Complex Analysis and Geometry

美国国家科学基金会提案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Grauerttube结构在大(最大)半径或区域上的刚性和唯一性，并检验这一观点是否增加了表示论和自同构形式中的旧问题。第二，我们将研究我们猜想成立的Kaehler-Einstein版本的Min-Oo刚性定理。我们将继续与X合作。关于具有奇点的Levi平坦超曲面，特别是具有孤立奇点的代数超曲面。它们的重要性被Siu等人最近的工作所证明，证明了Camacho关于复射影平面上不存在光滑的Levi平坦超曲面的猜想。最后，我们建议在PI以前的学生M.Walters和R.Bryant工作的基础上，研究旗形上特殊类Schubert圈的刚性。该建议解决了复分析和几何中的几个问题。大多数人从高中起就熟悉笛卡尔的解析几何：在大多数方面，这一研究路线都是那些早期思想的现代后裔。我们将研究几何轨迹和它的解析性质之间的关系，这些性质受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一种复杂化，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。该项目的另一部分试图了解与爱因斯坦方程有关的特殊度量几何的渐近或长程性质。众所周知，这两个性质都可以在物理应用中产生影响，最近的工作使人们希望这种复杂化结构和爱因斯坦度规的渐近性都可以用来理解理论物理中所谓的马尔达克定律的部分响应。

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.