School of Science Department of Mathematics 48 Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: POON Pak Hei Andersen / MATH-PMA Course: UROP 1100, Spring Geometric flow is a field in mathematical research, which by studying how geometric quantities such as the first fundamental form (metric tensor) and curvature tensors affect the topology of the manifold. Examples of geometric flow are mean curvature flow and Ricci flow. In this report, some basic differential geometry definitions and results are reviewed. Then by considering the settings in Hamilton’s Ricci Flow, where we work on a compact 3-manifold with strictly positive Ricci curvature, we then define some curvature related tensors and study their variation formulae. Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: QU Yuxian / MATH-PMA Course: UROP 1100, Summer After studying the third chapter of this paper, this report will be based on the article conditions, take the original Riemannian manifold = 1 with parameter (modulo 2 ) and write the curve as ( , )= ( ). We will rewrite the heat equation in the form of = ( ) and show that the conclusions in chapter 3 will stay true if ( ) and satisfies that 0< ( ) ≤ , >0, 2 2 ≥0. Same as the essay, we firstly define the arclength and the operator in the term of by = 1 . Where =�� �2 +� �2 =� �. The arclength parameter is = We let and be the unit tangent vector and the (inward pointing) unit normal vectors to the curve. The Frenet equations are = , =− .
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