Heat Flow for Closed Geodesics on Finsler Manifolds

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 $$\begin{aligned}\frac{\partial u^{i}}{\partial t}-\frac{\partial ^{2}u^{i}}{\partial x^{2}} &=\Gamma _{hk}^{i}\left( u,\frac{\partial u}{\partial x}\right) \frac{\partial u^{h}}{\partial x}\frac{\partial u^{k}}{\partial x}\text{ in }M\times \lbrack 0,T), \\ u\left( x,0\right) &=f\left( x\right);\ i=1,...,n.\end{aligned}$$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