Let $(M^n,g)$ be a complete Riemannian manifold with bounded geometry, that is, it has bounded curvature and positive injectivity radius. Given two disjoint smoothly embedded homotopically trivial closed curves $\gamma_1$ and $\gamma_2$, we consider the problem of minimizing annulus $\Sigma$ with $\partial \Sigma=\gamma_2\cup \gamma_2$.

More precisely, we consider all maps $u\in W^{1,2}(A,M)\cap C^0(\bar A,M)$, where $A:=\{z\in \mathbb C \,\mid 1 < |z| <2\}$, such that $u$ restricted to $\partial A$ are reparametrizations of $\gamma_1$ and $\gamma_2$.

Can we prove the existence of such $u$ whose image has the least area? Is there any reference for such existence?

Let $(M^n,g)$ be a complete Riemannian manifold with bounded geometry, that is, it has bounded curvature and positive injectivity radius. Given two disjoint smoothly embedded homotopically trivial closed curves $\gamma_1$ and $\gamma_2$, we consider the problem of minimizing annulus $\Sigma$ with $\partial \Sigma=\gamma_2\cup \gamma_2$.

More precisely, we consider all maps $u\in W^{1,2}(A,M)\cap C^0(\bar A,M)$, where $A:=\{z\in \mathbb C \,\mid 1 < |z| <2\}$, such that $u$ restricted to $\partial A$ are reparametrizations of $\gamma_1$ and $\gamma_2$. Here, we assume at least one such map $u$ exists.

Can we prove the existence of such $u$ whose image has the least area? Is there any reference for such existence?