Complex Analysis and Geometry

NSF提案dms -0104047从技术上讲，该提案旨在处理复杂几何和分析中的四个研究方向。我们正在研究grauttube结构在大（最大）半径或域上的刚性和唯一性，并研究这一观点是否增加了表征理论和自同构形式中的老问题。其次，我们将研究Kaehler-Einstein版本的Min-Oo刚性定理，我们推测它是成立的。我们将继续与x合作。具有奇异点的李维平面超曲面的功，特别是具有孤立奇异点的代数超曲面。Siu和其他人最近的工作证明了camacho关于复杂投影平面上不存在光滑的列维平面超曲面的猜想，这表明了它们的重要性。最后，我们建议在PI的前学生M. Walters和R. Bryant的工作之后，研究旗流形上特殊类别的Schubert环的刚性。该方案解决了复杂分析和几何中的几个问题。大多数人从高中就熟悉笛卡儿的解析几何：在大多数方面，这条研究路线是那些早期思想的现代后裔。我们将研究几何轨迹（有时由笛卡尔的原始情况下的方程定义）与其解析性质之间的关系，这些性质受到牛顿和莱布尼茨微积分的影响。特别地，我们研究了几何轨迹的复化，即在几何中加入与虚单位“i”相关的“虚点”，并研究了虚点对实点及其几何的影响。该项目的另一部分旨在了解与爱因斯坦方程相关的特殊几何形状的渐近性或长程性。众所周知，这两种性质在物理应用中都有影响，最近的工作使人们希望，这种复杂结构和爱因斯坦度尺的渐近性都可以用来理解理论物理中所谓的马尔达纳西对应的部分。

Abstract NSF Proposal DMS-0104047Speaking more technically, the proposal aims to treat four researchlines in complex geometry and analysis. We are studying the Grauerttube construction for its rigidity and uniquesness properties forlarge (maximal) radius or domain, and examining whether this point ofview adds to old questions in representation theory and automorphic forms.Second, we will study a Kaehler-Einstein version of Min-Oo's rigiditytheorem which we conjecture to hold. We will continue work withX. Gong on Levi flat hypersurfaces with singularities, especiallyalgebraic ones with isolated singularities. Their importance issuggested by recent work of Siu and others proving the conjecture ofCamacho on the non-existence of smooth, Levi-flat hypersurfaces inthe complex projective plane. Finally we propose to study the rigidityof special classes of Schubert cycles on flag manifolds, followingupon the work of the PI's former student M. Walters and,independently, R. Bryant.The proposal addresses several questions in complex analysis andgeometry. Most people are familiar with Descartes' analytic geometryfrom high school: in most respects, this line of investigation is themodern descendent of those early ideas. We will study the relationshipbetween a geometric locus, sometimes defined by equations as inDescartes' original case, and its analytic properties, thoseproperties influenced by the calculus of Newton and Leibniz. Inparticular we study a complexification of geometric locus, that is, weadd "imaginary points" to the geometry, related to the imaginary unit"i", and study the influence of the imaginary points on the realpoints and their geometry. Another portion of the project seeks tounderstand the asymptotic, or long range properties of special metricgeometries related to the Einstein equations. It is well known thatboth of these properties can be influential in physical applications,and there is recent work to lead one to hope that both thiscomplexification construction and the asymptotics of Einstein metricscan be used to understand parts of the so-called Maldacenacorrespondence in theoretical physics.