| Abstract |
Curvature flows are geometric evolutions of a hypersurface moved by curvature quantities such as the mean curvature and the Gauss curvature, which have been applied in material science and image processing. The main difficulty to treat curvature flows is development of singularities in finite time which arises in many case. In this thesis, we would like to propose a method to continue curvature flows for a long time by placing obstacles enclosed by the initial hypersurface. We apply the method to prevent the development of singularities for the mean curvature flow when the initial hypersurface is given by a graph and for the Gauss curvature flow when the initial hypersurface is strictly convex and closed. Moreover, we investigate the obstacle problem for the parabolic Monge-Amp`ere equation which is closely related to the Gauss curvature flow. Our approach is based on the penalization method by allowing the evolution of hypersurface can pass the obstacle, with the property that the more the hypersurface pass the obstacle, the more penalty is imposed on the velocity.
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