# Monthly Archives: July 2014

## 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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