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-Poincar\'{e} characteristic zero) in ${\bold R}^3$ 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 ${\bold R}^3$. 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 $\Pi_1$ 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 $\gamma_1+\gamma_2\le\pi+\alpha$, 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 {\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.