Mean Curvature Flow for Hypersurfaces in Euclidean Spaces

Specialeforsvar ved Pavlos Tsiakoumis

Titel: Mean Curvature Flow for Hypersurfacesin Euclidean Spaces

 

 

Abstract: The Mean Curvature Flow is a well known second-order, quasilinear and degenerate parabolic system. The studying of this geometric evolution problem is situated at the crossroads of several scientific disciplines: Geometric Analysis, Geometric Measure Theory, Differential Topology, PDE’s Theory, Mathematical Physics, Image Processing, Computer-aided Design, among others.
We say that a family of hypersurfaces evolves by mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. The aim of this thesis is to present some of the known results of this interesting evolution problem. We first discuss the definition of this geometric flow, give some basic examples of surfaces evolving under the mean curvature flow and show the existence of the flow for some positive interval of time. Next, we prove the maximum principle and its consequences. In particular, we examine whether at a singular time the curvature of the evolving hypersurface stays bounded or not. Finally, we introduce the level set approach of the flow and use the theory of viscosity solutions to give one more way of proving the short time existence.

 

 

Vejleder: Niels Martin Møller
Censor:   Steen Markvorsen, DTU