Symmetry via Spherical Reflection and Spanning Drops in a Wedge.

We consider embedded ring type surfaces ^* in R^3 of constant mean curvature which meet planes II_1 and II_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 II _1 meets II_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<=pi + alpha, then the problem admits no solution. The result contrasts with recent work of H. C. Wente who constructed, in the particular case gamma1 = gamma2 = 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, 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. For spanning surfaces satisfying gamma _1, gamma_2 <= pi/2, we are able to extend the nonexistence result to include surfaces of arbitrary topology. ftn^*Compact, connected, orientable surfaces with two boundary components and Euler -Poincare Characteristic 0.