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.

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