Stability of Near Cylindrical Stationary Solutions to Weighted-Volume Preserving Curvature Flows

Abstract

In this paper we consider a class of weighted-volume preserving curvature flows acting on hypersurfaces that are trapped within two parallel hyperplanes and satisfy an orthogonal boundary condition. In previous work the stability of cylinders under the flows was considered; it was found that they are stable provided the radius satisfies a certain condition. Here we consider flows where there is a critical radius such that the cylinders are only stable on one side of the critical value, and the weight function is a linear combination of the elementary symmetric functions, the reason for such a choice is made clear in the appendix. We find that in such instances bifurcation from the cylindrical stationary solutions occurs at the critical radius and we determine a condition on the speed and weight functions such that the nearby non-cylindrical stationary solutions are stable under the flow. In particular, we find that high dimensional half-period unduoids close to the cylinder are stable stationary solutions to the volume preserving mean curvature flow under axially symmetric, volume preserving perturbations. We will also highlight the specific cases of homogeneous speed functions and the mixed-volume preserving mean curvature flows.