dg.differential geometry – Plateau’s Problem from an annulus


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 $partial Sigma=gamma_2cup gamma_2$.

More precisely, we consider all maps $uin W^{1,2}(A,M)cap C^0(bar A,M)$, where $A:={zin mathbb C ,mid 1 < |z| <2}$, 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?