Journal Home Page

Cumulative Index

List of all Volumes

Complete Contents
of this Volume

Previous Article

Next Article

Journal of Convex Analysis 15 (2008), No. 4, 891--903
Copyright Heldermann Verlag 2008

Heat Flow for Closed Geodesics on Finsler Manifolds

Mamadou Sango
Dept. of Mathematics, University of Pretoria, Pretoria 0002, South Africa


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.

[ Fulltext-pdf  (146  KB)] for subscribers only.