Symmetry via Spherical Reflection and Spanning Drops in a Wedge

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.

No reviews yet.

No comments yet.