Convergence towards an elastica in a Riemannian manifold
Given a unit speed, closed curve \(\gamma\) of length \(L\) on a compact, Riemannian manifold \(M\), its elastic energy (or total squared curvature) is given by \[ E(\gamma) = \int_{\gamma}\left\|\nabla_{\gamma'}\gamma'\right\|^{2} ds, \] where \(\nabla_{\gamma'}\) denotes covariant differentiation with respect to the tangent vector \(\gamma'\). A critical point of this functional on the space of closed, unit-speed curves of fixed length \(L\) on \(M\) is called an elastic curve or an elastica. Geodesics are clearly elastic curves; the existence of solutions to the corresponding Euler-Lagrange equations is not so clear when there are no geodesics of length \(L\). In [Topology 24, 75-88 (1985; Zbl 0561.53004)] and later in [Osaka J. Math. 29, No. 3, 539-543 (1992; Zbl 0777.53041)], the authors showed the existence of an elastica of length \(L\). The main result is as follows. Suppose that there are no closed geodesics of length \(L\) in \(M\). Given a closed, unit-speed curve of length \(L\) in \(M\), the corresponding parabolic initial value problem has a unique solution \(\gamma(\cdot,t)\) for all time, and the solution has a subsequence that converges to an elastica as \(t\to\infty\). In the case where \(M\) is real analytic, the solution converges to an elastica as \(t\to\infty\).
scientific article; zbMATH DE number 1515487