In this paper we investigate the flow of hypersurfaces by a class of symmetric
functions of the principal curvatures with a mixed volume constraint. We consider
compact hypersurfaces 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 hypersurfaces that are sufficiently close to a sphere
and, using centre manifold analysis, the stability of the sphere as a stationary
solution to the flow is determined. We will find that for initial hypersurfaces
sufficiently close to a sphere, the flow will exist for all time and the hypersurfaces will
converge exponentially fast to a sphere. This result was shown for the case where
the symmetric function is the mean curvature and the constraint is on the
-dimensional
enclosed volume by Escher and Simonett (1998).
Keywords
curvature flows, mixed volume, stability, centre manifolds
