Metric completion of $Diff([0,1])$ with the $H1$ right-invariant metric

TL;DR

This paper characterizes the metric completion of the group of smooth increasing diffeomorphisms on the interval with an $H^1$ right-invariant metric, revealing it as a space of increasing maps with boundary conditions, and analyzes geodesic properties.

Contribution

It explicitly computes the metric completion of Diff([0,1]) with the $H^1$ metric and studies the relaxation of the geodesic problem, including the Eulerian and Lagrangian formulations.

Findings

The completion is the space of increasing maps with boundary conditions.

The lower-semicontinuous envelope of the length functional is characterized.

Smooth solutions of EPDiff are length minimizing for short times.

Abstract

We consider the group of smooth increasing diffeomorphisms Diff on the unit interval endowed with the right-invariant $H^1$ metric. We compute the metric completion of this space which appears to be the space of increasing maps of the unit interval with boundary conditions at $0$ and $1$. We compute the lower-semicontinuous envelope associated with the length minimizing geodesic variational problem. We discuss the Eulerian and Lagrangian formulation of this relaxation and we show that smooth solutions of the EPDiff equation are length minimizing for short times.