Abstract

The results presented in this thesis contribute to the understanding of the evolution of smooth, strictly convex, closed hypersurfaces in $\mathbb{R}^{n+1}$ driven by non-symmetric speeds on the principal curvatures. The preservation of convexity, the occurrence of singularities, and the asymptotic behavior of the flows are studied.
After an introduction to geometric flows, Chapter 3 focuses on the analysis of the short-term and long-term behavior of a contraction flow governed by a non-symmetric speed for rotationally symmetric hypersurfaces. Our investigation reveals two key findings. Firstly, we establish that the flow maintains convexity throughout the deformation process. Secondly, we observe the development of a singularity within a finite time, leading to the convergence of every such strictly convex hypersurface to a single point.
To investigate the asymptotic behavior of the flow, we employ a proper rescaling technique of the solutions. Through this rescaling, we demonstrate that the rescaled solutions converge subsequentially to the boundary of a convex body. In the fourth chapter, we extend our study to the short-term and long-term behavior of a non-symmetric expansion flow in $\mathbb{R}^{n+1}$. We show that, starting with a smooth, strictly convex, rotationally symmetric, closed hypersurface, the flow preserves convexity while expanding infinitely in all directions. Depending on certain parameters within the speed function, we establish that the existence time of the flow can be either finite or infinite. We also investigate the asymptotic behavior of the flow through a suitable rescaling process and demonstrate the subsequential convergence of the solutions to the boundary of a convex body in the Hausdorff distance. In the fifth chapter, we introduce the most general version of the flow studied in the Chapter 3. We address the barriers and challenges encountered when transitioning from a symmetric speed to a non-symmetric speed, and present our strategies to tackle some of these difficulties.