We consider embedded ring-type surfaces (that is, compact, connected, orientable surfaces with two boundary components and Euler-Poincare characteristic zero) in R^3 of constant mean curvature which meet planes \Pi_1 and \Pi_2 in constant contact angles \gamma_1 and \gamma_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 \Pi_1 meets \Pi_2 in an angle \alpha and if \gamma_1+\gamma_2>\pi+\alpha, then portions of spheres provide (explicit) solutions. In the present work it is shown that if \gamma_1+\gamma_2 is less than or equal to \pi+\alpha, then the problem admits no solution. The result contrasts with recent work of H.C. Wente who constructed, in the particular case \gamma_1 = \gamma_2 =\pi/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 [1], 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.
[1] John McCuan, Symmetry via spherical reflection, preprint, 1994.