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 . 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 , where c is a small positive constant, in the overtwisted contact manifold . 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 in , this is a quite general normal form (which we can obtain by trivializing along the y–lines of ). If we restrict to the sphere we can write H in terms of 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 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 with a given characteristic foliation ?
Answer: Consider a disk in the (z,x)–plane and a function H(z,x). The standard contact structure restricts to the graph of H in as .
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 , 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 in a 3–fold (which is given by Gromov’s h–principle, see previous posts) and that the characteristic foliation on the boundary has monodromy with fixed points (h stands for hole). Can we extend the contact structure ?
Suppose that there exists a sphere somewhere inside the manifold with an overtwisted monodromy (in blue, see above) in its characteristic foliation. Consider the annulus . Use the south poles of and to connect both and obtain an annulus 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 are drawn in pink (exterior boundary) and blue (interior boundary). The monodromy in green is that of . Connecting the spheres and 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 and . We have reduced the problem of extending the contact structure to the interior of 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 : in order to use the argument with the Hamiltonians we can cut the North pole of 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 in higher dimensions 2n+1 to be the object that appears when using the contact Hamiltonian on a simplex 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.