School of Science Department of Mathematics 41 Geometric Flows Supervisor: FONG, Tsz Ho / MATH Student: CHUNG, Tsun Ho Anson / MATH-PMA Course: UROP4100, Fall This progress report contains results derived during the last three months regarding monotone quantities of general curvature flow. Gerhardt (1990) was able to prove the long-time existence of the general curvature flow and the convergence to sphere under the assumption that the homogenous function is convex in each variable. A long-standing problem is to remove the assumption of convexity, where it is only used once for proving the boundedness of higher order derivatives of the second fundamental form. A possible tool to attack the problem is the use of monotone quantities that can related both the analytical aspect and the geometric intuition. This paper presents some results based on the monotone quantity found by Guan and Li (2009). Geometric Flows Supervisor: FONG, Tsz Ho / MATH Student: LAU, Kwun Chai Michael / MATH-PM Course: UROP2100, Fall This report is to give a short summary on Riemannian Manifolds and the motivation behind. It is inspired by the Theorema Engregium in Differential Geometry, which is a breakthrough in mathematics that shows Gauss Curvature of surface only depends on the first fundamental form. In other words, Guass curvature is an intrinsic properties of surfaces. On the other hand, by introducing Geodesic Equation and exponential map, we can derive the geodesic normal coordinates, which is a powerful tool to simplify tensor calculation. For example, all Christoffel Symbol will be zero at the point that Geodesic normal coordinate applied. Geometric Flows Supervisor: FONG, Tsz Ho / MATH Student: LO, Lik Kan Nicholas / SSCI Course: UROP1000, Summer In this report I will focus on the paper “The heat equation shrinking convex plane curves” written by M Gage and R.S.Hamilton, this is an essay that investigates the behavior of curvature , such as the maximum curvature and higher order derivative of curvature , given that heat equation shrinks a convex curve M to a point. In this report, firstly I will introduce the equations required , then I will provide outlines of proofs of those long term behaviors of curvature . The writer of this report would like to give thanks to Frederick Fong, supervisor of this project (geometric analysis)

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