Harmonic flow of geometric structures
type: Item
instance of: scholarly article
title: Harmonic flow of geometric structures
arXiv ID: 1907.06072
author: Eric Loubeau
author: Henrique N. Sá Earp
published in: Annals of Global Analysis and Geometry
publication date: 2023-10-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1907.06072
review text: Let \(M\) be a closed \(m\)-manifold. A geometric structure \(\xi\) on \(M\) can be viewed as a section of the homogeneous fiber bundle \(\pi : N=P/H \rightarrow M\), which emerges by reduction of the (oriented) frame bundle \(P\rightarrow M\). Given a suitable fiber metric \(\eta\) on \(N\), a Dirichlet energy can be assigned to such sections \(\sigma:M\rightarrow N\) by \[ E(\sigma)=\frac{1}{2}\int_{M}|d^{V}\sigma|^{2}_{\eta} \ , \] where \(d^{V}\sigma\) is the projection of \(d\sigma\) onto the distribution \(\mathrm{Ker}(\pi_{*})\) tangent to the fibers of \(\pi:N\rightarrow M\). A geometric structure is said to be harmonic if \(\sigma\) is a critical point of \(E\), and torsion-free if \(d^{V}\sigma=0\). Starting from any smooth section \(\sigma_{0}\) of \(\pi\), the Dirichlet energy gives rise to a natural gradient flow, which is reminiscent of the classical heat flow for harmonic maps. The paper establishes a priori estimates, uniqueness, short-time and long-time existence, regularity and convergence of solutions of the gradient flow, assuming the following data: \begin{itemize} \item \(G\subset \mathrm{SO}(m)\) is a compact semi-simple Lie group; \item \(H\subset G\) is a normal reductive Lie subgroup; \item \(P\) is a principal \(G\)-bundle over a closed Riemannian manifold \((M, g)\); \item \(\eta\) is an \(H\)-invariant fiber metric on \(P\), constructed from a compatible bi-invariant metric on \(G\). \end{itemize}
reviewed by: Andrea Tamburelli
Mathematics Subject Classification ID: 53E99
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 7755906
zbMATH Keywords: homogeneous fiber bundles
zbMATH Keywords: Dirichlet energy
zbMATH Keywords: gradient flow
zbMATH Keywords: harmonic flows
MaRDI profile type: Publication
OpenAlex ID: W2958168750
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DOI: 10.1007/S10455-023-09928-7
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description: scientific article; zbMATH DE number 7755906