The energy functional on the Virasoro-Bott group with the \(L ^{2}\)-metric has no local minima
type: Item
instance of: scholarly article
title: The energy functional on the Virasoro-Bott group with the \(L ^{2}\)-metric has no local minima
zbMATH Open document ID: 1267.35190
author: Martins Bruveris
published in: Annals of Global Analysis and Geometry
publication date: 2013-05-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.4326
review text: The geodesic equation for the right invariant \(L ^{2}\)-metric on each Virasoro-Bott group is equivalent to the Korteweg-de Vries equation. The main purpose of this paper is to prove that the energy functionals (when restricted to paths with fixed endpoints) for the Virasoro-Bott group and for the diffeomorphism group of \(\mathbb{R}\) have no local minima. In particular, the author gets the same result for the length functional. Given any curve there is a close by curve with the same endpoints that uses less energy. Hence no geodesic is locally energy or length minimizing.
reviewed by: Ahmed Lesfari
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 58D05
Mathematics Subject Classification ID: 58D15
Mathematics Subject Classification ID: 58E12
zbMATH DE Number: 6162251
zbMATH Keywords: diffeomorphism group
zbMATH Keywords: Virasoro group
zbMATH Keywords: geodesic distance
MaRDI profile type: Publication
OpenAlex ID: W3105500311
arXiv ID: 1106.4326
cites work: Sur la géométrie différentielle des groupes de Lie de dimension infinite et ses applications à l'hydrodynamique des fluides parfaits
cites work: Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation
cites work: On geodesic exponential maps of the Virasoro group
cites work: Geodesic flow on the diffeomorphism group of the circle
cites work: Groups of diffeomorphisms and the motion of an incompressible fluid
cites work: Q3052807
cites work: The Geometry of Infinite-Dimensional Groups
cites work: Q4360299
cites work: Q3596055
cites work: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms
cites work: Q4214030
cites work: Conjugate points in the Bott-Virasoro group and the KdV equation
cites work: Korteweg - de Vries suoerequation as an Euler equation
cites work: The geometry of the KdV equation
DOI: 10.1007/S10455-012-9350-0
description: scientific article