School of Science Department of Mathematics 47 Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: TAN Weile / SSCI Course: UROP1000, Summer In this paper, we summarize the basic stuff of plane curves and the curve shortening flow, and show that if we deform a convex smooth plane curve at a speed proportional to its curvature along its normal vector field, the isoperimetric ratio L2/A decreases to 4π. If we normalize the family of curves in this transformation by homothetic expansion so that all the lengths of the curves are 2π, then the family we got will converge to the unit circle. Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: WU Haoyu / SSCI Course: UROP1000, Summer This report focus on some basic knowledge about flow geometry. It’s a statement that there is a evolution, i.e. the heat equation, for a closed convex plain curve shrinking to a point with spect to time t ∈ [0,T), where t is finite. The curve is also becoming more and more circular in the evolution. Firstly, this report will state some basic lemma in the calculation of some value of the plane curve. Then mainly, this report gives a way to prove that the curve is becoming more and more circular during the evolution. Using some geometric techniques, we show that the ratio of the maximum curvature and minimum curvature of the curve goes to 1. Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: YEUNG Wai Ho / SSCI Course: UROP1000, Summer This report is about the comparison between Euclidean Geometry and Riemannian Geometry. How does it motivate some geometric idea, for example, define the arc-length and surface area/volume intrinsically. After explain how to generalise the concept of parallel and straight line from Euclidean space to an ambient manifold detailly. For example the intuition of parallel transport and geodesic etc. Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: CHUNG Tsun Ho Anson / MATH Course: UROP1100, Summer Gage and Hamilton first showed that convex curves shrink to a point under the curve shortening flow in 1986. This paper extends such results to a more general case of the curve shortening flow, where the curve’s flow is proportional to the th power of its curvature, using the same techniques with suitable modifications. The presented proof is much simpler than Huisken’s two-point estimates or methods involving the support function by Andrews, as it uses only curvature estimates to bound the curvatures. The paper aims to be an introductory essay to the simplest geometric flow without prior knowledge of differential geometry.