Stability of stationary solutions to curvature flows

2017-02-23T03:55:27Z (GMT) by Hartley, David James
In this thesis we study the evolution of hypersurfaces under weighted volume preserving curvature flows. Specifically we consider the stability of spheres and finite cylinders as stationary solutions to the flows. The flows are formulated as a partial differential equation for a height function and an existence result is obtained when the height function is small. Through further analysis we prove that the sphere and finite cylinder, provided the radius of the finite cylinder satisfies a certain condition, are stable. That is, we prove that if a graph over a sphere or cylinder has small height function its flow exists for all time and converges to a sphere or cylinder respectively. This is the first result proving that there exist non-axially symmetric hypersurfaces that converge to cylinders under the flows. In the case of volume preserving mean curvature flow near a cylinder, we improve the above results to obtain greater regularity of the flow and convergence with respect to a stricter norm. Analysing the condition on the radius in this situation we find it is necessary in order for the cylinder to be stable. The analysis also leads to the surprising result that certain constant mean curvature unduloids are stable stationary solutions to the axially symmetric flow in high dimensions. The last result of the thesis proves the instability of two dimensional catenoids under the classical mean curvature flow. The results in this thesis are obtained using functional analysis and semi-group methods, which can be applied since the linearised speed operators are sectorial. The stability results come from analysing the spectrum of the linearised operators and analysing the center manifold of the system.