[Paper Review] How to recognise a 4-ball when you see one

Hansjörg Geiges, Kai Zehmisch|arXiv (Cornell University)|Apr 8, 2011

Geometric and Algebraic TopologyMathematics43 references20 citations

TL;DR

This paper establishes a dichotomy for 4-dimensional symplectic cobordisms with a convex boundary component diffeomorphic to the standard contact 3-sphere: either the cobordism is diffeomorphic to the 4-ball, or it contains a periodic Reeb orbit of quantifiably short period in its concave boundary. The result is derived via holomorphic disc filling techniques and yields unified proofs of classical results in symplectic and contact topology, including non-squeezing, symplectic fillability, and non-existence of exact Lagrangian surfaces in R^4.

ABSTRACT

We apply the method of filling with holomorphic discs to a 4-dimensional symplectic cobordism with the standard contact 3-sphere as a convex boundary component. We establish the following dichotomy: either the cobordism is diffeomorphic to a ball, or there is a periodic Reeb orbit of quantifiably short period in the concave boundary of the cobordism. This allows us to give a unified treatment of various results concerning Reeb dynamics on contact 3-manifolds, symplectic fillability, the topology of symplectic cobordisms, symplectic non-squeezing, and the non-existence of exact Lagrangian surfaces in standard symplectic 4-space.

Motivation & Objective

• To establish a unified criterion for recognizing when a symplectic 4-manifold with a convex boundary component is diffeomorphic to the 4-ball.
• To extend Eliashberg's holomorphic disc filling method to general symplectic cobordisms with a standard contact S^3 boundary.
• To derive quantitative estimates on the periods of periodic Reeb orbits in the concave boundary when the cobordism is not a 4-ball.
• To unify and reprove classical results in 4-dimensional symplectic and contact topology using a single, coherent framework.

Proposed method

• Apply the method of filling a symplectic cobordism with holomorphic discs adapted to a contactomorphism of the boundary S^3.
• Use compactness and bubbling analysis of the moduli space of holomorphic discs to detect non-compactness, which implies the existence of periodic Reeb orbits.
• Employ energy estimates on holomorphic discs to derive lower bounds on the periods of such Reeb orbits.
• Introduce a symplectic capacity via the minimal period of Reeb orbits on contact type hypersurfaces, which recovers Gromov's non-squeezing theorem.
• Utilize a homotopical boundary condition and evaluation map to show that the truncated moduli space is diffeomorphic to S^1 × D^1, enabling topological classification.
• Apply the framework to weak fillings of S^2 × S^1, showing that such fillings are classified up to diffeomorphism.

Experimental results

Research questions

• RQ1 Under what conditions is a symplectic cobordism with a convex standard contact S^3 boundary component diffeomorphic to the 4-ball?
• RQ2 What topological or dynamical obstruction arises when the moduli space of holomorphic discs is non-compact?
• RQ3 Can the minimal period of a periodic Reeb orbit in the concave boundary of a symplectic cobordism be quantitatively bounded in terms of holomorphic disc energy?
• RQ4 How does the holomorphic disc filling method unify existing results in symplectic and contact topology, such as non-squeezing and non-existence of exact Lagrangian surfaces?
• RQ5 What is the value of the symplectic capacity defined via Reeb orbit periods in standard examples like the 4-ball and the cylinder over the 2-ball?

Key findings

• A symplectic cobordism with a convex boundary component diffeomorphic to the standard contact 3-sphere is either diffeomorphic to the 4-ball or contains a periodic Reeb orbit in its concave boundary with period bounded above by a quantity derived from holomorphic disc energy.
• The minimal period of such a Reeb orbit is at most π in the case of the 4-ball, and this bound is sharp, corresponding to the contractible Reeb orbit γ(t) = (cos 2t, sin 2t, 0, θ₀) for t ∈ [0, π].
• The symplectic capacity c(V, ω) for a minimal strong symplectic filling of (S^2 × S^1, α_st) is exactly π, and c₀(V, λ) = π for exact fillings.
• The method provides a unified proof of Gromov’s non-squeezing theorem via the symplectic capacity construction.
• The existence of a periodic Reeb orbit in the concave boundary is guaranteed when the moduli space of holomorphic discs is non-compact due to bubbling or breaking.
• The framework extends to classify weak symplectic fillings of S^2 × S^1 up to diffeomorphism, analogous to the S^3 case.