In this thesis, we derive various entropy upper bounds for self-shrinkers of the mean curvature flow which admit a symmetry, including several applications.
In our first paper, which is a joint work with Niels Martin Møller and John Ma, we study the space of complete embedded rotationally symmetric self-shrinkers. We first derive explicit entropy upper bounds for this class of self-shrinkers. The proof is purely geometric and relies on an application of the general Toponogov’s theorem from metric geometry to derive length upper bounds on simple closed geodesics in an incomplete surface with curvature bounded from below by a positive constant. We then apply the entropy bounds to first prove a smooth compactness theorem for this space of self-shrinkers. Second, we show that there are finitely many such selfshrinkers which additionally are symmetric with respect to the hyperplane perpendicular to the axis of rotation.
In our second paper, which is a joint work with John Ma, we generalize the entropy bounds obtained in our first work in two directions. We modify the proof of the embedded class to include entropy upper bounds for compact non-spherical immersed rotationally symmetric self-shrinkers. We also obtain entropy bounds for a larger class of embedded self-shrinkers which are constructed through the theory of isoparametric foliations of the sphere and which contain the space of complete embedded rotationally symmetric self-shrinkers as a special case.