唐納森流和平均曲率流在四維超凱勒流型的穩定性

Abstract

近幾年王慕道教授和蔡忠潤教授發表了一系列關於平均曲率流的論文[19][20][21]，他們證明當一個子流型若C1靠近一個嚴格穩定的極小子流型，那麼平均曲率流就會存在並收斂到此嚴格穩定極小子流型，另一方面，在1999 年Donaldson [6]構造出一系列的幾何流，Song和Weinkove [17]探討了四維超凱勒流型的情況並得到了一些結果，他們發現說在這個情況下唐納森流和平均曲率流是蠻相似的，所以在此碩士論文，我們將完整介紹唐納森流並證明一個類似於王教授和蔡教授的結果到唐納森流上。

In recent year, Wang and Tsai [19][20][21] proposed a series of paper about the stability of mean curvature flow about strongly stable submanifolds. They show that if a submanifold is C1 close to the strongly stable submanifold then the mean curvature flow exists for all time and converges smoothly. On the other hand, Donaldson [6] used moment map and diffeomorphism to construct lots of geometric evolution flows. In particular, the hyperkähler four manifold case was explicitly discussed by Song and Weinkove [17]. They found that Donaldson’s flow is similar to the mean curvature flow in this case. In this thesis, we discuss the Donaldson’s flow in detail and prove a result similar to the Wang and Tsai’s result.