EMBEDDED DISC-TYPE SURFACES WITH LARGE CONSTANT MEAN CURVATURE AND FREE BOUNDARIES

Abstract

Given Ω ⊂ ℝ3 an open bounded set with smooth boundary ∂Ω and H ∈ ℝ, we prove the existence of embedded H-surfaces supported by ∂Ω, that is regular surfaces in ℝ3 with constant mean curvature H at every point, contained in Ω and with boundary intersecting ∂Ω orthogonally. More precisely, we prove that if Q ∈ ∂Ω is a stable stationary point for the mean curvature of ∂Ω, then there exists a family of embedded [Formula: see text]-surfaces near Q, ε > 0 small, which, once dilated by a factor [Formula: see text] and suitably translated, converges to a hemisphere of radius 1 as ε → 0.