Some aspects of the dynamic of V=H-(H)over-bar

We consider the evolution of a surface Gamma(t) according to the equation V = H - (H) over bar, where V is the normal velocity of Gamma(t), H is the s um of the two principal curvatures and (H) over bar is the average of H on Gamma(t). We study rhs case where Gamma(t) intersects orthogonally a fixed surface Sigma and discuss some aspects of the dynamics of Gamma(t) under th e assumption that the volume of the region enclosed between Gamma(t) anti S igma is small. We show that, in this case, if Gamma(0) is near a hemisphere , Gamma(t) keeps its almost hemispherical shape and slides on Sigma crawlin g approximately along orbits of the tangential gradient del H-Sigma of the sum H-Sigma of the two principal curvatures of Sigma. We also show that, if (p) over bar epsilon Sigma, is a nondegenerate zero of del H-Sigma and a > 0 is sufficiently small, then there is a surface of constant mean curvatur e which is near a hemisphere of radius a with center near (p) over bar and intersects Sigma orthogonally. (C) 1999 Academic Press.