Abstract

We use the celebrated heat flow method of Eells and Sampson to the question of deformation of a smooth loop M∈R2M\in \mathbf{R}^{2} on a Finsler manifold (N,h)\left( N,h\right) to a closed geodesic in NN. This leads to the investigation of the corresponding heat equation which is the parabolic initial value problem ∂ui∂t−∂2ui∂x2=Γhki(u,∂u∂x)∂uh∂x∂uk∂x in M×[0,T),u(x,0)=f(x); i=1,...,n.\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(x,t)u\left( x,t\right) and its subsequent convergence to a closed geodesic u∞ ⁣:M→Nu_{\infty} \colon M\rightarrow N as t→∞t\rightarrow \infty, are dealt with. Appropriate concepts arising from the Finslerian nature of the problem are introduced

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Suggested citation

M. Sango. “Heat Flow for Closed Geodesics on Finsler Manifolds.” Journal of Convex Analysis 15 (2008), No. 4, 891–903. https://doi.org/10.68381/jca15057

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Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.