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 ${\bold R}^3$ 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 ${\bold R}^3$. 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 $γ_1+γ_2\leπ+α$, 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 {\it 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.