Kahler Manifolds with Prescribed Ricci Curvature

复流形是使用复数定义的高维曲面。这样的空间上的“度量”是指赋予该空间一个形状的物体，更准确地说，是具有Ricci曲率的物体。对这些指标的研究是复杂几何和几何分析的结合点。这个项目是关于在非紧的Kahler流形上，更具体地说，非紧的Calabi-Yau流形和Kahler-Ricci孤子上，构造具有给定Ricci曲率的完备的Kahler度量。这种流形的应用渗透到整个物理和数学领域。事实上，三维复数维的Calabi-Yau流形在弦理论中已经变得非常重要，在弦理论中，它们被认为是我们看不到的六个额外的实数维时空的模型，而Kahler-Ricci孤子提供了在一个重要的演化方程中形成奇点的模型，即Kahler-Ricci流。我们将努力构造这些流形的新例子，并用代数几何准则来表示它们的存在。这将加深我们对非紧Kahler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。第一部分的目的是构造更多的具有欧氏体积增长且在无穷远处具有奇异非平坦切锥的非紧致完备Calabi-Yau流形的例子，发展了Li、PI和Rochon以及Szekelyidi的工作。这种流形已被用作G2-流形的积木，并可能阐明Yau关于有限拓扑完备的Calabi-Yau流形的紧致性的猜想。此外，这些流形在物理中也有应用，例如在ADS/CFT对应和弦理论中。该方案的第二部分涉及研究完全非紧致梯度Kahler-Ricci孤子。这一部分的目的有三个：(1)将PI与Deruelle构造的扩展梯度Kahler-Ricci孤子粘合在紧致Kahler流形上，构造具有奇异初值的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相结合，并建立在Gianniotis和Schulze的工作基础上。(2)构造目前尚无已知例子的欧几里德体积增长的完全扩张梯度Kahler-Ricci孤子的例子。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。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.