Convergence to a geodesic
The equation \((d/dt) (c_ t)= D_{\dot c_ t}\dot c_ t\) (ES), a semi-linear heat equation, was studied by Eels and Sampson [\textit{J. Eels jun.} and \textit{J. H. Sampson}, Am. J. Math. 86, 109-160 (1964; Zbl 0122.401)]. They proved that for a compact Riemannian manifold \((M,g)\), with nonpositive sectional curvature, a solution \(c_ t\) exists for all time and a subsequence \(c_{t_ i}\) converges to a geodesic. The author shows that under a natural condition (\(C^ \infty\) initial data) the above curvature restriction is not necessary and the equation (ES) has a unique solution \(c_ t\); moreover, if \((M,g)\) is real analytic then the solution converges to a geodesic.
scientific article; zbMATH DE number 579213