Partial Differential Equations in Several Complex Variables

一个或多个变量的复分析在数学和数学物理中占有中心地位。复数的使用在数学的发展中是必不可少的。例如，复分析中的相位概念通常用于控制理论、图像分析和动力系统。物理学中某些重要的结构是由复分析与偏微分方程的结合而产生的。偏微分方程和几个复杂的变量也用于弦和扭的数学物理理论，试图统一不同的物理力场。对于所有这些应用，关于在某些几何约束下解这些偏微分方程的基本问题仍然没有完全理解，它们构成了数学中一些最具挑战性的问题。目前的研究不仅对数学的发展具有重要意义，而且可能导致对物理现象的新认识，并在其他科学技术中具有潜在的应用价值。本研究将集中于一些最重要的复杂变量方程，包括柯西-黎曼方程和诱导切向柯西-黎曼方程。本文讨论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、复投影空间上的Cauchy-Riemann算子和负弯曲流形。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复杂分析和几何中最具挑战性和最重要的问题之一。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。Cauchy-Riemann方程解的正则性与边界上的叶理理论有关。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。这些问题的解决将提高我们在上述所有领域的知识。复杂几何理论才刚刚开始发展，更多的努力将朝着这个方向。几何测量理论和谐波分析也将应用于研究非光滑域上的复杂变量。这些问题是分析、几何和拓扑学在应用数学和物理中的交叉应用。

Complex analysis in one and several variables occupies a central place in mathematics and mathematical physics. The use of complex numbers is essential in the development of mathematics. The concept of phase in complex analysis, for example, is commonly used in control theory, image analysis and dynamical systems. Certain important structures in physics arise from an integration of complex analysis with partial differential equations. Partial differential equations and several complex variables are also used in String and Twister mathematical physical theories that try to unify different physical force fields. For all these applications, foundational questions concerning the solution of such partial differential equations under certain geometric constraints are still not fully understood and they form some of the most challenging problems in mathematics. The current study is not only important for the development in mathematics, but it may lead to new understanding of physical phenomena with potential applications in other sciences and technology. This research will focus on some of the most important equations in several complex variables, including the Cauchy-Riemann equations and the induced tangential Cauchy- Riemann equations. The problems discussed in this proposal include function theory on complex manifolds, Hausdorff property of Dolbeault cohomology groups, Levi-flat hypersur- faces and complex foliation, the Cauchy-Riemann operators on complex pro jective 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 one of the most challenging and important problems in complex analysis and geometry. 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. Regularity of the solutions to the Cauchy-Riemann equations are related to the foliation theory on the boundary. 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. The solution of these questions will advance our knowledge in all the aforementioned areas. Complex geometric theory has only just begun to develop and more efforts will be directed in this direction. Geometric measure theory and harmonic analysis will also be applied to study several com- plex variables on non-smooth domains. These problems are at the intersection of analysis, geometry and topology with applications in applied mathematics and physics.