项目名称: 多复变中的L2估计
项目编号: No.11201347
项目类型: 青年科学基金项目
立项/批准年度: 2013
项目学科: 数理科学和化学
项目作者: 朱朗峰
作者单位: 武汉大学
项目金额: 22万元
中文摘要: 在多复变与复几何领域中, L2估计占有重要地位。在本项目中,我们将主要研究多复变与复几何中的L2延拓问题和L2除法问题。 在L2延拓问题上,我们将研究弱拟凸流形内的超曲面上的dbar闭的L2的光滑的向量丛值(0,q)形式的L2延拓性质。当q=0时,著名的Ohsawa-Takegoshi L2延拓定理以及后续工作给出了这个问题的满意回答。当q≧1时,这个问题至今还未在文献中完全解决,但是我们已有部分进展。在本项目中,我们将继续研究q≧1时的情况。 L2除法问题主要关心的是,弱拟凸流形上满足一定的L2积分条件的向量丛值的全纯截面是否具有某种L2除法性质。在这个问题中,L2积分条件以及最后的估计式是关键所在。在本项目中,我们将重点改进其中的L2积分条件和最后估计式中的一致常数,这也和其中包含的公开问题有关。
中文关键词: L2估计;L2延拓问题;全纯函数;多重次调和函数;最优估计
英文摘要: L2 estimates are important in the fields of several complex variables and complex geometry. In this project, we will mainly study the L2 extension problem and the L2 division problem in several complex variables and complex geometry. On the L2 extension problem, we will study the L2 extension properties of dbar-closed L2 smooth vector bundle-valued (0, q)-forms on a hypersurface in a weakly pseudoconvex manifold. When q=0, the famous Ohsawa-Takegoshi L2 extension theorem and later developments have given satisfying answers. When q≧1, this problem has not been solved completely in the literature. However, we has gotten some progress on this problem. In this project, we will continue to study the case q≧1. What the L2 division problem mainly concerns is whether the vector bundle-valued holomorphic sections which satisfy a certain L2 integral condition have some L2 division property. The L2 integral condition and the final estimate are the key of this problem. In this project, we will focus our attention on the improvements of the L2 integral condition and the uniform constant in the final estimate, which have some relation with an open question involved in the problem.
英文关键词: L2 estimates;L2 extension problems;holomorphic functions;plurisubharmonic functions;optimal estimates