School of Science Department of Mathematics 47 Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: HO Chi Man / MATH-PM Course: UROP 1000, Summer Throughout reading the paper “THE HEAT EQUATION SHRINKING CONVEX PLANE CURVES” by M. GAGE & R. S. HAMILTON, we have discussed the nature of curve shortening flow in ℝ2. Specifically, we have a closed curve represented by a smooth function : 1 →ℝ2, where 1 is modulo 2 with parameter , and then we assign a time variable ∈[0, ) such that = where and represents the curvature and unit normal vector of the curve respectively. The heat equation = implies for each point of the curve, it will move perpendicularly to the curve at a speed proportional to the curvature. In this report, we will shift our focus to a generalization of the curve shortening flow of simply closed strictly convex 2 plane curves, where the heat equation now becomes = ( ) , with : ℝ→ℝ being a function with respect to the curvature . In the rest of the paper, for simplicity we will replace ( ) by in most of the time. Similar to what the paper did, we will derive evolution equations for the length of the curve, curvature and area under the curve. We will also show that under certain constraints, if the initial curve is embedded, then the curve remains embedded as the curve evolves. Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: KHUC Dinh Toan / MATH-PMA Course: UROP 1000, Summer The curve shortening flow is a well-done problem in differential geometry, describing the evolution of a curve whose normal velocity depends on its curvature. This report investigates a generalized curve shortening flow where the normal velocity is prescribed by a function of the curvature. Under certain assumptions, we establish the preservation of embeddedness for closed curves during the evolution. This extends the classical result of Gage and Hamilton for the standard curve shortening flow. In this report, we will present the embeddedness, discuss some of the geometric evolutions of the flows, and then find the bounds for higher derivatives of the curvature. Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: LAM Ching Yiu / MATH-PM Course: UROP 1100, Summer This paper aims to generalize the equation = , where is curvature and is the unit normal vector for a simple closed curve ( , ) ∶ 1 ×[0, ) →ℝ2 by replacing with an arbitrary curvature function ( ). We argue that in what condition ( ) must be satisfied so that the previous theorems still hold. Intuitively, one can think of speeding up or down in a more complicated way, if the flow remains in the same direction (i.e. sign) at every point when compared to the original equation. The theorems should remain the same. We highly recommend the readers to read Gage M. & Hamilton R. S.’s paper first to gain a better picture or intuition.
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