Symmetry via spherical reflection and spanning drops in a wedge

Abstract

We consider embedded ring-type surfaces (that is, compact, connected, orientable surfaces with two boundary components and Euler-Poincaré characteristic zero) in R 3 of constant mean curvature which meet planes Π 1 and Π 2 in constant contact angles γ 1 and γ 2 and bound, together with those planes, an open set in R 3 . If the planes are parallel, then it is known that any contact angles may be realized by infinitely many such surfaces given explicitly in terms of elliptic integrals. If Π 1 meets Π 2 in an angle α and if γ 1 + γ 2 > π + α, then portions of spheres provide (explicit) solutions. In the present work it is shown that if γ 1 + γ 2 ≤ π + α, then the problem admits no solution. The result contrasts with recent work of H.C. Wente who constructed, in the particular case γ 1 = γ 2 = π/2, a self-intersecting surface spanning a wedge as described above.Our proof is based on an extension of the Alexandrov planar reflection procedure to a reflection about spheres [7], on the intrinsic geometry of the surface, and on a new maximum principle related to surface geometry. The method should be of interest also in connection with other problems arising in the global differential geometry of surfaces.

On the shape of a droplet in a wedge: new insight from electrowetting

“…However, there is no proof that such solution is unique [21,22]. Our experimental results are rather interesting in this light, as we directly observe the truncated sphere shape.…”

Computational Modelling of a Spanning Drop in a Wedge with Variable Angle

“…It is important to notice that Figure 4 shows a profile not in accordance with , who conclude that both γ 1 and γ 2 should be acute. However, soon we will present Theorem 1 of McCuan (1997), which states that under certain hypotheses we cannot have γ 1 + γ 2 ≤ π + α, and consequently γ 1 + γ 2 cannot be less than π. This contradiction occurs because the theorem refers to a mathematically ideal model that includes some simplifications.…”

“…Because of these particularities, and using the term ring type surface to denote a topological annulus, McCuan (1997) assumes that S is of ring type and also verifies the following hypotheses: it is an embedded surface of constant H which spans a wedge of opening angle α and meets the planes Π 1 and Π 2 of the wedge at constant contact angles γ 1 and γ 2 , respectively. With this setting he proves Of course, spherical rings S clearly verify the inequality of Theorem 2.…”

Free boundary minimal hypersurfaces with spherical boundary

“…The principal curvatures κ 1 and κ 2 are given by If , then Σ is flat. (See for more details.) Theorem Let Σ be a compact immersed minimal annulus lying in the domain bounded by two concentric spheres S 1 and S 2 .…”