
Journal of Convex Analysis 15 (2008), No. 4, 891903 Copyright Heldermann Verlag 2008 Heat Flow for Closed Geodesics on Finsler Manifolds Mamadou Sango Dept. of Mathematics, University of Pretoria, Pretoria 0002, South Africa mamadou.sango@up.ac.za [Abstractpdf] We use the celebrated heat flow method of Eells and Sampson to the question of deformation of a smooth loop $M\in \mathbf{R}^{2}$ on a Finsler manifold $\left( N,h\right) $\ to a closed geodesic in $N$. This leads to the investigation of the corresponding heat equation which is the parabolic initial value problem \begin{eqnarray*} \frac{\partial u^{i}}{\partial t}\frac{\partial ^{2}u^{i}}{\partial x^{2}} &=&\Gamma _{hk}^{i}\left( u,\frac{\partial u}{\partial x}\right) \frac{% \partial u^{h}}{\partial x}\frac{\partial u^{k}}{\partial x}\mbox{ in }% M\times \lbrack 0,T), \\ u\left( x,0\right) &=&f\left( x\right) ;\ i=1,...,n. \end{eqnarray*}% The existence of a global in time solution $u\left( x,t\right)$ and its subsequent convergence to a closed geodesic $u_{\infty} \colon M\rightarrow N$ as $t\rightarrow \infty$, are dealt with. Appropriate concepts arising from the Finslerian nature of the problem are introduced. [ Fulltextpdf (146 KB)] for subscribers only. 