Curvature-dependent flows have been successfully used for shape representation in two dimensions, principally due to a theorem on curve shortening flow stating that embedded curves evolving proportional to their curvature do not self-intersect and smoothly converge to a circular point. In this paper, we investigate properties of curvature-driven flows for surfaces and derive necessary conditions for the flows to avoid self-intersections. Since known curvature flows, such as mean and Gaussian flows, lead to self-intersections, we impose geometric constraints on the direction and magnitude of arbitrary flows to narrow down the space of candidates. Our main result is to establish necessary conditions for the direction of movement in order to avoid self-intersections: 1) convex elliptic points should move in, while concave elliptic points move out; 2) hyperbolic points should not move at all; and 3) parabolic points should also be stationary. In the process, we also establish monotonicity conditions for the magnitude of the curvature-dependent function. Finally, based on these conditions and previously studied literature on the flow of strictly convex surfaces, we select a magnitude proportional to square root of Gaussian curvature leading to:
$\Partial {\psi} t= sign(H)\sqrt{G + |G|}\vec{N}$.
Informally, the direction of deformation depends on mean curvature and the magnitude of deformation on Gaussian curvature. Our numerical simulations show thatfor a large class of non-convex surfaces, but not for all, this deformation takes the initial surface to a round point without developing self-intersections. The necessary conditions and the existence of surfaces that self-intersect for this flow indicate that a flow solely based on curvatures would lead to self-intersections for some surface.