Intersection theory of punctured pseudoholomorphic curves

Abstract

We study the intersection theory of punctured pseudoholomorphic curves in 4-dimensional symplectic cobordisms. Using the asymptotic results of the author [22], we first study the local intersection properties of such curves at the punctures. We then use this to develop topological controls on the intersection number of two curves. We also prove an adjunction formula which gives a topological condition that will guarantee a curve in a given homotopy class is embedded, extending previous work of Hutchings [14].We then turn our attention to curves in the symplectization R M of a 3-manifold M admitting a stable Hamiltonian structure. We investigate controls on intersections of the projections of curves to the 3-manifold and we present conditions that will guarantee the projection of a curve to the 3-manifold is an embedding.Finally we consider an application concerning pseudoholomorphic curves in manifolds admitting a certain class of holomorphic open book decomposition and an application concerning the existence of generalized pseudoholomorphic curves, as introduced by Hofer [7].

The Weinstein conjecture for stable Hamiltonian structures

“…Using arguments going back to Hofer, Wysocki and Zehnder [10; 11] and developed further by Siefring [21] and Wendl [26], one can show that under certain circumstances the projections of these curves to Y are also embeddings, and the corresponding moduli spaces of holomorphic curves locally give a foliation of Y . In particular, we will need the following proposition.…”

On nonseparating contact hypersurfaces in symplectic 4–manifolds

“…One then has to show that if u 0 intersects any trivial cylinder R 0 over an orbit 0 in N , then it also has an "asymptotic intersection" with R z , which cannot be true if windˆ.e z / D 0. This follows easily from the intersection theory of punctured holomorphic curves; see Siefring [31] and also Siefring-Wendl [32] for details. For the case where z is Morse-Bott, the fact that u M intersects the Morse-Bott submanifold means 0 D windˆ.e z / < windˆ.0/ due to Lemma 4.3.…”

“…Proof Since u 0 has only simply covered Reeb orbits and all of them are distinct, it satisfies the following somewhat simplified version of the adjunction formula from Siefring [31] and Siefring-Wendl [32]:…”

Sistemas de seções transversais próximos a níveis críticos de sistemas Hamiltonianos em $\mathbb{R}^4$

“…Neste apêndice, vamos usar a teoria de interseção de curvas pseudo-holomorfas, desenvolvida por Richard Siefring em [54], para garantir a unicidade dos planos rígidos u 1,E e u 2,E e dos cilindros rígidos v E e v ′ E na 3-esfera W E . Além disso, vamos provar algumas propriedades de interseção para semi-cilindrosJ E -holomorfos assintóticosàórbita periódica hiperbólica P 2,E , utilizando para isso a fórmula obtida por R. Siefring em [53], que descreve o comportamento assintótico da diferença de dois semi-cilindros pseudo-holomorfos distintos convergindo exponencialmente para uma mesmaórbita periódica do fluxo de Reeb.…”

“…Em seguida, obtemos a seguinte propriedade de enlaçamento deórbitas periódicas em S E , para E > 0 pequeno: se Qé umaórbita periódica de λ E contida em S E \ (∂S E ∪ P 3,E ), cuja ação não excede a ação de P 3,E , então Q deve estar enlaçada com P 3,E . No Apêndice B, usamos a teoria de interseção de curvas pseudo-holomorfas, desenvolvida por R. Siefring em [54], para obtermos resultados de unicidade para planos e cilindros rígidos assintóticosàórbita hiperbólica P 2,E . Além disso, provamos algumas propriedades de interseção para semi-cilindrosJ E -holomorfos que são assintóticos a algum recobrimento de P 2,E e se projetam em W E dentro da 3-bola aberta S E \ ∂S E .…”