Kahler Manifolds with Prescribed Ricci Curvature

复流形是使用复数定义的高维曲面。在这样一个空间上的“度规”是一个赋予空间一个形状的对象，或者更准确地说，是一个赋予空间一个里奇曲率的对象。这些度量的研究是复几何与几何分析的结合点。这个项目关注的是在非紧Kahler流形上构造具有指定Ricci曲率的完备Kahler度量;更具体地说，非紧Calabi-Yau流形和Kahler-Ricci孤子。这种流形的应用渗透在物理学和数学中。事实上，三维复维的卡-丘流形在弦论中已经变得非常突出，因为它们被认为是我们看不到的时空的另外六个真实的维的模型，而卡勒-里奇孤子则为一个重要的演化方程（即卡勒-里奇流）中奇点的形成提供了模型。将努力构建这些流形的新的例子，并制定其存在的代数几何标准。这将提高我们对非紧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。第一部分的目标是构造更多的例子，非紧完备卡-丘流形与欧几里德体积增长和奇异非平坦切锥在无穷远，发展工作的李，PI和Rochon，和Szekelyhidi。这样的流形已被用作G2-流形的构建块，并可能揭示了一个猜想的丘断言紧的一个完整的卡-丘流形有限拓扑。此外，这些流形在物理学中也有应用，例如AdS/CFT对应和弦理论。第二部分研究完全非紧梯度Kahler-Ricci孤子。这一部分的目标有三个方面。(1)在紧致Kahler流形上将PI利用Deruelle构造的扩展梯度Kahler-Ricci孤子粘在一起，构造了具有奇异初值的Kahler-Ricci流的解.这与宋和田提出的使用Kahler-Ricci流的分析最小模型程序有关，并建立在Gianniotis和Schulze的工作基础上。(2)构造具有无穷远奇异切锥的完全扩展梯度Kahler-Ricci孤子的例子，其具有欧几里得体积增长，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NSF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。

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.