Page 32 - Undergraduate Research Opportunities Program Proceedings 2017-18

School of Science
Department of Mathematics

Supervisor: FONG Tsz Ho / MATH
Student: LIANG Shu / PHYS Course: UROP1100, Summer

In this report, I’ll discuss the paper Geometric Flow written by Hamilton and Gage. The main purpose of this report is to
fulfil some missing parts in the proofs of the paper. Since chapter 1 and 2 are not very relevant to the proof of the result
of the paper, I’ll discuss the details of chapter3 and 4. Chapter3 derives some useful evolution equations of the curve,
and chapter 4 shows that the curve will evolve until it shrinks to a point. Some techniques used in the proof such as
maximum principle, doing estimation using geometric properties are very useful in the analysis of geometric flow
problems.

Supervisor: FONG Tsz Ho / MATH
Student: LONG Zhong / MATH-PMA Course: UROP1100, Fall
UROP2100, Summer

This report focus on Chapter 3,4,5,9 of the the paper Three-Manifolds with Positive Ricci Curvatureby Richard S.
Hamilton. By filling the details of the computation which lead to important result, we try to get a point of view on how
to connect the Ricci curvature with the constant curvature of a compact 3-manifold.

Supervisor: FONG Tsz Ho / MATH
Student: TANG Tianchen / MATH-IRE Course: UROP1100, Fall

In the paper The Heat Equationn Shrinking Convex Curves by M.Gage and R.S.Hamilton, they consider the case of a plane
curve shrinking when acting on a quasi-heat equation.By using geometric analysis methods, they first derived the time
evolution equations of the length of the curve,the curvature of the curve and the area it encloses. With these equations
one can show that a curve remains embedded in finite time. After gaining these results they turned to the more specific
case of convex curves and showed the long time existance of the solution to evolution equation. Quite a lot of technical
calculation is involved in the process and the authors of that paper omitted some unimportant details. In this report, I
will try to go through the important calculations in that paper and summarize the main idea.

Supervisor: FONG Tsz Ho / MATH
Student: WAN Jingbo / MATH-IRE Course: UROP2100, Fall
UROP3100, Summer

The Harnack type estimate is very important in geometric analysis. This report aims at introducing some results about
gradient estimate and Harnack inequalities and their applications which I have read in this summer time. I will mainly
focus on the Harnack estimates of Laplacian equation which is elliptic and heat equation which is parabolic. Moreover, I
will also cover their applications on eigenvalue estimate and heat kernel estimate. In the end, further results for
different geometric flows will be mentioned.

Supervisor: FONG Tsz Ho / MATH
Student: WONG Sing Lam / MATH-PMA Course: UROP1100, Fall

In this project, we mainly concern the paper titled “Curvature Bound for Curve Shortening Flow via Distance Comparison
And A Direct Prof of Grayson’s Theorem” by Ben Andrews and Paul Bryant. In this report, I would try to fill in some non-
trivial details in the paper. The main objective of this paper is the same as the paper that I have read in previous project

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