Given a Riemannian manifold $M$ and two smooth loops $\gamma_0, \gamma_1: S^1 \longrightarrow M$ in it, I am looking for maps $\phi: [0, T] \times S^1 \longrightarrow M$ which minimize the energy $$E[\phi] = \iint \|\mathrm{d} \phi(\theta, t) \|^2\, \mathrm{d}t\,\mathrm{d}\theta$$ under the boundary condition $$ \phi(0, \,\cdot\,) = \gamma_0, ~~~~~~\phi(1, \,\cdot\,) = \gamma_1.$$
Can we say anything about existence, assuming that for each $\theta$, $\gamma_1(\theta)$ is in an $\varepsilon$-neighborhood of $\gamma_0(\theta)$? What about uniqueness? Naively, one would expect the image not to leave the given $\varepsilon$-tube.
A very naive way to get this could be to connect each point $\gamma_0(\theta)$ in $M$ with $\gamma_1(\theta)$ using the unique minimizing geodesic $l_\theta$ connecting these points. This will work if for each $\theta$, $\gamma_1(\theta)$ is in some small enough $\varepsilon$-neighborhood of $\gamma_0(\theta)$, and then the map $$\tilde{\phi}(\theta, t) := l_\theta(t)$$ will be smooth. Now what happens if we let it flow with the harmonic map flow? Are there convergence results?
Feel free to make as many assumptions on anything as you please.
