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.

ContactHam

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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One response to “Contact Hamiltonians II

  1. Pingback: Overtwisted disks and filling holes | The Electric Handle Slide

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