Invariant probability measures from pseudoholomorphic curves II: Pseudoholomorphic curve constructions

TL;DR

This paper extends previous work by using advanced symplectic geometry techniques, including Gromov-Witten and Seiberg-Witten theories, to construct pseudoholomorphic curves that generate invariant measures for volume-preserving flows on odd-dimensional manifolds.

Contribution

It introduces new analytical tools and methods to establish the existence of pseudoholomorphic curves, broadening the class of systems where invariant measures can be constructed.

Findings

Constructed large classes of pseudoholomorphic curves using Gromov-Witten and Seiberg-Witten theories.

Applied neck stretching and new analytical techniques to prove existence of curves.

Extended the framework for invariant measure construction to more complex flows.

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.