping classes look like?

Here are two sensible ways to study random elements of bounded complexity in a finitely-generated group.

• Fix a generating set. Look at all words of length N or less in those generators and their inverses. (word ball)
• Fix a generating set and the associated Cayley graph. Look at all vertices within distance N of the identity. (Cayley ball)

A property of elements in a group is generic if a random element has the property with probability, so the meaning of “generic” differs with the meaning of “random.” For example, consider the group $G = \langle a, b \rangle \oplus \mathbb{Z}$ with generating set $\{(a,0), (b,0), (id,1)\}$.  The property “is zero in the second coordinate” is generic for the first notion but not the second.  So we are stuck/blessed with two different notions of genericity.

Recall that the mapping class group of a surface is the group of orientation-preserving homeomorphisms modulo isotopy. Thurston and Nielsen showed that a mapping class $\phi$ falls into one of three categories:

• Finite order: $\phi^n = id$ for some $n$.
• Reducible: $\phi$ fixes some finite set of simple closed curves.
• Pseudo-Anosov: there exists a transverse pair of measured foliations which $\phi$ stretches by $\lambda$ and $1/\lambda$.

The first two classes are easier to define, but the third is generic.

Theorem: (Rivin and Maher, 2006) Pseudo-Anosov mapping classes are generic in the first sense.

Question: Are pseudo-Anosov mapping classes generic in the second sense?

The braid group on n strands can be understood as the mapping class group of the disk with n punctures. But the braid group is not just a mapping class group; it admits an invariant left-order and a Garside structure. Tetsuya Ito gave a great minicourse on both of these structures!

Fast algorithms for the Nielsen-Thurston classification

Question: Is there a polynomial-time algorithm for computing the Thurston-Nielsen classification of a mapping class?

Matthieu Calvez has described an algorithm to classify braids in $O(\ell^2)$ where $\ell$ is the length of the candidate braid. The algorithm is not yet implementable because it relies on knowledge of a function $c(n)$ where $n$ is the index of the braid. These numbers come from a theorem of Masur and Minsky and are thus difficult to compute. These difficulties, as well as the power of the Garside structure and other algorithmic approaches, are described in Calvez’s linked paper.

Challenge: Implement Calvez’s algorithm, perhaps partially, without knowing $c(n)$.

Mark Bell is developing Flipper which implements a classification algorithm for mapping class groups of surfaces.

Question: How fast are such algorithms in practice?2

### John Etnyre

Contactomorphism and isotopy of unit cotangent bundles

For background on all matters symplectic and contact see Etnyre’s notes.

Let $M$ be a manifold of any (!) dimension.  The total space of the cotangent bundle $E = T^*M$ is naturally symplectic:  the cotangent bundle of $E$ supports the Liouville one-form $\lambda$ characterized by $\alpha^*(\lambda) = \alpha$ for any one-form $\alpha \in T^*M$; the pullback is along the canonical projection $T^* T^* \to T^*M$.  The form $d\lambda$ is symplectic on $T^*M$.

Inside the cotangent bundle is the unit cotangent bundle $S^*M = \{(p,v) \in T^*M : |v| = 1\}$. (This is not a vector bundle!) The form $d\lambda$ restricts to a contact structure on the $S^*M$.

Fact: If the manifolds $M$ and $N$ are diffeomorphic, then their unit cotangent bundles $S^*M$ and $S^*N$ are contactomorphic

Hard question: In which dimensions greater than two is the converse true?

This question is attributed to Arnol’d, perhaps incorrectly.  The converse is known to be true in dimensions one and to and also in the case that $M$ is the three-sphere (exercise!).

Tractable (?) question: Does contactomorphism type of unit cotangent bundles distinguish lens spaces from each other?

Also intriguing is the relative version of this construction. Let $K$ be an Legendrian embedded (or immersed with transverse self-intersections) submanifold of $M$. Define the unit cosphere bundle of $K$ to be $L_K = \{w \in T^*M : w(v) = 0, \forall v \in TK\}$. You can think of it as the boundary of the normal bundle to $K$. It is a Legendrian submanifold of the unit cotangent bundle $T^*M$.

Fact: If $K_1$ is Legendrian isotopic to $K_2$ then $L_{K_1}$ is Legendrian isotopic to $L_{K_2}$.

Relative question: Under what conditions is the converse true?

Etnyre noted that contact homology may be a useful tool here.  Lenny Ng’s “A Topological Introduction to Knot Contact Homology” has a nice introduction to this problem and the tools to potentially solve it.

### Sucharit Sarkar

How many Szabó spectral sequences are there, really?

Ozsváth and Szabó constructed a spectral sequence from the Khovanov homology of a link to the Heegaard Floer homology of the branched double cover of $S^3$ over that link. (There are more adjectives in the proper statement.) This relates two homology theories which are defined very differently.

Challenge: Construct an algorithm to compute the Ozsváth-Szabó spectral sequence.

Sarkar suggested that bordered Heegaard Floer homology may be useful here. Alternatively, one could study another spectral sequence, combinatorially defined by Szabó, which also seems to converge to the Heegaard Floer homology of the branched double cover.

Question: Is Szabó’s spectral sequence isomorphic to the Ozsváth-Szabó spectral sequence?

Again, the bordered theory may be useful here. Lipshitz, Ozsváth, and D. Thurston have constructed a bordered version of the Ozsváth-Szabó spectral sequence which agrees with the original under a pairing theorem.

If the answer is “yes” then Szabó’s spectral sequence should have more structure. This was the part of Sarkar’s research talk which was unfortunately scheduled after the problem session. I hope to return to it in a future post (!).

Question: Can Szabó’s spectral sequence be defined over a two-variable polynomial ring? Is there an action of the dihedral group $D_4$ on the spectral sequence?

### András Stipsicz

Knot Floer Smörgåsbord

Link Floer homology was spawned from Heegaard Floer homology but can also be defined combinatorially via grid diagrams. Lenny Ng explained this in the second part of his minicourse. However you define it, the theory assigns to a link $L$ a bigraded $\mathbb{Z}[U]$-module $HFK^-(L)$. From this group one can extract the numerical concordance invariant $\tau(L)$. Defining $HFK^-$ over $\mathbb{Q}[U]$ or $\mathbb{Z}/p\mathbb{Z}[U]$ one can define invariants $\tau_0$ and $\tau_p$.

Question: Are these invariants distinct from $\tau$?

Harder question: Does $HFK^-$ have $p$-torsion for some $p \in \mathbb{Z}$? (From a purely algebraic perspective, a “no” to the first question suggests a “no” to this one.)

Stipsicz noted that there are complexes of $\mathbb{Z}[U]$-modules for which the answer is yes, but those complexes are not known to be $HFK^-(L)$ of any link. Speaking of which,

“A shot in the dark:” Characterize those modules which appear as $HFK^-$.

In another direction, Stipsicz spoke earlier about a family of smooth concordance invariants $\Upsilon_t$. These were constructed from link Floer homology by Ozsváth, Stipsicz, and Szabó. Earlier, Hom constructed the smooth concordance invariant $\epsilon$. Both invariants can be used to show that the smooth concordance group contains a $\mathbb{Z}^\infty$ summand, but their fibers are not the same: Hom produced a knot which has $\Upsilon_t = 0$ for all t and $\epsilon \neq 0$.

Conversely: Is there a knot with $\epsilon = 0$ by $\Upsilon_t \neq 0$?

Stipsicz closed the session by waxing philosophical: “When I was a child we would get these problems like ‘Jane has 6 pigs and Joe has 4 pigs’ and I used to think these were stupid. But now I don’t think so. Sit down, ask, do calculations, answer. That’s somehow the method I advise. Do some calculations, or whatever.”

1. An analogous result holds for arbitrary number fields — I make no claims about the cuteness of such generalizations.
2. An old example: the simplex algorithm from linear programming runs in exponential time in the worst-case, but in

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## Overtwisted disks and filling holes

This post is on the end of the proof by Borman, Eliashberg, and Murphy that there is an overtwisted contact structure in every homotopy class of almost contact structures in higher dimensions and via the parametric version, any two overtwisted contact structures which are homotopic as almost contact structures, are isotopic as contact structures. There are a number of other posts preceding this one that are meant to be read first, and there are a few pieces of the proof that we skipped, but I think this will be my last post on this topic.

Overtwisted disks in higher dimensions and filling the holes

In dimension three, an overtwisted disk is a certain model germ of a contact structure on a two dimensional disk. The key property of this overtwisted disk which generalizes in higher dimensions, is its role in the proof of the h-principle: after connecting the codimension zero “holes” where the almost contact structure resists becoming genuinely contact, with a neighborhood of the overtwisted disk, one is able to extend the contact structure. One useful feature of overtwisted disks in dimension three, is that they can be recognized simply by finding an embedded unknotted circle with Thurston-Bennequin number 0 (the contact planes along the unknot do not twist relative to the Seifert framing determined by the disk that is bounded by the unknot). This is not true in higher dimensions: there are quantitative properties of the contact structure on the interior of the disk which are needed for the h-principle proof to work.

Recall, from Roger’s post, that in the presence of an overtwisted disk, we can reduce the problem of extending the contact structure over the hole, to extending the contact structure over an annulus (interval times sphere) whose germ on one boundary component is modelled by the contact Hamiltonian obtained by concatenating the Hamiltonian modelling the hole with the overtwisted model Hamiltonian, and whose germ on the other boundary component is given by the overtwisted Hamiltonian. (Remember this picture?)

This is because we can connect each hole to an overtwisted annulus by a tunnel, and then forget that we already had a genuine contact structure on the tunnel and the overtwisted annulus and just look at the contact germs on the two boundary components of the boundary sum of the ball with the annulus, like in this schematic picture.

This is the key point where we use the overtwistedness of the contact structure. The arguments to get to this point are made in a relative way that just fixes the contact structure in the overtwisted regions. At this point, we need to change the contact structure on the overtwisted annulus. In order to fill in the larger annulus (the overtwisted annulus connected to the hole) with a genuine contact structure, we need to show that, up to conjugation, the overtwisted Hamiltonian is less than the connect sum of the Hamiltonian for the hole with the overtwisted Hamiltonian. We are assuming at this point, that we know how to homotope the almost contact structure so that it is genuinely contact in the complement of holes, and each of the holes has its almost contact structures given by a circle model. Moreover, by doing this extra carefully (using equivariant coverings), we can assume that there are finitely many different types of contact Hamiltonians defining the circle models for the holes. The number of types of contact Hamiltonians needed a priori depends on the dimension. An easier reduction is to assume that the Hamiltonian $K: \Delta\times S^1 \to \mathbb{R}$ is independent of the $S^1$ (time) direction since the circle is compact so $\overline{K}(x)=\min_{\theta\in S^1} K(x,\theta)$ is well-defined and satisfies $\overline{K}\leq K$ so there is a genuine contact annulus extending the contact structure from the boundary of the circle model for $K$ inward to the boundary of the circle model for $\overline{K}$.

In order to prove the key lemma that we can fill in the appropriate annuli, we need a more concrete family of contact Hamiltonians. Consider a contact Hamiltonian $K_{\varepsilon}$ on the cylinder $\Delta_{cyl}=\{(z,u_i,\theta_i): |z|\leq 1, u=\sum u_i\leq 1\}\subset \mathbb{R}^{2n-1}$ which is negative on the region where $|z|$ and $u$ are both less than $1-\varepsilon$, and which increases linearly from 0 in $z$ and $u$ towards the boundary with slope 1. These are called special Hamiltonians . The main thing which is special about such a Hamiltonian $K_{\varepsilon}$ is that there is a contact embedding $\Theta$ of $\Delta_{cyl}$ with the standard contact form, into the boundary sum of $\Delta_{cyl}$ with itself, such that $\Theta_*K_{\varepsilon}$ is less than the connected sum of $K_{\varepsilon}$ with itself. Given this, if the hole and the overtwisted annulus are both modelled by such Hamiltonians with the same $\varepsilon$, we can fill in the holes by genuine contact structures.

Notice that any contact Hamiltonian which is positive on $\partial \Delta_{cyl}$ must dominate (is greater than) some special Hamiltonian for sufficiently small $\varepsilon$. It is important that it is possible to reduce to assuming that the holes are modelled by finitely many types of contact Hamiltonian circle models, therefore in a given dimension, there is a certain universal $\varepsilon_{univ}$, such that for any $\varepsilon<\varepsilon_{univ}$, every hole dominates a circle model for a special $K_{\varepsilon}$. Therefore, the key overtwisted annuli are given by circle models for special Hamiltonians corresponding to such an $\varepsilon$.

To get from overtwisted annuli to overtwisted disks, we use the fact that the main lemma embedding $\Theta$ fixes the end where $z\in[1-\varepsilon,1]$. Therefore we do not need the full annulus (neighborhood the boundary of the cylinder), only the topological disk obtained but cutting off the end of the cylinder.

The overtwisted disk is thus defined to be the disk with the contact germ on the boundary of a circle model over a cylinder (excluding one end) defined by a special contact Hamiltonian $K_{\varepsilon}$ for some $\varepsilon<\varepsilon_{univ}$ where $\varepsilon_{univ}$ depends only on the dimension. I think that dependence on the dimension is not really understood at this point, but the idea is that $\varepsilon_{univ}$ probably gets smaller as the dimension increases, so the region where the contact Hamiltonian is negative would be larger.

Proving the main lemma

We want to show that there is a contact embedding $\Theta:\Delta_{cyl}\to \Delta_{cyl}\# \Delta_{cyl}$ such that for a special Hamiltonian $K_{\varepsilon}$, $\Theta_*K_{\varepsilon} (where here $\#$ denotes the boundary sum obtained by tubing the two cylinders together so that the contact Hamiltonian is positive on the tube). For the parametric version, the main lemma shows there is a family $\Theta_s$ interpolating between the identity and $\Theta$.

Recall the things we know how to do with contactomorphisms from the previous post:
(1) We can reorder contact Hamiltonians however we want in regions where they are negative by the disorder lemma.
(2) We have transverse scaling contact embeddings which shrinks/expands $\Delta_{cyl}$ in the $z$ direction by a diffeomorphism $h:\mathbb{R}\to \mathbb{R}$ at the cost of correspondingly shrinking/expanding $\Delta_{cyl}$ in the $u$ direction by rescaling by $h'(z)$. The effect on the contact Hamiltonian is $(\Phi_h)_*K(h(z),h'(z)u,\theta)=h'(z)K(z,u,\theta)$.
(3) We have twist embeddings which shrink/expand $\Delta_{cyl}$ in the radial $u$ direction by rescaling by $\frac{1}{1+g(z)u}$ if you allow the angular $\theta$ directions to be twisted. The effect on the contact Hamiltonian if we ignore the angular coordinate is $(\Psi_g)_*K(z,\frac{u}{1+g(z)u})=(1-g(z)u)K(z,u)$.

To prove the main lemma, we want to stretch out the $z$ direction of $\Delta_{cyl}$ so that it spreads the length of the connected sum. We can do this with a transverse scaling contactomorphism, but the $u$ directions will expand: $(z,u)\mapsto (h(z),h'(z)u)$. Since we don’t want to mess with the contact structure on the $z$ ends, we choose $h$ to look like a translation so $h'(z)=1$ when $z$ is within $\varepsilon$ of the ends. We can compensate for the expansion in the $u$ directions away from the ends with a twist embedding which rescales the expanded $u$ directions to fit back inside a (longer) cylinder where $u\leq 1$, by choosing $g(z)=1-\frac{1}{h'(h^{-1}(z))}$. The total effect of composing these two maps is an embedding $\Gamma$ mapping $(z,u)\mapsto (h(z),\frac{h'(z)u}{1+(h'(z)-1)u})$ (the angular directions get twisted some amount but we don’t care). $\Gamma$ sends a short cylinder $\Delta_{cyl}$ to a longer cylinder $\Delta_{cyl}\# \Delta_{cyl}$, so that the points where $u=1$ are sent to points where $u=1$, but points where $u<1$ are sent to points with $u$-coordinate $\frac{h'(z)u}{1+(h'(z)-1)u}> u$. So this contactomorphism inflates the cylinder in the $u$ directions towards the boundary. By choosing a family of diffeomorphisms $h_s$ starting with a basic translation we get a family of embeddings $\Gamma_s$ which look like this:

Now, we want to see the effect on the contactomorphisms on a special Hamiltonian $K_{\varepsilon}$. We find that

$(\Gamma_s)_*K_{\varepsilon}(h_s(z),\frac{h_s'(z)u}{1+(h_s'(z)-1)u})=(h_s'(z)-(h_s'(z)-1)u)K(z,u)$

which can be rewritten as

$(\Gamma_s)_*K_{\varepsilon}(h_s(z),u)=(h_s'(z)-(h_s'(z)-1)u)K\left(z,\frac{u}{h_s'(z)-(h_s'(z)-1)u}\right)$.

When $z$ is within $\varepsilon$ of the ends, we have chosen $h$ to be a translation, so $(\Gamma_s)_*K_{\varepsilon}(h_s(z),u)=K_{\varepsilon}(z,u)$, i.e. the Hamiltonian is basically fixed to be standard on these ends. When we reach $s=1$, the ends of $\Gamma_1(\Delta_{cyl})$ coincide with the ends of $\Delta_{cyl}\#\Delta_{cyl}$ so in these regions $(\Gamma_1)_*(K_\varepsilon)=K_{\varepsilon}\#K_{\varepsilon}$.

The rescaling factor for the Hamiltonian, $(h_s'(z)-(h_s'(z)-1)u)$ is always greater than or equal to 1, so the region where $(\Gamma_s)_*K \leq 0$ is the image under $\Gamma_s$ of the region where $K\leq 0$ and similarly $\{(\Gamma_s)_*K\geq 0\}=\Gamma_s(\{K\geq 0\})$. Since we can use the disorder lemma, we don’t care much about the exact negative values of $(\Gamma_s)_*K$, but we do need $(\Gamma_1)_*K(z,u)\leq K\#K(z,u)$ wherever $(\Gamma_1)_*K\geq 0$. Therefore we need to check this inequality on points $\Gamma_1(z,u)$ where $u\in [1-\varepsilon,1]$ and $z$ is more than $\varepsilon$ away from the ends (since we already understand the behavior when $z$ is within $\varepsilon$ of the boundary). On this region, the special Hamiltonian $K_{\varepsilon}$ is just a linear function of $u$ with slope 1. Therefore

$(\Gamma_s)_*K_{\varepsilon}(h_s(z),u)=(h_s'(z)-(h_s'(z)-1)u)\left(\frac{u}{h_s'(z)-(h_s'(z)-1)u}-(1-\varepsilon) \right)$

which as a function of $u$ is linear, has the value $0$ when $\frac{u}{h_s'(z)-(h_s'(z)-1)u}=1-\varepsilon$, and the value $\varepsilon$ at $u=1$. Notice that $\frac{u}{h_s'(z)-(h_s'(z)-1)u}=1-\varepsilon$ when $u=\frac{h_s'(z)(1-\varepsilon)}{1+(h_s'(z)-1)(1-\varepsilon)}>1-\varepsilon$ so in this region $(\Gamma_s)_*K_{\varepsilon}$ compares to $K\#K$ like this:

Therefore $(\Gamma_1)_*K_{\varepsilon}(z,u)\leq K\#K(z,u)$ wherever $(\Gamma_1)_*K_{\varepsilon}\geq 0$. Then we can use the disorder lemma to produce a contactomorphism which fixes everything on this positive region but makes the Hamiltonian sufficiently negative in the region where $K_\varepsilon\#K_\varepsilon\leq 0$ so that after composing $\Gamma_s$ with this disorder contactomorphism we get the embedding $\Theta_s$ such that $(\Theta_1)_*K_{\varepsilon}\leq K_{\varepsilon}\#K_{\varepsilon}$ as required. Notice that $\Theta_s$ fixes the end where $z\in[1-\varepsilon,1]$ so we do not actually need to use that end of the overtwisted annulus to fill in the hole.

It is worth noting that an overtwisted disk could be modelled using any Hamiltonian for which the main lemma could be proven, not just the ones that increase linearly near the boundary. The tricky part to check for a more general function is the inequality near the $u$-boundary. When the contact Hamiltonian was linear, the contactomorphism transformation and the rescaling factor cancelled in just the right way so that the pushed forward contact Hamiltonian was still linear in $u$ so the inequality could be determined simply by understanding the values near end points. For more general contact Hamiltonians you would probably need to do more work to get the required estimates.

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## Contact Hamiltonians II

This post is a continuation from Roger’s last post on Contact Hamiltonians about Borman, Eliashberg, and Murphy’s h-principal result on higher dimensional overtwisted contact structures. Here we will start to get into some of the main pieces of the proof.

First lets recall what we are trying to prove: given an almost contact structure that contains a particular model “overtwisted disk”, this almost contact structure can be homotoped through almost contact structures to a genuine contact structure. A parametric version of this theorem implies that homotopic overtwisted contact structures are isotopic through contact structures. So far, we still have not actually defined an overtwisted disk in higher dimensions (but will soon); for now just keep in mind that there is a model piece of contact manifold that we assume is embedded in the almost contact manifold from the start. The broad idea of the proof is to modify the almost contact structure to be genuinely contact on larger and larger pieces of the manifold until all the “holes” (pieces where the almost contact structure has not been made contact yet) are filled in. Gromov’s (relative) h-principal for open contact manifolds implies that the almost contact structure can be homotoped to be contact in the complement of a compact codimension zero piece (while fixing the structure near the overtwisted disk). A technical argument which keeps track of the angles between the contact planes and the boundary of the hole reduces the argument to extending the contact structure over holes which near the boundary agree with a certain circular model. We put off this technical argument for now, but mention that it is analogous to the argument in the 3-dimensional case called part 1 in this earlier post.

Refer to section 6 of the BEM paper for more details on the first half of this post, and to section 8 for the second half.

The circular model

The goal here is to define a model almost contact structure on a ball, which near the boundary is a genuine contact structure encoded by a contact Hamiltonian. View the 2n+1 dimensional ball as the product of a 2n-1 dimensional ball $\Delta$ with a 2-dimensional disk $D^2$, viewed as a subset of $\mathbb{C}$. The contact Hamiltonian is a function

$K: \Delta \times S^1 \to \mathbb{R}$

Using the standard contact structure $\lambda_{st}=dz+\sum_i r_i^2d\theta_i$ on $\Delta\subset \mathbb{R}^{2n-1}$, recall that an extension of this function $\widetilde{K}: \Delta \times D^2 \to \mathbb{R}$ defines an almost contact structure $\alpha = \lambda_{st}+\widetilde{K}d\theta$ on $\Delta\times D^2$ which is genuinely contact wherever $\partial_r\widetilde{K}>0$ (compute $\alpha\wedge d\alpha>0$). Using the conventions from the BEM paper, we will use the coordinate $v=r^2$. If $K$ is everwhere positive, we can realize this contact structure near the boundary of the following embedded subset of the standard contact $(\mathbb{R}^{2n+1},\ker(\lambda_{st}+vd\theta)$

$B^{S^1}_{K}:=\{(x,v,\theta)\in \Delta\times \mathbb{C} : v\leq K(x,\theta)\}$

If $K$ is negative anywhere, then we need to look at a modified version. We can still encode the shape of $K$ by shifting everything up by a sufficiently large constant $C$ so that $K+C$ is positive. Then define

$B^{S^1}_{K,C}:=\{(x,v,\theta)\in \Delta \times \mathbb{C} : v\leq K(x,\theta)+C \}$.

In order to have the contact form encodes the contact Hamiltonian $K$ near the boundary, we want to shift the contact form from $\lambda_{st}+vd\theta$ to $\lambda_{st}+(v-C)d\theta$ near the boundary. However, because the polar coordinates degenerate near $v=0$, in a neighborhood of $v=0$, we need to keep the form standard: $\lambda_{st}+vd\theta$. Define a family of functions $\rho_{(x,\theta)}(v)$ to interpolate between these two, and then define the almost contact structure on $B^{S^1}_{K,C}$ by the form $\eta_{\rho}=\lambda_{st}+\rho d\theta$. We want this almost contact form to be genuinely contact near the boundary since we are looking for a model for the holes. You can compute $\eta_{\rho}\wedge d\eta_{\rho}$ to see that $\eta_{\rho}$ defines a genuine contact form exactly when $\partial_v\rho_{(x,t)}(v)>0$. The boundary of the ball $B^{S^1}_{K,C}$ has two pieces: the piece where $v=K(x,\theta)+C$ and the piece where $x\in \partial \Delta$. In a neighborhood of the former piece, $\rho(v)=v-C$ so it has positive derivative, but on the latter piece we have to impose the condition directly that $\partial_v\rho_{(x,t)}>0$ in an open neighborhood of points where $x\in \partial\Delta$.

One can show that different choices for $C,\rho$ which satisfy these conditions do not yield genuinely different almost contact forms $\eta_\rho$ because up to diffeomorphism, different choices do not change the contact structure near the boundary or the relative homotopy type of the almost contact structure on the interior.

The key point is that this almost contact structure on $B^{S^1}_{K,C}$ can be chosen to be a genuine contact structure only along $x$ slices where $K$ is positive. Remember that $\rho_{(x,\theta)}$ says how much the almost contact planes are rotating in the radial direction, and if $\partial_r\rho_{(x,\theta)}=0$ this means the twisting has stopped. If $K(x,\theta)$ is negative then since $\rho_{(x,\theta)}(K(x,\theta)+C)=K(x,\theta)<0$ and $\rho_{(x,\theta)}(v)=v$ near 0, $\rho_{(x,\theta)}$ must have a critical point and the almost contact planes must stop twisting and thus fail to be genuinely contact. In particular, to define the circle model for a contact Hamiltonian $K$ we need $K(x,\theta)>0$ near points where $x\in \partial \Delta$, so we only consider such Hamiltonians.

Here is a 3-dimensional example. The arrows indicate the twisting of the almost contact planes defined by $\rho$. Note that where K fails to be positive the planes start twisting counterclockwise as you move radially outward, but then have to switch to turning clockwise at some point. The functions $\rho_{(x,\theta)}$ are indicated by the graphs above–they start having critical points when K fails to be positive.

If we have two contact Hamiltonians $K_0$ on $\Delta_0$ and $K_1$ on $\Delta_1$ such that $\Delta_0\subset \Delta_1$ and $K_0 \leq K_1$, then it is not hard to see that we can choose circle models for each such that $(B^{S^1}_{K_0,C},\eta_{\rho_0})$ embeds into $(B^{S^1}_{K_1,C},\eta_{\rho_1})$ and so that $(\rho_1)_{(x,\theta)}(v)=v-C$ in a neighborhood of the entire region where $K_0(x,\theta)\leq v \leq K_1(x,\theta)$. In other words, the almost contact structure is contact and twisting in the standard way along the radial direction on the region between $K_0$ and $K_1$. In the terminology of the BEM paper, $K_1$ directly dominates $K_0$. View of the extendability a contact structure from one contact germ defined by a contact Hamiltonian $K_1$ to another germ defined by $K_0$, as an ordering. The thing that makes this ordering interesting is that using contactomorphisms to change coordinates, a contact germ can be modelled by a different contact Hamiltonian. Therefore if $K_0$ and $K_1$ cannot be directly compared (i.e. at some points $K_0\leq K_1$ but at others $K_0>K_1$), then there may be a different contact Hamiltonian $\widetilde{K}_0$ which corresponds to the same contact germ in different coordinates such that $\widetilde{K}_0$ can be compared to $K_1$. This will be the subject of the rest of this post.

Contactomorphisms and conjugating the Hamiltonian

Given a contactomorphism on the domain $(\Delta,\lambda)$, we want to construct an induced contactomorphism on $(\Delta\times \mathbb{C},\lambda+\rho d\theta$. Because contactomorphisms only preserve the contact planes, and not the contact form, a contactomorphism $\Phi: (\Delta,\ker(\lambda))\to (\Delta, \ker(\lambda))$ satisfies $\Phi^*(\lambda)=c_{\Phi}\lambda$ where $c_{\Phi}$ is a positive real valued function on $\Delta$. Because the pull-back rescales $\lambda$, we need to rescale the Hamiltonian on the image as well so that it fits together with $\Phi^*\lambda$ to give a contact form for the same contact structure. Therefore define $\Phi_*K$ by $(\Phi_*K)(\Phi(x),\theta)=c_{\Phi}(x)K(x,\theta)$.

$\Phi$ natural induces an extension on $\Delta\times \mathbb{C}$ defined by $\widehat{\Phi}(x,v,\theta)=(\Phi(x),\phi_{(x,\theta)}(v),\theta)$ for any family of functions $\phi_{(x,\theta)}$. If $\widetilde{\rho}$ defines the contact structure on the image $\lambda+\widetilde{\rho}d\theta$ then

$\widehat{\Phi}^*(\lambda+\widetilde{\rho}d\theta)=\Phi^*\lambda+\widetilde{\rho}\circ\phi d\theta=c_{\Phi}\lambda+\widetilde{\rho}\circ \phi d\theta$

Therefore the function defining the contact Hamiltonian on the image must satisfy $\widetilde{\rho}_{(x,\theta)}\circ \phi_{(x,\theta)}(v)=c_{\Phi}(x)\rho_{(x,\theta)}$.

Why did we include the function $\phi_{(x,\theta)}$ in the above definition of $\widehat{\Phi}$? This is to allow us to reparameterize $\widetilde{\rho}_{(x,\theta)}$ so that it satisfies the required conditions to define the circular model (should look like the identity near $v=0$, and should look like the identity shifted by the constant near $v=K+C$). Before the contactomorphism, to define the circular model, you choose a constant $C$ and then $\rho_{(x,\theta)}$ is considered on the domain $[0, K(x,\theta)+C]$ and is required to have certain behavior near the endpoints of this interval. After rescaling, we have a new Hamiltonian $(\Phi_*K)(\Phi(x),\theta)=c_{\Phi}(x)K(x,\theta)$, so we pick a new constant $\widetilde{C}$ so that $\Phi_*K+\widetilde{C}>0$. Then we consider $\widetilde{\rho}_{\Phi(x),\theta}$ on the interval $[0,c_{\Phi}(x)K(x,\theta)+\widetilde{C}]$ and require it to have particular behavior near $0$ and $c_{\Phi}(x)K(x,\theta)+\widetilde{C}$. Because $\widetilde{\rho}_{(x,\theta)}=c_{\Phi}(x)\rho_{(x,\theta)}\circ \phi_{(x,\theta)}^{-1}$, modifying the functions $\phi_{(x,\theta)}$ allows us to make $\widetilde{\rho}_{\Phi(x),\theta}$ have the desired behavior near the end points of the interval $[0,c_{\Phi}(x)K(x,\theta)]$ so that $\widetilde{\rho}$ can be used to define a circular model for $\Phi_*K$.

The action of the contactomorphism $\Phi$ on the contact Hamiltonian $K$ is referred to as conjugating the Hamiltonian for the following reason. If the contact Hamiltonian is generated by a contact isotopy $\phi^t_K$ in the sense that $\lambda(\partial_t\phi^t_K)=K(\phi^t_K,t)$, then you can compute that $\Phi\phi^t_K\Phi^{-1}=\phi^t_{\Phi_*K}$.

Important types of contactomorphisms and their effects on the Hamiltonian

What kinds of changes can we make in the contact Hamiltonian through contactomorphisms? A key lemma is that in a (star-shaped) region where the contact Hamiltonian is negative, contactomorphisms can be used to make the values arbitrarily close to zero. This basically means that the exact negative values of a contact Hamiltonian do not matter in the ordering, since a contactomorphism can make any given negative values larger than any other given negative values. This indicates that the key difficulty in filling in the contact structure on holes whose boundary looks like a contact Hamiltonian circular model, is where and how large are the regions where the contact Hamiltonian is positive.

The idea of the proof of this “disorder lemma” (Lemma 6.8 in the BEM paper) is as follows. Let $\Delta$ be the region where the contact Hamiltonian $K$ is defined and let $\widetilde{\Delta}$ be a subset containing the piece where $K$ is negative. Construct a contactomorphism $\Phi$ which shrinks $\widetilde{\Delta}$ into itself a lot, but fixes the points of $\Delta$ sufficiently away from $\widetilde{\Delta}$. (You can do this by looking at the flow of an inward pointing contact vector field–this is where the star-shaped condition comes in–cut off to zero sufficiently away from $\widetilde{\Delta}$.) Because $\widetilde{\Delta}$ is being shrunk, the rescaling function $c_{\Phi}(x)$ for the contact form defined by $\Phi^*\lambda=c_{\Phi}\lambda$ is a positive function with very tiny values close to 0, for $x\in \widetilde{\Delta}$. The more $\widetilde{\Delta}$ is shrunk, the tinier the values of $c_{\Phi}$. The corresponding Hamiltonian $(\Phi_*K)(\Phi(x),\theta)=c_{\Phi}(x)K(x,\theta)$ has values rescaled by $c_{\Phi}$. Therefore, by choosing a contactomorphism which shrinks $\widetilde{\Delta}$ enough, the values of $c_{\Phi}$ can be made sufficiently small so that $c_\Phi K(x,\theta)>-\varepsilon$ for $x\in \widetilde{\Delta}$.

In addition to the disorder lemma, we need two types of contactomorphisms of $(\mathbb{R}^{2n-1},\xi_{st})$ which rescale in certain directions. Choose cylindrical coordinates $(z,r_i,\theta_i)$ on $\mathbb{R}^{2n-1}$ and let $u_i=r_i^2$ so $\xi_{st}=\ker(dz+\sum_i u_i\theta_i)$.

A transverse scaling contactomorphism $\Phi_h$ is defined by a diffeomorphism $h:\mathbb{R}\to \mathbb{R}$ by $\Phi_h(z,u_i,\theta_i)=(h(z),h'(z)u_i,\theta_i)$. You can check directly that this diffeomorphism is a contactomorphism which rescales the standard contact form by $h'(z)$. Therefore this contactomorphism modifies a contact Hamiltonian by

$(\Phi_h)_*K(z,u_i,\theta_i)=h'(h^{-1}(z))K\circ\Phi_h^{-1}(z,u_i,\theta_i)$

The tagline for this type of contactomorphism is you can “trade long for thin”. By choosing a shrinking $h$, you can shrink a domain which is long in the $z$ direction at the cost of shrinking the radial $u_i$ directions.

A twist embedding contactomorphism $\Psi_g$ allows you to rescale the radial directions $u_i$ by $\frac{1}{1+g(z)u}$ at the cost of twisting in the angular directions by an amount that depends on $g$ (see section 8.2 of the BEM paper for the exact formulas). The points at radii where $g(z)u>-1$ get sent to points where $g(z)u<1$ since $g(z)\frac{u}{1+g(z)u}-1$. The rescaling factor for the contact form is $(1-g(z)u)$, so the contact Hamiltonian is rescaled accordingly. For positive functions $f_1,f_2$, setting $g=\frac{1}{f_1}-\frac{1}{f_2}$ gives $\Psi_g$ taking the region where $u\leq f_2(z)$ to the region where $u\leq f_1(z)$. Therefore twist embeddings allow you to modify the radial directions however you want to, with basically no cost (just twisting the angular directions).

By composing these two types of contactomorphisms we can use transverse scaling to stretch or shrink in the $z$ direction at the cost of stretching or shrinking radially. Then we can use a twist embedding to counteract the stretching or shrinking in the radial directions, with only the cost of twisting in the angular direction, which does not significantly change the shape of the region.

These contactomorphisms are the key ingredients towards filling in circular model holes connected summed with neighborhoods overtwisted disks, as will be discussed in the next post.

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## Contact Hamiltonians (Part I)

This entry follows the post Contact Hamiltonians (Introduction), where we discussed normal forms for contact forms and the appearance of contact Hamiltonians. In this entry we will focus on the 3–dimensional situation and hence we will be able to write formulas and draw (realistic) pictures.

Consider a 2–sphere of radius 1 in the standard tight contact Euclidean space $(\mathbb{R}^3,\lambda_{st}=dz+r^2d\theta)$. Its characteristic foliation (defined by the intersection of the tangent space and the contact distribution) has two elliptic singular points in the north and south poles and all the leaves are open intervals connecting the north and the south pole. Take a transversal segment I=[0,1] connecting the poles (a vertical segment will do). Given a point in the segment we can consider the unique leaf through that point and move around the leaf until we hit the interval I=[0,1] again. This defines a diffeomorphism of the interval [0,1] fixed at the endpoints. We will call this diffeomorphism the monodromy of the foliation (and note that conversely any diffeomorphism will give a foliation on the 2–sphere via a mapping torus construction and collapsing the boundary). This is drawn in the following figure:

In the figure the monodromy map is represented by the orange arrow. This monodromy does not have fixed points (this is crucial). Let us look at the monodromy in the sphere of radius $\pi+c$ , where c is a small positive constant, in the overtwisted contact manifold $(\mathbb{R}^3,dz+rtg(r)d\theta)$. The overtwisted monodromy is drawn in the next figure:

There are 3 types of points in the vertical transverse interval I=[0,1]. The Type 1 points belong to a leaf, Leaf I in the figure, such that the points move down in the segment. The Type 2 points are the points between the unique pair of closed leaves, these belong to Leaf II and move up. The Type 3 points are fixed points, there are two leaves of this type (Leaf III). The monodromy is represented by the blue arrows.

Hence, we can encode the tight and the overtwisted foliations on the 2–sphere in terms of their monodromies in the following figure:

In the last entry we explained a relation between monodromies and contact Hamiltonians. Consider a contact form $dz-H(x,y,z)dx$ in $\mathbb{R}^3$, this is a quite general normal form (which we can obtain by trivializing along the y–lines of $\mathbb{D}^2(x,y)$). If we restrict to the sphere $x^2+y^2+z^2=R^2$ we can write H in terms of $H=H(x,z)$ at points where the implicit function theorem works. Then the characteristic foliation is nothing else than the solution of the time–dependent (x is the time) differential equation $dz-Hdx=0$ on the interval I=[-1,1] given by the coordinate z. Hence the contact Hamiltonian yields the ODE  to which the monodromy is a solution.

Tool: How do we obtain a piece of a disk in standard contact $(\mathbb{R}^3,dz-ydx)$ with a given characteristic foliation ?

Answer: Consider a disk in the (z,x)–plane and a function H(z,x). The standard contact structure $dz-ydx$ restricts to the graph of H in $\mathbb{R}^2(z,x)\times\mathbb{R}(y)$ as $dz-ydx|_{\{y=H\}}=dz-Hdx$.

For instance, let us consider the following function H(z) for z=[-1,1]:

This function H can be considered as a function on the polydisk (x,z) which is represented by the lower square in the third figure (the whole figure is PL immersed in the standard contact 3–space). Its image is the bumped square drawn above it, and we may consider the PL sphere obtained by adding the vertical annulus connecting the domain and the graph. The characteristic foliation on the bottom piece is by the horizontal z–lines, on the annulus the foliation is vertical and on the top piece the foliation is drawn on the left. Note that the characteristic foliation in this immersed PL sphere has a closed leaf (in red) coming from the fixed point (or zero, if we look at it horizontally) of H.

Let us briefly focus on the existence of a contact structure in a region bounded by a domain and a graph as in the previous paragraph.

Exercise: Does there exist a contact structure filling the following pink region ?

(The contact structure should restrict to the germs (in purple) already defined on the boundary.)

Answer: Yes. This is already embedded in $\mathbb{R}^3$, hence we just need to restrict the ambient contact structure. (This should be compared with the previous post where this question was also formulated and answered in terms of the positivity of the function H).

The second exercise we need to solve is as simple as the previous one, let us however draw the figures in order to keep them in mind.

Annulus Problem (weak): Does there exist a contact structure in the (yellow) annulus ?

The contact structure should also restrict to the germs (in purple and green) already defined on the boundary.

Answer: Yes, again this is already embedded in standard contact Euclidean space. This is yet another instance of the relevance of order. If one Hamiltonian is less than another one, then we can obtain a contact structure on the annulus.

This will be formalized in subsequent posts using the notion of domination of Hamiltonians and their corresponding contact shells. We shall not use this language right now.

We are now going to prove Eliashberg’s existence theorem in dimension 3 from the contact Hamiltonian perspective (i.e. from the monodromy viewpoint). The fundamental fact is that we only need to extend contact structures up to contactomorphism and this is translated to the fact the Hamiltonians can be conjugated.

Annulus Problem (strong): Does there exist a contact structure on the following region ?

Answer: If we are able to conjugate the bottom Hamiltonian (in green) strictly below to the upper one (in purple), then we can use the contact structure of the embedded annulus (weak version of the annulus problem). Hence, it all reduces to the order (or rather, the lack thereof).

Fundamental Fact: There exists a conjugation of the bottom Hamiltonian such that it is strictly less than the upper one. In general, given two Hamiltonian with fixed points which are positive at the endpoints of the interval, there exists a conjugation bringing one of them below the other.

(This is an exercise with functions in one variable, in higher dimensions this is no longer simple and this is precisely the main point that M.S. Borman, Y. Eliashberg and E. Murphy have understood).

Let us prove Eliashberg’s 3–dimensional existence theorem, we focus on the extension part (part 2 according to the post three entries ago).

Extension Problem (Version I): Suppose that there exists a contact structure on the complement of a ball $B^3$ in a 3–fold (which is given by Gromov’s h–principle, see previous posts) and that the characteristic foliation on the boundary $S_h^2$ has monodromy with fixed points (h stands for hole). Can we extend the contact structure ?

Suppose that there exists a sphere $S_{ot}^2$ somewhere inside the manifold with an overtwisted monodromy (in blue, see above) in its characteristic foliation. Consider the annulus $A_{ot}=S_{ot}^2\times(-\tau,\tau)$. Use the south poles of $S_{ot}^2\times\tau$ and $S_h^2$ to connect both and obtain an annulus $A$ such that the monodromy in the exterior boundary sphere is the concatenation of the contactomorphisms of the intervals (green#pink). Hopefully this figure helps:

The monodromies of the foliations in the two spheres bounding the annulus $A_{ot}$ are drawn in pink (exterior boundary) and blue (interior boundary). The monodromy in green is that of $S_h^2$. Connecting the spheres $S_h^2$ and $S_{ot}^2\times\{\tau\}$ yields a sphere with the monodromy green#pink (the transition area is purple, this has some relevance but it is not essential). Consider the annulus A bounded by $S_h^2\#(S_{ot}^2\times\{\tau\})$ and $S_{ot}^2\times\{-\tau\}$. We have reduced the problem of extending the contact structure to the interior of $S_h^2$ to the problem of extending the contact structure in the annulus A. In the exterior boundary of A the characteristic foliation is green#pink and on the interior is red (which comes from moving blue).

Extension Problem (Version II): Does there exists a conjugation such that (the graph of) any contactomorphism can be conjugated to lie beneath any other (graph) ?

Answer:  No. Fixed Points are an obstruction. However, if we restrict ourselves to the same question in the class of contactomorphisms with fixed points the answer is yes. This is exactly the Fundamental Fact stated above.

How do we conclude the proof ? Conjugate the red Hamiltonian to lie beneath the green#pink Hamiltonian and use the contact structure in the resulting annulus (as embedded in standard contact space). Assuming Gromov’s h–principle and the technical work in order for the foliation to be controlled, this argument concludes the theorem.

(We have disregarded some details, but the idea of the argument is the one described above. Observe that the parametric version of the existence problem in dimension 3 is quite immediate from the Hamiltonian perspective.)

Note also that we do not need the whole sphere $S^2_{ot}$: in order to use the argument with the Hamiltonians we can cut the North pole of $S^2_{ot}$ and retain just the remaining disk, which is an overtwisted disk.

There is a substantial advantage in this proof of the 3–dimenisonal case: we can define an overtwisted disk $\mathbb{D}^{2n}$ in higher dimensions 2n+1 to be the object that appears when using the contact Hamiltonian on a simplex $\Delta^{2n-1}$ given by

(We will give precise definitions in the subsequent entries.)

The strategy of the argument works in higher dimensions if we can prove the Fundamental Fact stating that there is enough disorder for contact Hamiltonians. In the next entries we will focus on this crucial step in higher dimensions and conclude existence.

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## Contact Hamiltonians (Introduction)

This entry is part of the series of posts on the recent work of  M. S. Borman, Y. Eliashberg and E. Murphy on the existence and classification of overtwisted contact structures in all dimensions. In the previous two entries the construction in the 3–dimensional case and Gromov’s h–principle for the open case have been explained.

The essential fact in Eliashberg’s 3–dimensional argument (this is part 2, two entries ago) is the control on the characteristic foliation: the extension problem is reduced to being able to fill a 2–sphere with a contact ball given a particular characteristic foliation on the boundary. This can be done explicitly by deforming the local model provided by the standard overtwisted contact ball in order to have the given characteristic foliation on the boundary. The construction in higher dimensions is not quite the same and it relies on the use of contact Hamiltonians, hence this and subsequent entries. This first introductory entry should help the reader to follow the next entries, each entry should however be readable on its own.

Consider a contact manifold of dimension 2n+1. The information of a contact structure is contained in a 1–form (locally this is the data of 2n+1 functions, plus another one as a conformal factor, satisfying 1 non–degeneracy equation). This data can be drastically reduced when restricted to simple topological subsets if we choose appropriate coordinates: for instance, the Darboux theorem tells us that the normal form of a contact 1–form around a point is $\alpha=dz-\sum_{n=1}^{2n}y_idx_i$. It will be helpful for the reader to understand the geometric proof of the Darboux theorem, see Theorem 2 in Page 5 of Topological methods in 3-dimensional contact geometry. The strategy is finding a suitable flow to obtain the desired coordinates.

Suppose that we have a contact manifold $(M,\alpha_0)$ and a codimension–1 distribution $\xi=\ker(\alpha)$ on the manifold $M\times\mathbb{D}^2(r,\theta)$ such that it restricts to $\ker(\alpha_0)$ on each slice $M\times\{pt.\}$. In these hypotheses:

Lemma: There exists the following normal form for the 1–form $\alpha$, we can write $\alpha=\alpha_0+H(p,r,\theta)d\theta$ for some smooth function $H:M\times\mathbb{D}^2\longrightarrow\mathbb{R}$.

Proof: Consider the product manifold $M\times\mathbb{D}^2$ as a trivial fibre bundle over the disk $\mathbb{D}^2$. The data in the hypothesis gives a connection in this bundle whose parallel transport is by contactomorphisms, it is defined as the skew–orthogonal complement of the symplectic subspace $\ker(\alpha_0)$ in the bundle $(\xi,d\alpha)$ with respect to the 2–form $d\alpha$ (which is not necessarily symplectic). We can then consider the radial vector field in the base $\mathbb{D}^2$ and lift it to the total space $M\times\mathbb{D}^2$. The pull–back of the contact form by this flow is (conformally) of the form $\alpha_0+H(p,r,\theta)d\theta$ for some function $H:M\times\mathbb{D}^2\longrightarrow\mathbb{R}$. The reason being that the radial factor $dr$ cannot appear because in the trivializing coordinates (provided by the flow of the lift), the lift of the radial vector field belongs to the distribution. $\hfill\Box$

There are a couple of technical details regarding the existence of the flow, which can be translated into the size of the base disk. Let us not focus on that. Thanks to the Lemma we have the following reduction of the extension problem.

Suppose that on a given almost contact (2n+1)–fold V we have a contact structure on all of V except on a neighbourhood $Op(M)\cong M\times\mathbb{D}^2$ of a codimension–2 submanifold M with trivial normal bundle. If the almost contact structure $\xi$ satisfies the hypothesis for the Lemma in $Op(M)$, then the extension problem for the contact structure is reduced to:

Problem: Given a germ of a contact structure on $M\times S^1$ described by a function $H:M\times S^1\times(1-\varepsilon,1+\varepsilon)\longrightarrow\mathbb{R}$, does there exist a contact structure on $M\times\mathbb{D}^2$ such that it restricts to the given germ on $M\times S^1$ ?

There are two remarks at this point. First, the meaning of the function H is really geometric. It describes the angle of rotation of the contact structure in the radial direction, in particular the condition for $\alpha_0+Hd\theta$ to be a contact structure on $M\times S^1\times(1-\varepsilon,1+\varepsilon)$ reads $\partial_r H>0$ (this is often stated as the contact structure has to rotate). Second, the extension does not need to be of the form $\alpha_0+Hd\theta$, we just need a contact structure on $M\times\mathbb{D}^2$.

Example 1 (Tight): Consider $(M,\alpha_0)=((-1,1),dz)$ and the function $H(p,r,\theta)=r^2$. The contact form is $\alpha=dz+r^2d\theta$ and since the function H verifies the contact condition on $B^3=(-1,1)\times\mathbb{D}^2$ this defines a contact structure on $B^3$. This is the standard contact structure on the ball.

Example 2 (Overtwisted): Consider $(M,\alpha_0)=((-1,1),dz)$ and $H(p,r,\theta)=r\cdot tg(r)$. The contact form is then $\alpha=dz+rtg(r)d\theta$, which should be read as $\alpha=cos(r)dz+rsin(r)d\theta$. This is the standard overtwisted contact structure on the ball $B^3=(-1,1)\times\mathbb{D}^2$ if the radius of the disk is larger than $\pi$.

This second example has the following very nice feature: the function $H(r)=rtg(r)$ is negative at r=2. This provides a solution to the problem of extending a germ in $(-1,1)\times S^1\times\{2\}$ to the interior $(-1,1)\times\mathbb{D}^2$ if this germ is everywhere negative. Although a priori it seems non–sense to go from 0 to a negative value growing (in order to preserve the contact condition) this can be done by inserting a pole, i.e. going to infinity (and then continuing from minus infinity). This phenomenon underlies many h–principles, try to solve for instance Section 4.1.1 from Chapter 4 in Eliashberg–Mishachev book.

The functions H appearing in the above constructions are called contact Hamiltonians.

Problem (Easy Case): Given a germ of a contact structure on $M\times S^1$ described by a positive function $H:M\times S^1\times(1-\varepsilon,1+\varepsilon)\longrightarrow\mathbb{R}$, does there exist a contact structure on $M\times\mathbb{D}^2$ such that it restricts to the given germ on $M\times S^1$ ?

Answer: Yes. In this case the extension can be a contact structure of the form $\alpha_0+\widetilde{H}d\theta$ where $\widetilde{H}$ extends H and is such that $\partial_r\widetilde{H}>0$. Certainly, we just need to construct a function which at the origin looks like $\widetilde{H}=r^2$ and then it grows in the radial direction until we reach the value given by H on the boundary $M\times S^1$. The existence of such a function is immediate. $\hfill\Box$

The difficult case is that of a germ of a contact structure defined by a Hamiltonian which is negative in some points and positive in others (the presence of such negativity requires overtwistedness). The situation described above is quite hard because we may not even understand the (contact) topology of M. The first step is to focus on $M=\Delta^{2n-1}$ a (2n-1)-ball, or star–shaped domain, in $\mathbb{R}^{2n-1}$.

In the next entry, Contact Hamiltonians (Part I) we will continue to use contact Hamiltonians and relate them to Eliashberg’s 3–dimensional argument using the characteristic foliation. The essential word will be monodromy.

In the context above, monodromy arises as follows: consider the contact germ on $M\times S^1(\theta)$ and lift the vector field $\partial_\theta$ to the connection defined before. Its flow at time equal to the length of the circle (say 1) defines a contactomorphism of the fibre $M\times\{0\}$. This is the monodromy contactomorphism.

There is however another way to obtain a contactomorphism of $(M,\alpha_0)$ if we have a  function $H:M\times S^1\longrightarrow\mathbb{R}$ (referred to as a time–dependent contact Hamiltonian). Indeed, compute the Hamiltonian contact vector field X associated to H, which is the unique solution of

$\alpha_0(X_\theta)=H_\theta$ and $d\alpha_0(X_\theta,\cdot)=-dH_\theta+dH(R_{\alpha_0})\cdot\alpha_0$

where $R_{\alpha_0}$ is the Reeb vector field. Then the time–1 flow of the Hamiltonian vector field is a contactomorphism of M. This contactomorphism is said to be generated by the contact Hamiltonian H.

Lemma: Given the contact germ $\alpha_0+Hd\theta$ on $M\times S^1$, the monodromy contactomorphism coincides with the contactomorphism generated by H.

The proof of this lemma is a nice exercise on linear algebra using the defining equation of the connection. This setup can be explicitly studied in 3–dimensions where the monodromies (and the functions H) can be drawn and they correspond to ODEs in the plane. In the next post we will proof Eliashberg’s theorem in dimension 3 from the contact Hamiltonian perspective.