Vertex theorems for capillary drops on support planes
Robert Finn and John McCuan
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Submission date: 11. Aug. 1997
published in: Mathematische Nachrichten, 209 (2000), p. 115-135
MSC-Numbers: 76B45, 53A10, 53C42, 49Q10
Keywords and phrases: capillarity, mean curvature, liquid drops, wedges, polyhedral angles
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We consider a capillary drop that contacts several planar bounding walls so as to produce singularities (vertices) in the boundary of its free surface. It is shown under various conditions that when the number of vertices is less than or equal to three, then the free surface must be a portion of a sphere. These results extend the classical theorem of H. Hopf on constant mean curvature immersions of the sphere. The conclusion of sphericity cannot be extended to more than three vertices, as we show by examples.