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-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...

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Bibliographic Details
Published in:arXiv.org 1995-09
Main Author: McCuan, John
Format: Article
Language:English
Subjects:
Angles (geometry)
Contact angle
Curvature
Differential geometry
Elliptic functions
Geometry
Maximum principle
Planes
Reflection
Surface geometry
Wedges
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Summary: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.
ISSN:2331-8422