Invariant probability measures from pseudoholomorphic curves Ⅱ: Pseudoholomorphic curve constructions

Abstract

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.DEFINITION 1.2. The Hamiltonian vector field associated to a framed Hamiltonian structure (λ, ω) is the unique vector field X satisfying λ(X ) ≡ 1 and ω(X , −) ≡ 0.

Invariant probability measures from pseudoholomorphic curves Ⅰ

“…On the other hand, by combining our Theorem 1.6 and the existence theorem for pseudoholomorphic curves proved in the sequel [19], it is known that all such hypersurfaces admit explicit X -invariant probability measures that are not equal to the normalized volume measure. Any invariant probability measure has support equal to a non-empty, closed invariant subset.…”

“…The following proposition sums up this discussion. It follows from more general results proved in [19]. PROPOSITION 1.17 ([19,Proposition 1.7]).…”

“…It follows from more general results proved in [19]. PROPOSITION 1.17 ([19,Proposition 1.7]). Fix an integer G ≥ 0.…”

“…At first glance, this requirement seems to complicate matters significantly. However, there are many rich pseudoholomorphic curve-based invariants for symplectic manifolds, which we leverage in [19] to construct a large class of framed Hamiltonian manifolds satisfying the conditions of Theorem 1.6. More precisely, Hamiltonian hypersurfaces (Example 1.4) in closed symplectic manifolds with enough nonvanishing Gromov-Witten invariants, satisfy the conditions of Theorem 1.6.…”

“…This is explained in more detail below. Another application to the question of unique ergodicity in symplectic 4-manifolds, using Seiberg-Witten theory to construct ω-finite pseudoholomorphic curves, is discussed in [19,Proposition 1.7]. The proofs are elementary and we defer them to Section 6.…”