Free boundaries surfaces and Saddle towers minimal surfaces in S2×R

Bounded and unbounded capillary surfaces derived from the catenoid

“…As in [3], [4], [5], the proof of Theorem 1.1 consists in two steps. First we shall show that for τ sufficiently small, for each choice of the function f, there exist functions u ∈ C 2,α δ such that their normal graph over a piece of C τ satisfies the first equation in (3).…”

“…More precisely we have the following statement. It is easily obtained using (5). It describes the region of the rescaled catenoid C τ which is a graph over the annular domain A = {(r, θ) | |r − 1| τ } of the z = 0 plane.…”