Partial Differential Equations in Several Complex Variables

单变量和多变量复分析在数学和数学物理中起着特殊的作用。复数的使用在数学的发展中是必不可少的。偏微分方程和几个复变量被用于弦理论和扭转理论，这些物理理论试图统一不同的物理力场。这些偏微分方程的解的存在性和规律性仍然没有被完全理解，它们构成了数学分析中一些最具挑战性的问题。目前的研究不仅对数学的发展很重要，而且还可能导致对物理现象的新理解，并在其他科学和技术中具有潜在的应用。本文主要研究了几种复杂变量中最重要的方程，即Cauchy-Riemann方程和诱导切向Cauchy-Riemann方程。本文主要研究了复流形上的函数理论、Dolbeault上同群的Hausdorff性质、列维平面超曲面和复叶理、复射影空间和负弯曲流形上的Cauchy-Riemann算子。理解这些方程在曲率条件下的几何方面及其与复杂流形中函数理论的关系是一个具有挑战性和重要的问题。引入了新的方法来研究复杂流形中域拓扑与Dolbeault上同群拓扑之间的联系。在几何环境中对几个复杂变量的研究为拓扑学、叶理理论、复杂动力学、代数和复杂几何等问题提供了有趣的新问题和新的见解。该项目旨在加深对这一领域的理解。

Complex analysis in one and several variables plays a special role in mathematics and mathematical physics. The use of complex numbers has been essential in the development of mathematics. Partial differential equations and several complex variables are employed in string theory and twistor theory, physical theories that try to unify different physical force fields. The existence and regularity of solutions to such partial differential equations are still not fully understood, and they form some of the most challenging problems in mathematical analysis. The current study is not only important for the development of mathematics, but it may lead to new understanding of physical phenomena as well, with potential applications in other sciences and technology. This research focuses on some of the most important equations in several complex variables, the Cauchy-Riemann equations and the induced tangential Cauchy-Riemann equations. The topics investigated in this research project include function theory on complex manifolds, Hausdorff property of Dolbeault cohomology groups, Levi-flat hypersurfaces and complex foliation, and the Cauchy-Riemann operators on complex projective spaces and negatively curved manifolds. Understanding the geometric aspects of these equations under the curvature conditions and their relations with function theory in complex manifolds is a challenging and important problem. New approaches have been introduced to study these problems which connect the topology of domains in complex manifolds with the topology of Dolbeault cohomology groups. The study of several complex variables in a geometric setting has provided interesting new questions with fresh insight to problems in topology, foliation theory, complex dynamics, algebraic and complex geometry. This project aims to deepen understanding in this area.