陈伯勇

论文题目：关于Bergman核的一些研究

作者简介：陈伯勇，男，1971年08月出生，1996年09月师从于复旦大学张锦豪教授，于1999年07月获博士学位。

摘要

Bergman 核的概念是波兰著名数学家Stephen Bergman 于20世纪20年代在研究平面区域之间的共形映照时所引进的。它恰好是平方可积函数空间到平方可积的全纯函数子空间的正交射影的再生核。Bergman 核有着非常丰富的性质，例如，它可表示为任何一组平方可积函数空间的完备正交基之和；由它可诱导一些双全纯不变量如Bergman度量及表示域等等；更进一步，Bergman 核的某些极值性质使它成为研究偏微分方程理论的一个强有力的工具，因此关于Bergman核的研究吸引了大量学者。另外，在数学物理中，Bergman核也有广泛的应用，例如，F.A. Brezin构造了机械系统的一个漂亮的量子化过程其相空间正好是带有Bergman度量的有界对称域。国内外一些著名的数学家如L. Hörmander, C. Fefferman 以及华罗庚，陆启铿等在此领域都有杰出的贡献。正是由于Bergman核在复分析中扮演着如此重要的角色，以至于在Bergman逝世不久，美国数学会专门设立了一个Bergman奖，授予那些在Bergman核的研究中作出突出贡献的学者。

然而，研究Bergman核, 例如它的边界行为等等并非易事。这是因为除了球和多圆柱之外很少有区域上的Bergman核可给出显式表达。从历史上讲，关于Bergman核的现代理论始于20世纪60年代，随着偏微分方程理论在多复变函数论中逐渐广泛的应用，一些著名的问题如Levi问题，Cousin问题等等得到了解决或是更简单的证明，从而使多复变函数论的研究进入了一个黄金时期。J.J. Kohn与Hörmander关于Cauchy-Riemann方程的开创性工作，使得Bergman核的研究也有了突飞猛进的发展。Hörmander给出了强拟凸域上Bergman核在对角线上即所谓Bergman核函数趋向于无穷大的增长率。基于Kohn的工作，N. Kerzman 证明了强拟凸域上的Bergman核除去对角线外可光滑地延拓到边界。关于Bergman核在强拟凸域边界附近的渐近行为的深刻描述则由C.Fefferman在1974年的一篇重要论文中给出。作为一个应用, 他证明了如下一个基本定理: 任何两个强拟凸域之间的双全纯映射可光滑延拓到边界。这是单复变函数论中一个众所周知的经典定理, 即任何两个光滑边界的平面区域之间的双全纯映射可光滑延拓至边界, 的一个推广。这是因为任何光滑边界的平面区域必定是强拟凸域。关于Fefferman定理的简化证明由S. Bell等人给出。Bell还将之推广至: 任意两个弱拟凸有限型区域之间的逆紧全纯映照可光滑延拓到边界。然而, 更多的拟凸域的的边界是非光滑的。

本篇博士学位论文主要是来研究非光滑边界拟凸域上Bergman核的性质。

在第一章, 我们介绍Bergman核的定义, 一些基本性质, 以及一些预备知识。

在第二章, 我们研究Bergman穷竭性, 即Bergman核函数是否在边界附近一致地趋向于无穷。一个Bergman穷竭的区域必拟凸, 这是因为Bergman核函数本身就是一个强多次调和的穷竭函数。在另一方面, 由Riemann可去奇点定理, 一些拟凸域如有孔圆盘不是Bergman穷竭的。因此研究什么样的拟凸域是Bergman穷竭就变得很有意义。在1976年, 应用Skoda的一个定理, P. Pflug证明了任何一个Lipschitz边界的拟凸域是Bergman穷竭的。本文的第一个结果是将Pflug的定理推广至只须要求边界可局部地表示为一个连续函数的图。要证明我们的定理, 光用Skoda定理是不够的, 我们必须用Ohsawa-Takegoshi深刻的延拓定理。

第三章是本文的重点。我们研究S.Kobayashi的一个著名问题, 即什么样的拟凸域上的Bergman度量是完备的? 根据Bremermann的一个定理, Bergman完备性隐含拟凸性。另外一些简单的例子如有孔圆盘表明反之则不成立。早在1959年, Kobayashi证明了任何解析多面体是Bergman完备的。Pflug证明了光滑边界的拟凸域也是Bergman完备的。随后, 在1981年, T.Ohsawa将边界条件减弱为。同年, Kerzman-Rosay证明了任何一个边界拟凸域是超凸的。一个区域称为超凸, 如果其上存在一个负的连续多次调和穷竭函数。自那以后, 人们就猜想超凸域是Bergman完备的。但是由于在边界上没有什么限制, 要验证它就相当困难。令人鼓舞的是, Ohsawa在1993年证明了超凸域是Bergman穷竭的。这似乎预示着上面的猜想成立。Ohsawa的证明由两方面构成: 首先利用Green函数的性质证明平面上的超凸域的Bergman穷竭性, 然后应用延拓定理证明一般维数情形也成立。然而Ohsawa的方法并不适合来证明超凸域的Bergman完备性。在1998年, 我注意到一种称为pluricomplex Green的函数在研究Bergman完备性时所起到的重要作用。Pluricomplex Green函数是平面区域的经典的(负)Green函数在高维时的一种推广, 他的奇性仍是logarithmic。正如经典的Green函数在位势理论中所起的作用一样, pluricomplex Green 函数在多位势理论中也有着十分重要的作用。我证明了如果一个拟凸域的子集 {}的体积在当w趋向边界时趋向0, 则这个区域是Bergman完备的。由此即得: 如果一个超凸域的pluricomplex Green函数是对称的, 则它是Bergman完备的。特别地, 由 Green函数的对称性即得平面上的超凸域是Bergman完备的。同时我还举了一个反例，其Bergman完备却不是超凸域。在我的论文刚被Ann. Polon. Math. 接收不久, P. Pflug来信告诉我, 其实我的论文本身就隐含着证明超凸域的Bergman完备性的关键所在。随后, 在他与Z. Blocki合作的一篇文章中, 上述猜想的到了解决。从另一个方向推广Ohsawa的结果是将边界条件由减为局部可表为一个连续函数的图。这个推广在 On Bergman Completeness and Bergman Stability 一文中得以实现。该文已发表在Math. Ann. 上。在证明Bergman完备性的过程中我们都用到了区域的Bergman穷竭性。人们自然会对两者之间的关系产生浓厚兴趣。我证明了对于平面区域Bergman穷竭性包含Bergman完备性, 而对高维情形则存在反例。该结果发表在Complex Variables: Theory and Applications 上。在另一方面，Kobayashi又提出如下猜想：Bergman完备性隐含 Bergman穷竭性。然而，受我的两篇论文 (Ann.Polon.Math.和Complex Variables)的启发，W. Zwonek最近已经给出了一个反例。另外, 我还构造了一个区域非Bergman穷竭, 但Bergman度量在边界附近却一致地趋向于无穷。尽管如此, 这并不能保证Bergman完备性。

在第四章, 我们研究Bergman核的稳定性。在1967年, I.-P. Ramadanov证明了一列非减区域上的Bergman核局部一致地收敛为这些区域的并集上的Bergman核。这个定理很快就找到了许多应用，其中包括著名的陆启铿猜想。一个区域上的Bergman核的零点使得在其上不能定义一个整体的所谓Bergman表示坐标，这就是陆启铿提出如下问题的动机：什么样的区域上Bergman核无零点？我们称这样的区域为陆启铿域。然而，除了球和多圆柱外，很难判断一个区域是否陆启铿域。在1981年，Green-Krantz证明了球的任何一个很小的光滑扰动仍是陆启铿域。在很长一段时间，人们猜想所有强拟凸域均为陆启铿域。直到1985年，H.-P. Boas给出一个反例。通过简单的计算，容易验证区域 {} 的Bergman核存在零点。由于它是拟凸的，因而可从内部被一列递增, 强拟凸的完全Reinhardt域逼近。由Ramadanov定理和Hurwitz定理，马上可得所需反例。相应于Ramadonov定理，人们自然会问：若一个区域在外部被一列区域逼近，那么情形如何？在1996年，Boas证明了若一列拟凸域的边界一个具有边界的区域D的边界之间的Hausdorff 距离趋向于零，则Bergman核的稳定性定理仍成立。同时他提出了二个问题：1. D 的边界的正则是否可减为正则？2. 区域的拟凸性假设是否可去掉？对问题1我们给出一个肯定的回答，更确切地说，D的边界只须局部可表示为一个连续函数的图即可。相关结果已发表在Proc. Amer. Math. Soc.。对问题2 我们了给出一个反例。在本章中还有大量的关于Bergman核的稳定性的例子。

近来，与博士论文相关的工作取得了较大的进展：1. 尽管超凸域是Bergman完备的, 然而这并不意味着它的 Bergman度量一致的趋向于无穷．多圆柱就是一个最简单的例子．为了排除这种情形, 人们自然地考虑在每个边界点都存在多次调和Peak 函数这样一个条件。我证明了若一个拟凸域 (并不要求边界的光滑性) 在任意一个边界点都有一个Hölder连续的多次调和peak函数, 那么Bergman度量一致地趋向无穷。该文的修改稿已经寄给 Nagoya Math. J. 。2. 在 1956年，Sommer-Mehring证明了若一个区域与它的outhull (所有包含该区域的闭包的拟凸域的交集) 重合时则其Bergman核不能连续地延拓过任何一个边界点。受其启发，Kobayashi在1959年猜想在上述条件下区域是 Bergman完备的。近来我举了一个反例，即将发表在 Internat. J. Math.上。3. 基于 pluricomplex Green 函数的重要性，人们自然会联想到它与经典的Green函数之间的联系。近来，Carehed证明了在强拟凸域上两者是可比较的。我把该结果推广到局部凸的有限型弱拟凸域上。所用的方法是非平凡的，因而该文已被Michigan Math. J. 录用。4. 在1979年，Greene-Wu证明了若一个单连通的完备Kaehler流形的截曲率满足一定的Pinch条件，则其是Bergman完备的。他们猜想下界可去掉。最近我们成功地解决了这的一个猜想。其证明主要基于复流形上的pluricomplex Green函数的某些性质。5. 我发现了延拓定理的一些新的应用。我们可用它来研究陆启铿猜想以及Bergman核是否可作为一个区域到Riemann球的连续映照等一系列问题。最近我收到了 Nagoya大学Ohsawa教授的邀请, 在他们的讨论班上介绍我的工作。

ABSTRACT

Stephen Bergman first introduced the concept of the Bergman kernel in the 1920’s. He used it to study the conformal maps between planar domains. It is just the reproducing kernel of the orthogonal projection from the space of square-integrable functions to the subspace of square-integrable holomorphic functions. The idea of the Bergman kernel is such a rich and beautiful one, for example, it can be expressed by the sum of any complete orthonormal system for the space of square-integrable holomorphic functions; it behaves functorially under biholomorphic maps, it gives rise to biholomorphic invariants such as the Bergman metric and representative domains; moreover, the Bergman kernel has certain extremal properties that make it a powerful tool in the theory of partial differential equations. The Bergman kernel has also found applications in Mathematics Physics. F.A.Berezin constructed a nice quantization procedure for mechanical systems whose phase-space is a bounded symmetric domain with the Bergman metric. Hence the study of the Bergman kernel has attracted a lot of attention.. Some famous mathematicians such as L. Hörmander, C. Fefferman, Hua Luokeng and Lu Qi-Keng have their own contributions to this area. The Bergman kernel plays such an important role in complex analysis that the American Mathematics Society sets up a prize so-called the Bergman prize, which is awarded to mathematicians who has made outstanding works on the Bergman theory shortly after the death of S. Bergman.

However, to study the Bergman kernel such as the boundary behavior is quite difficult since few examples except balls or polydiscs of which the Bergman kernel can be computed explicitly. Historically, the modern theory of the Bergman kernel began in the 1960’s since the application of partial differential equation in several complex variables. The pioneer work of J. J. Kohn’s work on the Neumann problem and of L. Hörmander on the square–integrable solution to the Cauchy-Riemann equation on pseudoconvex domains in the complex Euclidean space made the study of the complex analysis develop rapidly, so is the Bergman kernel. Hörmander obtained the boundary limits of the Bergman kernel on the diagonal, which is called the Bergman kernel function, for strongly pseudoconvex domains. Based on the work of Kohn, N. Kerzman showed the boundary behavior of the Bergman kernel off the diagonal. A deep description for the Bergman kernel near the boundary of a strongly pseudoconvex domain has obtained by C. Fefferman in his important paper in 1974. With an application, he proved the following fundamental theorem: Any biholomorphic map between two strongly pseudoconvex domains with smooth boundaries extends smoothly to the boundary. This extends a well-known result in one variable which states that any biholomorphic map between to smooth domains in complex plane extends smoothly to the boundary since any smooth domain in plane is just strongly pseudoconvex. Significant simplification and improvements of Fefferman’s result were made by S. Bell and many others based on Kohn’s work. For weakly pseudoconvex cases, the situation is much difficult. Only on a type of smoothly weakly pseudoconvex domains called finite type domains the behavior of the Bergman kernel is similar to strongly pseudoconvex cases. Nevertheless, there also exist a lot of pseudoconvex domains with non-smooth boundaries.

The aim of the article is to study the boundary behavior of the Bergman kernel on non-smooth pseudoconvex domains.

In Chapter 1, we give the definition and some basic properties of the Bergman kernel.

In Chapter 2, we study the Bergman exhaustion, that is, the Bergman kernel function tends uniformly to infinity near the boundary. A Bergman exhausting domain must be pseudoconvex because the Bergman kernel function provides a strictly plurisubharmonic exhaustion function. On the other hand, by the Riemann’s removable singularity theorem, one cannot expect some pseudoconvex domains such as the puncture discs being Bergman exhausting. So it is interesting to invest which pseudoconvex domain is Bergman exhausting. In 1976, with an application of a theorem of Skoda, P. Pflug proved that any bounded pseudoconvex domain with Lipschitz boundary is Bergman exhausting. Our first result is to extend Pflug’s result to the case of which the boundary only assumed to being described locally as the graph of a continuous function. In proving our theorem, Skoda’s theorem is not sufficient; one has to use the extension theorem of Ohsawa-Takegoshi.

The main part of the article is contained in Chapter 3. We deal with a famous problem of S. Kobayashi: Which pseudoconvex domain is complete with respect to the Bergman metric? Due to a theorem of Bremermann we know that Bergman completeness implies pseudoconvexity. On the other hand, as simple examples (the puncture disc for instance) show that the converse is in general not true. Kobayashi himself proved that any bounded analytic polyhedron is Bergman complete. Pflug proved the Bergman completeness for smooth pseudoconvex domains. This result has extended by Ohsawa to the case of boundary in 1981. It was proved by Kerzman-Rosay in the same year that any bounded pseudoconvex domain with boundary is hyperconvex. A domain is called hyperconvex if there exists a negative continuous plurisubharmonic exhaustion function. It has been conjectured that any bounded hyperconvex domain is Bergman complete since then. However, it is difficult to verify since there is no assumption on the boundary. Inspiringly, Ohsawa proved in 1993 that any bounded hyperconvex domain is Bergman exhausting. This suggests that the above conjecture is very likely true. Ohsawa’s proof consists of an application of the classical Green function in plane, which leads to the exhaustion property of the Bergman kernel function in plane, and then by extension theorem he obtained the Bergman exhaustion for high dimension. However, this method is not fit for proving Bergman completeness. In 1998, I firstly notice that the importance of the so-called pluricomplex Green function in studying the Bergman completeness. The pluricomplex Green function is an extension of the (negative) classical Green function in plane to high dimension whose singularity is still logarithmic. It plays a similar role in the pluripotential theory as the classical Green function in the potential theory. The author proved that if the volume of the subset {} of a pseudoconvex domain tends to zero as tends to the boundary, and then the domain is Bergman complete. An immediately consequence is that any bounded pseudoconvex domain whose pluricomplex Green function is symmetric is Bergman complete. In particular, any bounded hyperconvex domain in plane is Bergman complete. Shortly after my paper being accepted by Ann. Polon. Math., P. Pflug wrote to me that my paper itself contains the key to prove the Bergman completeness for any hyperconvex domain. Thus the above conjecture was finally settled in a paper of Blocki-Pflug. Another approach to extend Ohsawa’s result is to weaken the boundary assumption of boundary to the case that the boundary can be locally described as the graph of a continuous function. It was proved in the paper On Bergman Completeness and Bergman Stability, which has been published in Math. Ann.. In this chapter, we also give a negative answer to another problem of Kobayashi: Is a bounded domain Bergman complete if it coincides with its outhull, i.e., the intersection of all pseudoconvex domains containing the closure of the domain? In the procedure or showing the Bergman completeness, we always do under the assumption that the domain is Bergman exhausting. A nature question is then arising: Is a Bergman exhausting domain just Bergman complete? We give a positive answer for bounded domains in plane and provided a counterexample for higher dimensions. The result has been published in Complex Variables: Theory and Application. On the other hand, Kobayashi made the following conjecture: The Bergman completeness implies the Bergman exhaustion. Recently, inspired by two papers of mine (Ann. Polon. Math. And Complex Variables), W. Zwonek was able to give a counterexample. In this chapter, I have also constructed a domain, which is not Bergman exhausting, but the Bergman metric tends uniformly to infinity near the boundary. However, this does not guarantee Bergman completeness.

In Chapter 4, we study the stability property of the Bergman kernel. In 1967, Ramadanov proved that if a domain is exhausted by a sequence of non-decreasing subdomains then the Bergman kernels of these subdomains converge locally uniformly to the Bergman kernel of the union domain. It soon found many applications, among them includes the famous Lu Qi-Keng conjecture. Zeros of a Bergman kernel poses an obstruction to the global definition of so-called Bergman representative coordinates. This observation was Lu Qi-Keng’s motivation for ask the following question: Which domains has zero-free Bergman kernels? A domain of which the Bergman kernel is zero free is called the Lu Qi-Keng domain. The explicit expression of the Bergman kernel follows immediately that balls and polydiscs are Lu Qi-Keng domains. However, little is known besides these cases. In 1981, Green-Krantz proved that the Bergman kernel of every smooth, small perturbation of a ball is still zero free, the stability property of the Bergman kernel plays an essential role in their proof. It has been conjectured that all strongly pseudoconvex domain with smooth boundary are Lu Qi-Keng for a long time until H. P. Boas gave a counterexample in 1985. After a simple computing, the Bergman kernel of the domain {} does have zeros. Since it is pseudoconvex, it can be approximated from inside by a sequence of increasing strongly pseudoconvex, complete Reinhardt domains. By Ramadanov’s theorem and Hurwitz’s theorem, one immediately gets the desired counterexample. Corresponding to Ramadanov’s theorem, it is natural to ask what happens when the domain is approximated from outside by a sequence of domains. Boas proved that if a domain D has smooth boundary such that it can be approximated by a sequence of pseudoconvex domains in the sense that the Hausdorff distances between the boundaries of D and these pseudoconvex domains tend to zero. He asks the following two questions: 1. Can smooth regularity be replaced by smooth regularity? 2. Can the pseudoconvexity assumption of the domains be removed? In this chapter, we give an affirmative answer to the first question. More precisely, we only need the hypothesis that the boundary can be described locally as the graph of a continuous function. This result is contained in paper A Remark Bergman completenesson, which has been published in Proc. Amer. Math. Soc. For problem 2 we provide a counterexample in the Math. Ann. Paper. Stability property of the Bergman kernels on various other cases has also been studied in this chapter.

Recently, the work related to the doctorial thesis has made rapid progress: 1. Despite the Bergman completeness of hyperconvex domains, their Bergman metric will, in general, not go to infinite uniformly. The polydiscs are the simplest examples. To remove this unexpected case, a natural condition considered is the existence of a plurisubharmonic peak function at each boundary point. I have proved that the Bergman metric of a pseudoconvex domain (does not assume the smoothness of the boundary) tends uniformly to infinity if there exists a Horlder continuous plurisubharmonic peak function at each boundary point. The result was contained in a revision of the preprint Boundary behavior of the Bergman metric which has been resubmitted to Nagoya Math. J. 2. In 1956, Sommer-Mehring proved that the Bergman kernel function can not be continuously extended through the boundary if the outhull of the domain coincides with itself. Inspired by this fact, Kobayashi conjectured that such domains are Bergman complete. However, I constructed a counterexample. It will appear in Internat. J. Math. 3. The interesting properties of the pluricomplex Green function makes one relates it to the classical Green function associated with Laplacian. M. Carehed proved that these two Green functions can be compared on bounded strongly pseudoconvex domains. In a recently paper accepted by Michigan Math. J., I extend his result to the case of locally convex domains of finite type provided that the type is no larger than the dimension of the domain, otherwise, we have found some counterexamples. 4. In 1979, Greene-Wu proved that if the Reimann sectional curvature of a simply connected Kaehler manifold is suitably pinched, then it carries a complete Bergman metric. They conjectured that the lower bound can be removed. This conjectured was solved recently. The idea was based on some properties of the pluricomplex Green function. 5. The author also noticed some new applications of the extension theorem of Ohsawa-Takegoshi, such as the Lu Qi-keng conjecture and the problem whether a domain can be continuously mapped to the Riemann sphere by its Bergman kernel. Recently, Professor Ohsawa in Nagoya University invited me to give a talk about the above work in their seminar.

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