Some New Results on Hyperbolic Gauss Curvature Flows

In this thesis we study the hyperbolic curvature flows. Qusilinear hyperbolic equations are derived and studied for the motion of hypersurfaces under the hyperbolic mean curvature flows. As contrast to this, a new hyperbolic curvature flow (Gauss curvature flow) is proposed for convex hypersurfaces. The equations satisfied by the graphs of the hypersurfaces under these flows give rise to a new class of Euclidean invariant fully nonlinear hyperbolic equations. Based on this, we investigate the local solvability, finite time blow-up and asymptotic behavior for these flows. Group invariant solutions of the flows are also concerned.;In Chapter 2, we present a leisure study on the reducibility of a geometric motion to a differential equation for its graph for plane curves. It serves as a motivation for the introduction of normal and normal preserving flows. We show that any Euclidean invariant quasilinear equation arises as the associated equation of some normal flow and all fully nonlinear Euclidean invariant equations arise from normal preserving flows. We further study Affine type hyperbolic motion. Finally, some properties of these flows are presented.;In Chapter 3, the symmetry groups of the hyperbolic flows are determined and the corresponding group invariant solutions are discussed.;In Chapter 4, the motions for hypersurfaces are studied. Besides the equations satisfied by the graphs, we shall derive the equations for the support function of convex hypersurface. Based on this, we establish the local solvability of the hyperbolic curvature flow. A preliminary discussion on topics such as finite time blow-up and asymptotic behavior will be given.;In the final part of this thesis, motion of free elastic curves is discussed. Conservation laws are derived by using the Noether's Theorem. We also consider group invariant solutions of this flow.