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.