Consider the stationary
capillary problem of a drop of liquid attached to a fixed surface, so that the drop
minimizes an energy functional subject to a volume constraint. There are many such
capillary problems in which, due to the symmetry of the fixed surface, one cannot
hope for a capillary surface which is a strict local minimum for energy. A weaker
concept which is sensible to consider is that of local minimality modulo the
isometries of space which map the fixed surface into itself. In other words, it is
reasonable to attempt to show that, given a capillary surface, any nearby
comparison surface will have energy greater than or equal to the given surface,
and if the energy is equal, the comparison surface is simply a translation or
rotation of the given surface. Eigenvalue conditions are derived which imply
that a capillary surface is a strict local minimum modulo isometries, and
are applied to the specific example of a liquid bridge between two parallel
planes.
