# 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?