Harmonic flow of geometric structures
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}
scientific article; zbMATH DE number 7755906
scientific article