  In this paper we investigate the flow of surfaces by a class of symmetric
functions of the principal curvatures with a mixed volume constraint. We
consider compact surfaces without boundary that can be written as a graph over
a sphere. The linearisation of the resulting fully nonlinear PDE is used to
prove a short time existence theorem for a large class of surfaces that are
sufficiently close to a sphere and, using center manifold analysis, the
stability of the sphere as a stationary solution to the flow is determined. We
will find that for initial surfaces sufficiently close to a sphere, the flow
will exist for all time and converge exponentially to a sphere. This result was
shown for the case where the symmetric function is the mean curvature and the
constraint is on the (n+1)-dimensional enclosed volume by Escher and Simonett
(1998).
