Kahler Manifolds with Prescribed Ricci Curvature

具有规定 Ricci 曲率的卡勒流形

• 批准号: 1906466
• 负责人: Ronan Conlon
• 金额: $ 17.88万
• 依托单位: Florida International University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2021-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906466&HistoricalAwards=false
• 关键词: Kahler; Manifolds; Prescribed; Ricci; Curvature

Complex manifolds are higher-dimensional surfaces that are defined using the complex numbers. A "metric" on such a space is an object that endows that space with a shape or more precisely, with Ricci curvature. The study of these metrics lies at the nexus of complex geometry and geometric analysis. This project is concerned with the construction of complete Kahler metrics with prescribed Ricci curvature on non-compact Kahler manifolds; more specifically, non-compact Calabi-Yau manifolds and Kahler-Ricci solitons. Applications of such manifolds permeate throughout physics and mathematics. Indeed, Calabi-Yau manifolds of complex dimension three have become prominent in string theory where they supposedly model the six additional real dimensions of space-time that we do not see, whereas Kahler-Ricci solitons provide models for the formation of singularities in an important evolution equation, namely the Kahler-Ricci flow. Efforts will be made to construct new examples of these manifolds and to formulate their existence in terms of algebraic geometric criteria. This will enhance our understanding of the geometry of non-compact Kahler manifolds and will involve applications of techniques from analysis, in particular partial differential equations, and algebraic geometry.In more technical terms, the proposed research project comprises two parts. The goal of the first part is to construct more examples of non-compact complete Calabi-Yau manifolds with Euclidean volume growth and with a singular non-flat tangent cone at infinity, developing the work of Li, the PI and Rochon, and Szekelyhidi. Such manifolds have been used as building blocks for G2-manifolds and may shed light on a conjecture of Yau asserting the compactifiability of a complete Calabi-Yau manifold with finite topology. Moreover, these manifolds have applications in physics such as in the AdS/CFT correspondence and string theory. The second part of the proposal deals with studying complete non-compact gradient Kahler-Ricci solitons. The objectives of this part are threefold.(1) To glue the expanding gradient Kahler-Ricci solitons that the PI has constructed with Deruelle together on a compact Kahler manifold to construct a solution of the Kahler-Ricci flow with singular initial data. This ties in with the analytic Minimal Model Program using the Kahler-Ricci flow proposed by Song and Tian and builds on the work of Gianniotis and Schulze. (2) To construct examples of complete expanding gradient Kahler-Ricci solitons with a singular tangent cone at infinity with Euclidean volume growth of which there are currently no known examples. This builds on the work of the PI and Deruelle.(3) To study the existence and uniqueness of non-compact complete shrinking gradient Kahler-Ricci solitons for a fixed holomorphic vector field. This is a natural extension of work by the PI, Deruelle, and Sun.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

复流形是用复数定义的高维曲面。这样一个空间上的“度规”是一个物体，它赋予空间一个形状，或者更准确地说，是利玛窦曲率。这些度量的研究是复杂几何和几何分析的结合。本课题研究非紧化Kahler流形上具有规定Ricci曲率的完全Kahler度量的构造；更具体地说，是非紧化的Calabi-Yau流形和Kahler-Ricci孤子。这种流形的应用遍及整个物理和数学。事实上，三维复杂的Calabi-Yau流形在弦理论中已经变得很突出，它们被认为是我们看不到的额外的六个真实时空维度的模型，而Kahler-Ricci孤子为一个重要的进化方程中的奇点形成提供了模型，即Kahler-Ricci流。我们将努力构建这些流形的新例子，并根据代数几何准则来表述它们的存在性。这将增强我们对非紧卡勒流形几何的理解，并将涉及分析技术的应用，特别是偏微分方程和代数几何。用更专业的术语来说，拟议的研究项目包括两个部分。第一部分的目标是构造更多具有欧几里得体积增长和无穷远处具有奇异非平坦切锥的非紧完全Calabi-Yau流形的例子，发展Li、PI、Rochon和Szekelyhidi的工作。这样的流形已经被用作g2流形的构建块，并可能阐明Yau的一个猜想，该猜想断言具有有限拓扑的完全Calabi-Yau流形的紧致性。此外，这些流形在物理上也有应用，如AdS/CFT对应和弦理论。第二部分研究完全非紧梯度Kahler-Ricci孤子。本部分的目标有三个方面。(1)将PI与Deruelle构造的膨胀梯度Kahler- ricci孤子粘合在紧致Kahler流形上，构造具有奇异初始数据的Kahler- ricci流的解。这与Song和Tian提出的使用Kahler-Ricci流的解析最小模型程序有关，并以Gianniotis和Schulze的工作为基础。(2)构造在无穷远处具有奇异切锥的完全扩展梯度Kahler-Ricci孤子的欧几里得体积增长的例子，目前还没有已知的例子。这建立在PI和Deruelle的工作基础上。(3)研究了固定全纯向量场的非紧完全收缩梯度Kahler-Ricci孤子的存在唯一性。这是PI、Deruelle和Sun工作的自然延伸。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。