In the previous work, we introduced a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds with pseudoholomorphic curve techniques from symplectic geometry. The technique requires existence of certain pseudoholomorphic curves satisfying some weak assumptions. In this work, we appeal to Gromov–Witten theory and Seiberg–Witten theory to construct large classes of examples where these pseudoholomorphic curves exist. Our argument uses neck stretching along with new analytical tools from Fish–Hofer's work on feral pseudoholomorphic curves. 

Invariant probability measures from pseudoholomorphic curves Ⅱ: Pseudoholomorphic curve constructions

We introduce a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds using pseudoholomorphic curve techniques from symplectic geometry. These flows include any non-singular volume preserving flow in dimension three, and autonomous Hamiltonian flows on closed, regular energy levels in symplectic manifolds of any dimension. As an application, we use our method to prove the existence of obstructions to unique ergodicity for this class of flows, generalizing results of Taubes and Ginzburg–Niche. 

Invariant probability measures from pseudoholomorphic curves Ⅰ

We introduce a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds using pseudoholomorphic curve techniques from symplectic geometry. These flows include any non-singular volume preserving flow in dimension three, and autonomous Hamiltonian flows on closed, regular energy levels in symplectic manifolds of any dimension. As an application, we use our method to prove the existence of obstructions to unique ergodicity for this class of flows, generalizing results of Taubes and Ginzburg-Niche.

Invariant probability measures from pseudoholomorphic curves I.

In the previous work, we introduced a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds with pseudoholomorphic curve techniques from symplectic geometry. The technique requires existence of certain pseudoholomorphic curves satisfying some weak assumptions. In this work, we appeal to Gromov-Witten theory and Seiberg-Witten theory to construct large classes of examples where these pseudoholomorphic curves exist. Our argument uses neck stretching along with new analytical tools from Fish-Hofer's work on feral pseudoholomorphic curves.

Invariant probability measures from pseudoholomorphic curves II: Pseudoholomorphic curve constructions

Geodesic and conformally Reeb vector fields on flat 3-manifolds

We prove a conjecture of Irie, stating that a strong version of the smooth closing lemma holds for Reeb flows on ellipsoids in any dimension. The proof involves analyzing a constrained cobordism map in contact homology, using holomorphic intersection theory.

Contact homology and the strong closing lemma for ellipsoids

Spectral invariants arising from twisted periodic Floer homology have recently played a key role in resolving various open problems in two-dimensional dynamics. We resolve in great generality a conjecture of Hutchings regarding the relationship between the asymptotics of the twisted PFH spectral invariants and the Calabi homomorphism for area-preserving disk maps. Our result, by an argument similar to that of Irie in the setting of Reeb flows, allows us to prove the smooth closing lemma for area-preserving diffeomorphisms of a closed surface.

The smooth closing lemma for area-preserving surface diffeomorphisms.

Right-handed and Reeb vector fields are two rich classes of vector fields on closed, oriented three-manifolds. Prior work of Dehornoy and Florio–Hryniewicz has produced many examples of Reeb vector fields which are right-handed. We prove a result in the other direction. We show that the closed two-form associated with a volume-preserving right-handed vector field is contact-type. This implies that any volume-preserving right-handed vector field is equal to a Reeb vector field after multiplication by a positive smooth function. Combining our result with theorems of Ghys and Taubes shows that any volume-preserving right-handed vector field has a global surface of section. 

/pdf/volume-preserving-right-handed-vector-fields-are-conformally-1s0a6n7f.pdf

Volume-preserving right-handed vector fields are conformally Reeb

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