The exponential map of the group of area-preserving diffeomorphisms of a surface with boundary

TL;DR

This paper proves that the exponential map of the right-invariant L^2 metric on the group of volume-preserving diffeomorphisms of a 2D surface with boundary is a nonlinear Fredholm map of index zero, advancing understanding of geometric analysis on such groups.

Contribution

It establishes the Fredholm property of the exponential map for volume-preserving diffeomorphisms on surfaces with boundary, a novel result in geometric analysis.

Findings

Exponential map is a nonlinear Fredholm map of index zero.

Advances understanding of the geometry of diffeomorphism groups.

Provides tools for further analysis of fluid dynamics on surfaces with boundary.

Abstract

We prove that the Riemannian exponential map of the right-invariant $L^2$ metric on the group of volume-preserving diffeomorphisms of a two-dimensional manifold with a nonempty boundary is a nonlinear Fredholm map of index zero.