Page 33 - Undergraduate Research Opportunities Program Proceedings 2017-18

School of Science
Department of Mathematics

i.e. proving the Grayson’s theorem by proving a new isometric bound. This paper uses a different approach by
considering firstly normalised the curve to 2π and then get an estimate lower bound for the chord length in terms of arc
length Applying the estimate the authors get, the maximum curvature of the curve would be decay exponentially to 1
which eventually leads to the same result in the papers of Grayson. This method avoid the use of monotonicity formula
or the classification of singularities thus simplify the analysis.

Supervisor: FONG Tsz Ho / MATH
Student: XIE Meng / MAEC Course: UROP1000, Summer

This report focuses on the paper, The Heat Equation Shrinking Convex Plane Curves, outlining its structure of proof with
some explanations and alternative proofs omitted in the paper from the beginning to corollary 5.5, with an emphasis on
the second part of chapter 4 where proofs of many lemmas are omitted. It will also focus on the geometric proof used in
chapter 5, providing a more detailed illustration of the two graphs. However, some parts of the paper which are already
complete with few things omitted will not be discussed in detail in this report. In particular, the details of the three
estimates used to prove the boundness of k will not appear in this report.

Supervisor: FONG Tsz Ho / MATH
Student: YE Jeff York / SSCI Course: UROP1000, Summer

The mean curvature flow according to the heat equation has been a recent topic of interest in differential geometry. It
has been shown by Gage and Hamilton that convex plane curves converge smoothly to a circle under this flow. Another
geometric flow of interest is the inverse mean curvature flow, which is used in the proof of the Riemannian Penrose
inequality. In this paper we imitate the method by Gage and Hamilton, to study the inverse curvature flow on convex
plane curves. We will prove the long term existence of the solution, then show that the isoperimetric ratio converges to
4π and the curve converges to a circle smoothly.

Supervisor: FONG Tsz Ho / MATH
Student: YUAN Tong / MATH-IRE Course: UROP1100, Fall

In the paper The Heat Equation Shrinking Convex Plane Curves, M.Gage and R. S. Hamilton considered the evolution of a
plane curve under the heat equation, and derived some nice results on the existence and properties of the evolution
equation. This report mainly focuses on section 3 and 4 of the original paper, concerning the heat equation acting on
embedded and especially convex plane curves. Some ideas in the original paper will be discussed, and some steps of
argument and calculation omitted in the original paper will be completed. Parts that are already covered in detail in the
original paper will be briefly described of interest in differential geometry. It has been shown by Gage and Hamilton that
convex plane curves converge smoothly to acircle under this flow. Another geometric flow of interest is the inverse
mean curvature flow, which is used in the proof of the Riemannian Penrose inequality. In this paper we imitate the
method by Gage andHamilton, to study the inverse curvature flow on convex plane curves. We will prove the long term
existence of the solution, then show that the isoperimetric ratio converges to 4π and the curve converges to a circle
smoothly.

Weather Prediction and Air Quality

Supervisor: FUNG Jimmy Chi Hung / MATH Co-Supervisor: LU Xingcheng / ENVR
Student: CHAN Siu Chung / MAEC Course: UROP1000, Summer

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