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 with a 2-dimensional disk , viewed as a subset of . The contact Hamiltonian is a function
Using the standard contact structure on , recall that an extension of this function defines an almost contact structure on which is genuinely contact wherever (compute ). Using the conventions from the BEM paper, we will use the coordinate . If is everwhere positive, we can realize this contact structure near the boundary of the following embedded subset of the standard contact
If is negative anywhere, then we need to look at a modified version. We can still encode the shape of by shifting everything up by a sufficiently large constant so that is positive. Then define
In order to have the contact form encodes the contact Hamiltonian near the boundary, we want to shift the contact form from to near the boundary. However, because the polar coordinates degenerate near , in a neighborhood of , we need to keep the form standard: . Define a family of functions to interpolate between these two, and then define the almost contact structure on by the form . 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 to see that defines a genuine contact form exactly when . The boundary of the ball has two pieces: the piece where and the piece where . In a neighborhood of the former piece, so it has positive derivative, but on the latter piece we have to impose the condition directly that in an open neighborhood of points where .
One can show that different choices for which satisfy these conditions do not yield genuinely different almost contact forms 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 can be chosen to be a genuine contact structure only along slices where is positive. Remember that says how much the almost contact planes are rotating in the radial direction, and if this means the twisting has stopped. If is negative then since and near 0, 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 we need near points where , so we only consider such Hamiltonians.
Here is a 3-dimensional example. The arrows indicate the twisting of the almost contact planes defined by . 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 are indicated by the graphs above–they start having critical points when K fails to be positive.
If we have two contact Hamiltonians on and on such that and , then it is not hard to see that we can choose circle models for each such that embeds into and so that in a neighborhood of the entire region where . In other words, the almost contact structure is contact and twisting in the standard way along the radial direction on the region between and . In the terminology of the BEM paper, directly dominates . View of the extendability a contact structure from one contact germ defined by a contact Hamiltonian to another germ defined by , 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 and cannot be directly compared (i.e. at some points but at others ), then there may be a different contact Hamiltonian which corresponds to the same contact germ in different coordinates such that can be compared to . This will be the subject of the rest of this post.
Contactomorphisms and conjugating the Hamiltonian
Given a contactomorphism on the domain , we want to construct an induced contactomorphism on . Because contactomorphisms only preserve the contact planes, and not the contact form, a contactomorphism satisfies where is a positive real valued function on . Because the pull-back rescales , we need to rescale the Hamiltonian on the image as well so that it fits together with to give a contact form for the same contact structure. Therefore define by .
natural induces an extension on defined by for any family of functions . If defines the contact structure on the image then
Therefore the function defining the contact Hamiltonian on the image must satisfy .
Why did we include the function in the above definition of ? This is to allow us to reparameterize so that it satisfies the required conditions to define the circular model (should look like the identity near , and should look like the identity shifted by the constant near ). Before the contactomorphism, to define the circular model, you choose a constant and then is considered on the domain and is required to have certain behavior near the endpoints of this interval. After rescaling, we have a new Hamiltonian , so we pick a new constant so that . Then we consider on the interval and require it to have particular behavior near and . Because , modifying the functions allows us to make have the desired behavior near the end points of the interval so that can be used to define a circular model for .
The action of the contactomorphism on the contact Hamiltonian is referred to as conjugating the Hamiltonian for the following reason. If the contact Hamiltonian is generated by a contact isotopy in the sense that , then you can compute that .
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 be the region where the contact Hamiltonian is defined and let be a subset containing the piece where is negative. Construct a contactomorphism which shrinks into itself a lot, but fixes the points of sufficiently away from . (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 .) Because is being shrunk, the rescaling function for the contact form defined by is a positive function with very tiny values close to 0, for . The more is shrunk, the tinier the values of . The corresponding Hamiltonian has values rescaled by . Therefore, by choosing a contactomorphism which shrinks enough, the values of can be made sufficiently small so that for .
In addition to the disorder lemma, we need two types of contactomorphisms of which rescale in certain directions. Choose cylindrical coordinates on and let so .
A transverse scaling contactomorphism is defined by a diffeomorphism by . You can check directly that this diffeomorphism is a contactomorphism which rescales the standard contact form by . Therefore this contactomorphism modifies a contact Hamiltonian by
The tagline for this type of contactomorphism is you can “trade long for thin”. By choosing a shrinking , you can shrink a domain which is long in the direction at the cost of shrinking the radial directions.
A twist embedding contactomorphism allows you to rescale the radial directions by at the cost of twisting in the angular directions by an amount that depends on (see section 8.2 of the BEM paper for the exact formulas). The points at radii where get sent to points where since . The rescaling factor for the contact form is , so the contact Hamiltonian is rescaled accordingly. For positive functions , setting gives taking the region where to the region where . 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 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.