Henry C. Wente

Twisted Tori of Constant Mean Curvature in R3

In an earlier paper [11] we showed the existence of a countable number of isometricly distinct closed bounded surfaces of genus one in R3 with constant mean curvature. These provided counterexamples to a conjecture attributed to H. Hopf [5].

The symmetry of rotating fluid bodies

Consider an incompressible fluid body (in outer space) rotating about an axis with a given angular velocity ω, and which is in equilibrium relative to the potential energy of its own gravitational field and the surface energy due to surface tension. We show that such a body possesses a plane of symmetry perpendicular to the axis of rotation such that any line parallel to the axis and meeting the body cuts it in a line segment whose center lies on the plane of symmetry. This extends an earlier result of L. Lichtenstein [4].

The capillary problem for an infinite trough

Consider an infinite trough (or wedge) with dihedral angle 2α, 0 < α <π and a quantity of fluid inside contacting the edge. In equilibrium the free interface of the fluid will be a surface of constant mean curvature meeting the planar walls at a constant angleγ determined from physical considerations. One obvious configuration is for the free surface to be a section of a round circular cylinder parallel to the axis of the wedge whose position is determined by the angles α and γ. For α + γ > π/2 the cylinder configuration is unstable and bifurcation occurs. We exhibit the full family of bifurcating solutions starting with the round cylinder solution and proceeding through a “beading up” process into a series of spherical sections suitably positioned. Furthermore, if the edge of the wedge is a re-entrant corner (α > π/2) then there are further bifurcating families. One is a secondary bifurcation from the family initially constructed while the other is a primary bifurcation from the cylinder which are less symmetric than the initial families.

The Floating Ball Paradox

Abstract.An earlier work by the present author questioned a concept of solid/liquid surface tension introduced by Thomas Young in 1805. The paper just preceding this one presents an explicitly computable example as supporting evidence by analogy that Young’s concept could be correct. The present note suggests a different view, and offers an alternative interpretation of the example.