# 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 $$\partial \Sigma=\gamma_2\cup \gamma_2$$.

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?

• Under the conditions that you stated there may be no annulus bounded by $\gamma_1\cup\gamma_2$. For example when your two curves are boundaries of disjoint disks in a torus. Oct 10, 2020 at 3:46
• @AlexandreEremenko Yes. The problem only concerns the case when at least one such map $u$ exists. Oct 10, 2020 at 4:18
• The Douglas' condition says that if there is an annulus with less area the the sum of the areas of the two minimizing disks, then there is an area minizimizing annulus -- this is the most general situation, but also not so easy to check. Sometimes you can make an argument using a barrier or topological condition (e.g., if the two curves are homotopic but not null-homotopic). Oct 10, 2020 at 14:51
• @RBega2 Is there any reference for the Douglas' condition? Thanks. Oct 10, 2020 at 15:50
• @Totoro The article "Conformal mappings and the Plateau-Douglas problem in Riemannian manifolds" by Jurgen Jost is I believe the most modern treatment. Struwe has a book on Plateau's problem that might also cover it (it's been a while since I looked at his book). Oct 10, 2020 at 16:06

Such an annulus need not exist. For example, consider two circles in $$\mathbb{R}^3$$ defined by $$x^2+y^2 = 1$$ and $$z = \pm R$$. If $$R$$ is sufficiently large, then there cannot be a minimizing annulus (or, indeed, any minimizing connected surface) with these two circles as boundary.
The reason is the following: First, one can certainly find an annulus with these two surfaces as boundary whose area is as close to $$2\pi$$ as desired: Just take the two flat disks with these circles as boundary and join them by a very thin tube $$x^2+y^2 = \epsilon^2>0$$ and $$|z|\le R$$ for $$\epsilon$$ very small and smooth the result to get an annulus.
Meanwhile, if there were a connected minimal surface $$A$$ with these circles as boundary, then $$A$$ would have to pass through the plane $$z=0$$ at some point $$p = (x_0,y_0,0)$$, and hence the ball $$B$$ of radius $$r centered on $$p$$ would not meet the two circles. Then the monotonicity formula for minimal surfaces implies that the part of the surface $$A$$ inside the ball $$B$$ would have to have area at least $$\pi r^2$$. Letting $$r$$ approach $$R$$, one sees that the area of $$A$$ would be at least $$\pi R^2$$.
Thus, if $$R^2>2$$, then no connected minimal surface with these two circles as boundary can achieve the lower bound of $$2\pi$$.