UROP Proceedings 2020-21

School of Science Department of Mathematics 46 Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: WU Ruirui / MATH-PMA Course: UROP3100, Spring This report will present a generalization of results in “Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders” by Frederick Tsz-Ho Fong and Peter Mcgrath in 2016 concerning the rotational symmetry of asymptotically conical mean curvature flow self-expanders. The technique of proving main theorem in “Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders” will be emphasized and some detailed calculations are very similar as in [2]. [2] Tsz-Kiu Aaron Chow & Ka-Wing Chow & Frederick Tsz-Ho Fong. “Self-Expanders to Inverse Curvature Flows by Homogeneous Functions”. In: arXiv:1701.03995 (2017). Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: LI Ka Ki / SSCI Course: UROP1000, Summer The report is based on the paper “The Heat Equation Shrinking Convex Plane Curves” written by M. Gage and R. S. Hamilton in 1986. In the paper, the curve shortening flow is defined such that the rate of change of point is proportional to curvature. This report is going to explain the details of some of the theorems and proofs in the paper, and to generalize some of the results to the flow that the rate of change of points are not proportional to curvature. Due to the requirement and limitation of content, the report only covers the theorem until the boundedness of k. Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: MA Xiangyu / SSCI Course: UROP1000, Summer In this paper we show that as a smooth convex plane curve is deformed along its normal vector field at a rate proportional to its curvature, the isoperimetric ratio L2/A of the curve approaches 4π as the enclosed area approaches zero. If the one parameter family of curves evolving in this fashion is normalized by homothetic expansion of the plane so that each curve encloses area g, then the resulting family will converge to the unit circle.

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