Volume-Preserving Mean Curvature and Willmore Flows with Line Tension

We show the short-time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by volume-preserving Mean Curvature Flow (MCF) and line tension effect on the boundary. Difficulties arise due to the non-local nature of the resulting second order, nonlinear PDE, which will be overcome by a perturbation result from semigroup theory. In addition, we prove the same result for the Willmore flow with line tension, which results in a nonlinear PDE of fourth order. For both flows we will use a Hanzawa transformation to write the flows as graphs over a fixed reference hypersurface. We finish the thesis with an application of the generalized principle of linearized stability to prove stability of spherical caps under the volume-preserving Mean Curvature Flow with line tension