It is shown that, with the exception of very particular cases, any tubular liquid bridge con guration joining parallel plates in the absence of gravity must change discontinuously with tilting of the plates, thereby proving a conjecture of Concus and Finn Phys. Fluids 10 (1998) 39{43]. Thus the stability criteria that have appeared previously in the literature, which t a k e no account of such tilting, are to some extent misleading. Conceivable con gurations of the liquid mass following a plate tilting are characterized, and conditions are presented under which stable drops in wedges, with disk-type or tubular free bounding surfaces, can be expected. As a corollary of the study, a new existence theorem for H-graphs over a square with discontinuous data is obtained. The resulting surfaces can beinterpreted as generalizations of the Scherk minimal surface in two senses: (a) the requirement o f z e r o m e a n c u r v ature is weakened to constant mean curvature, and (b) the boundary data of the Scherk surface, which alternate between the constants +1 and ;1 on adjacent sides of a square, are replaced by capillary data alternating between two constant values, restricted by a geometrical criterion.

We consider embedded ring-type surfaces (that is, compact, connected, orientable surfaces with two boundary components and Euler-Poincaré characteristic zero) in 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 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 ≤ π + 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 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 [7], 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.

We consider some of the complications that arise in attempting to generalize a version of Archimedes' principle concerning floating bodies to account for capillary effects. The main result provides a means to relate the floating position (depth in the liquid) of a symmetrically floating sphere in terms of other observable geometric quantities.A similar result is obtained for an idealized case corresponding to a symmetrically floating infinite cylinder.These results depend on a definition of equilibrium for capillary systems with floating objects which to our knowledge has not formally appeared in the literature. The definition, in turn, depends on a variational formula for floating bodies which was derived in a special case earlier (Pacific J. Math. 231:1 (2007), 167-191) and is here generalized to account for gravitational forces.A formal application of our results is made to the problem of a ball floating in an infinite bath asymptotic to a prescribed level. We obtain existence and nonuniqueness results.

Well known first order necessary conditions for a liquid mass to be in equilibrium in contact with a fixed solid surface declare that the free surface interface has mean curvature prescribed in terms of the bulk accelerations acting on the liquid and meets the solid surface in a materially dependent contact angle. We derive first order necessary conditions for capillary surfaces in equilibrium in contact with solid surfaces which may also be allowed to move. These conditions consist of the same prescribed mean curvature equation for the interface, the same prescribed contact angle condition on the boundary, and an additional integral condition which may be said to involve, somewhat surprisingly, only the wetted region.An example of the kind of system under consideration is that of a floating ball in a fixed container of liquid. We apply our first order conditions to this particular problem.

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