Higher genus capillary surfaces in the unit ball of R 3

Abstract

We construct the first examples of capillary surfaces of positive genus, embedded in the unit ball of R 3 with vanishing mean curvature and locally constant contact angles along their three boundary curves. These surfaces come in families depending on one parameter and they converge to the triple equatorial disk. Such surfaces are obtained by deforming the Costa-Hoffman-Meeks minimal surfaces. MSC: 53A10; 35R35; 53C21

Bounded and unbounded capillary surfaces derived from the catenoid

“…In [3] and [4] I produced new examples of smooth capillary minimal surfaces (H = 0) which are not graphs by a perturbation method. Furthermore, in the second work the examples are unbounded.…”

“…In [3] I showed the existence of positive genus capillary minimal surfaces in B 3 , a unit ball of R 3 . More precisely for each k ∈ [1, .…”

“…. , +∞) there exists a one parameter family of minimal surfaces of genus k, embedded in B 3 with non-empty boundary which consists in three simple closed curves c t , c m , c b which lie in ∂B 3 and which make locally constant contact angle with ∂B 3 . The parameter τ can vary in (0, τ 0 ), with τ 0 small.…”

“…We shall show how to deform the bounded piece of catenoid contained in the ball B 3 in order to create surfaces (with possibly non-zero mean curvature) which meet ∂B 3 making a non-constant angle α. The function α oscillates around a constant value given by the constant angle α 0 at which the original catenoid meets ∂B 3 . The difference |α − α 0 | is small.…”

“…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). To do that we need to find the expression of the mean curvature operator for normal graphs of functions defined on C τ .…”