On analysis of the exponential map of volume-preserving diffeomorphism group on closed orientable surfaces through the vorticity

TL;DR

This paper investigates the exponential map of volume-preserving diffeomorphisms on closed surfaces using vorticity, providing a fluid dynamics proof of its Fredholm properties and extending rigidity results to all orientable closed surfaces.

Contribution

It offers an alternative proof of the exponential map's Fredholm nature and extends Shnirelman's rigidity result from flat tori to all orientable closed surfaces.

Findings

Exponential map is a nonlinear Fredholm mapping of index zero.

The exponential map is Fredholm quasiregular on closed orientable surfaces.

Extension of rigidity results from flat tori to arbitrary orientable closed surfaces.

Abstract

We study the exponential map of group of volume-preserving diffeomorphisms on closed orientable surfaces via the vorticity formulation of the incompressible Euler equation. We present an alternative, fluid dynamical proof of the theorem of Ebin--Misio\l{}ek--Preston: the exponential is a nonlinear Fredholm mapping of index zero. We extend Shnirelman's rigidity result for the exponential map from 2-dimensional flat torus to arbitrary orientable closed surfaces. That is, we prove that the exponential map is Fredholm quasiregular.