On the structure of embedded minimal annuli

We show that any embedded minimal annulus in a ball (with boundary in the boundary of the ball and) with a small neck can be decomposed by a simple closed geodesic into two graphical subannuli. Moreover, we give a sharp bound for the length of this closed geodesic in terms of the separation (or height) between the graphical subannuli. This will illustrate our “pair of pants” decomposition in the special case where the embedded minimal planar domain is an annulus. The catenoid $\{x_1^2+x_2^2=\text{cosh}^2x_3\}$ is the prime example of an embedded minimal annulus.