SYMMETRY VIA SPHERICAL REFLECTION

AND SPANNING DROPS IN A WEDGE

John McCuan

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, 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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