We consider an annular
region Ω ⊂ ℝ2 and analyze the capillary surface z = u(x,y) formed within an annular
cylinder Ω × ℝ. Assuming identical contact angles γ along the inner and outer
boundaries, we determine several qualitative properties of the surface. In particular,
we examine the behavior of u in the limiting cases of Ω approaching a disk, a thin
ring, and the exterior of a disk.