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We consider embedded
ring-type surfaces (that is, compact, connected, orientable surfaces with two
boundary components and Euler-Poincaré characteristic zero) in R3 of constant
mean curvature which meet planes Π1 and Π2 in constant contact angles
γ1 and γ2 and bound, together with those planes, an open set in R3. 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 Π1 meets Π2 in an angle α and if γ1 + γ2 > π + α, then portions of
spheres provide (explicit) solutions. In the present work it is shown that if
γ1 + γ2 ≤ π + α, then the problem admits no solution. The result contrasts
with recent work of H.C. Wente who constructed, in the particular case
γ1 = γ2 = 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 [McCuan, John. Symmetry via spherical reflection. J.
Geom. Anal. 10 (2000), no. 3, 545–564], 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.
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