論文中英文摘要格式.doc

附件2論文中英文摘要格式作者姓名：尹萬科論文題目：實(shí)流形在復(fù)流形中的全純不變量作者簡介：尹萬科，男，1980年01月出生，2005年09月師從于武漢大學(xué)黃孝軍教授，于2008年08月獲博士學(xué)位。中 文 摘 要多復(fù)變函數(shù)論是當(dāng)今數(shù)學(xué)中一個非常重要的分支，這一學(xué)科由Poincare、 Hartogs、 E. Levi和 E.Cartan等著名數(shù)學(xué)家在20世紀(jì)初創(chuàng)立，且其后一直備受關(guān)注。在多復(fù)變的眾多基本問題中，有一類重要的問題是實(shí)流形在復(fù)流形中的雙全純等價性。在這一研究領(lǐng)域里，由于包括Cartan、Moser、Bishop、陳省身和Webster等一流數(shù)學(xué)家在內(nèi)的許多人的共同努力，已經(jīng)取得了相當(dāng)大的進(jìn)展；然而直到現(xiàn)在，這一領(lǐng)域里仍有許多基本的問題沒有解決。本論文主要研究了Bishop曲面在復(fù)空間中的雙全純等價性問題。本文的第一部分包括第二章和第三章，研究的是C2中Bishop不變量為0的橢圓Bishop曲面，這一工作可以看成是Moser-Webster在1983年的著名工作和其后Moser在1985年的著名工作的繼續(xù)，我們解決了Moser在1985年的工作中提出的著名問題；論文的第二部分為第四章，我們把Moser的工作推廣到高維的情形，利用KAM快速迭代法，我們證明了一個余維數(shù)為二的實(shí)解析子流形與一個對稱模型雙全純等價當(dāng)且僅當(dāng)他們形式等價。具體說來，本文由下面四部分組成：在第一章中，我們介紹了多復(fù)變中的一些整體和局部雙全純等價問題，詳細(xì)介紹了Bishop曲面研究的發(fā)展過程，直至最新的進(jìn)展；然后介紹了高維復(fù)空間中復(fù)切點(diǎn)的CR維數(shù)大于等于二的實(shí)子流形的最新發(fā)展?fàn)顩r；最后對本文的主要工作進(jìn)行了綜述。在第二章中，我們引入了一個不同于以往的權(quán)理論，這就使得我們可以成功的求解一個復(fù)雜的非線性方程，最終得到Bishop不變量為0的Bishop曲面的形式正規(guī)型，進(jìn)而得到了此類曲面在形式變換群下的完全不變量。特別地，我們證明了當(dāng)Bishop不變量為0，Moser不變量為某個固定的s時，他的模空間是一個無窮維Frechet空間。由此我們馬上可以得到Moser問題的一個否定答案，即Bishop不變量為0時Bishop曲面不能被其Taylor展開的一個有限截?cái)嗨鶝Q定。結(jié)合Poincare在1907年的一項(xiàng)工作中的證明思想，我們得到Bishop不變量為0, Moser不變量為有限時Bishop曲面一般不能雙全純等價于一個代數(shù)曲面。在第三章中，我們首先對所研究的曲面進(jìn)行復(fù)化，利用Moser-Webster的投影化思想和Huang-Krantz中的平坦化定理，得到了一簇貼于曲面上的圓盤，從而建立了Bishop不變量為0、Moser不變量為有限時的Bishop曲面上的雙曲幾何。接著我們證明了兩個Bishop不變量為0，Moser不變量為某個固定s的Bishop曲面雙全純等價當(dāng)且僅當(dāng)他們的形式正規(guī)型經(jīng)過一個旋轉(zhuǎn)變換后相同，從而解決了這一曲面的全純等價分類問題。最后我們研究了一類特殊Bishop曲面的一些等價性質(zhì)。在第四章中，我們研究了Moser定理的一個高維推廣。對于高維復(fù)空間中的一類余維數(shù)為2的奇異CR子流形，我們給出了它的擬正規(guī)型，同時得到了擬正規(guī)型能平坦化的充要條件。然后運(yùn)用著名的KAM快速迭代法，證明了如果這個子流形能雙全純等價于正規(guī)型中一個對稱的二次曲面當(dāng)且僅當(dāng)它們形式等價。關(guān)鍵詞： Bishop曲面，CR奇異流形，消失的Bishop不變量，全純等價，正規(guī)型Holomorphic Invariants Of Real Submanifolds In Complex ManifoldsYin WankeABSTRACTSeveral complex variables is an important branch of modern mathematics. The subject started with the work of Poincare, Hartogs, E. Levi, E. Cartan, and others at the beginning of the 20th century. Since then, it has attracted much attention. Among many fundamental problems in Several Complex Variables is the biholomorphic equivalence problem for real submanifolds in a complex manifold. Along these lines of research, much progress has been made due to the important work of many leading mathematicians of their time, that include at least E. Cartan, Moser, Bishop, Chern, Webster, etc. However, there are still many basic problems left unsolved.This thesis will study the biholomorphic equivalent problem for Bishop submanifolds in a complex space.The first part of this dissertation consists of Chapter two and Chapter three, which is to study the non-degenerate elliptic Bishop surfaces with a vanishing Bishop invariant in C2. Our work here can be viewed as the continuation of the celebrated work of Moser-Webster in 1983 and the late famous work of Moser in 1985. Among many things, we will present a solution to a well-known problem of Moser. The second part of the thesis is the fourth chapter, which is to establish a result of Moser to higher dimensions. We will show that a real analytic submanifold of codimension two with a symmetric model is biholomorphic to the model if it is formally equivalent to the model, by applying the KAM theory. More precisely, the dissertation is organized as follows:In the first chapter, we discuss some global and local biholomorphic equivalent problems in several complex variables. We will give a through description of the history and new development of the research on the Bishop surfaces. Next we discuss the new development of the real submanifolds in higher dimensional complex space with the codimension at the complex tangent points greater than or equal to two.In the second chapter, we derive a formal normal form for a Bishop surface near a vanishing Bishop invariant, by introducing a quite different weighting system. Next we obtain a complete set of invariants under the action of the formal transformation group. We show, in particular, that the modular space for Bishop surfaces with a vanishing Bishop invariant and with a fixed (finite) Moser invariant s is an infinitely dimensional manifold in a Frechet space. This then immediately provides an answer, in the negative, to Mosers problem concerning the determination of a Bishop surface with a vanishing Bishop invariant from a finite truncation of its Taylor expansion. Furthermore, it can be combined with some ideas of Poincare to show that most Bishop surfaces with Bishop invariant zero and Moser invariant finite are not holomorphically equivalent to algebraic surfaces.In the third chapter, we discuss the complexification of the surface under consideration, by making use of the Moser-Webster polarization and the Huang-Krantz flatting theorem. We then obtain a family of holomorphic discs attached to the surface. Starting with these discs, we introduce a new hyperbolic geometry. Based on the obtained geometry, we show that two Bishop surfaces with Bishop invariant zero and Moser invariant finite are biholomorphically equivalent if and only if their formal normal forms are the same up to a trivial rotation map. Hence, the formal normal form that we derive provides a solution to the equivalence problem also in the holomorphic category. At last we derived some equivalence properties of a family of Bishop surfaces.In the fourth chapter, we generalize the work of Moser to the higher dimensional case. For a certain codimension two CR singular submanifold in higher dimensional complex space, we first derive a pseudo-normal form for M near the origin, then use it to give a necessary and sufficient condition when (M,0) can be formally flattened. Finally we pro