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Motion By Mixed Volume Preserving Curvature Functions Near Spheres, David James Hartley
Motion By Mixed Volume Preserving Curvature Functions Near Spheres, David James Hartley
Senior Deputy Vice-Chancellor and Deputy Vice-Chancellor (Education) - Papers
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 …
Motion By Volume Preserving Mean Curvature Flow Near Cylinders, David James Hartley
Motion By Volume Preserving Mean Curvature Flow Near Cylinders, David James Hartley
Senior Deputy Vice-Chancellor and Deputy Vice-Chancellor (Education) - Papers
We investigate the volume preserving mean curvature flow with Neumann boundary condition for hypersurfaces that are graphs over a cylinder. Through a center manifold analysis we find that initial hypersurfaces sufficiently close to a cylinder of large enough radius, have a flow that exists for all time and converges exponentially fast to a cylinder. In particular, we show that there exist global solutions to the flow that converge to a cylinder, which are initially non-axially symmetric. A similar case where the initial hypersurfaces are spherical graphs has previously been investigated by Escher and Simonett [8].