Complex Analysis and Geometry

复杂分析和几何

• 批准号: 0104047
• 负责人: Daniel Burns
• 金额: $ 7.86万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104047&HistoricalAwards=false
• 关键词: Complex; Analysis; Geometry

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.

摘要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条研究线。我们正在研究的Grauerttube结构的刚性和uniquesness性质为大（最大）半径或域，并检查是否这一观点增加了老问题的表示论和自守形式。第二，我们将研究Kaehler-Einstein版本的Min-Oo的刚性定理，我们猜想举行。我们将继续与X合作。Gong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的研究.他们的重要性issuggested最近的工作萧和其他人证明猜想ofCamacho的不存在光滑，列维平坦超曲面在复杂的射影平面。最后，我们建议研究旗流形上特殊类Schubert圈的刚性，这是继PI以前的学生M. Walters和R.这个建议解决了复分析和几何学中的几个问题。大多数人从高中就熟悉笛卡尔的解析几何：在大多数方面，这条研究路线是那些早期思想的现代后代。我们将研究一个几何轨迹，有时定义的方程在笛卡尔的原始情况下，和它的分析性质之间的关系，这些性质的影响，由牛顿和莱布尼茨演算。特别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有关的虚点，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理解与爱因斯坦方程相关的特殊度量几何的渐近或长程性质。众所周知，这两个性质在物理应用中都是有影响力的，最近的工作使人们希望，这种复杂化结构和爱因斯坦度量的渐近性都可以用来理解理论物理中所谓的马尔达塞对应的一部分。