Canonical metrics on four dimensional manifolds, and orbifold structures

AbstractAward：DMS 1309029，首席研究员：Ioana Suvaina拟议研究的主要目标是研究四维orbifold空间的黎曼几何，以及它们如何作为光滑流形上的正则度量空间的边界点出现。主要研究者提出分析一个家庭的问题，解决研究的空间常数标量曲率Kaehler曲面。所提出的项目既涉及奇点的局部发展问题，也涉及对紧致光滑空间的理解。首席研究员的目的是建立之间的联系代数几何方面的问题和他们的同行在Kaehler几何。这些项目的决议将提供建设的新的例子常数标量曲率Kaehler表面，或常数标量曲率几乎Kaehler 4-流形。在另一个方向上，主要研究者提出了一系列关于曲率分量的估计，4-orbifold的拓扑和微分不变量以及爱因斯坦度量的存在或不存在之间的关系的项目。四维空间的研究在数学和物理学上都有很长的历史，而时空模型的研究是四维空间研究的动力。在数学中，有强有力的证据表明，四维空间是非常特殊的，因为它们允许无限多的光滑结构。本研究计划计划使用数学中不同领域的技术：微分拓扑，黎曼和凯勒几何，代数几何来研究四维空间。特别是，首席研究员旨在建立这些不同领域之间的新联系。

AbstractAward: DMS 1309029, Principal Investigator: Ioana Suvaina The main goal of the proposed research is the study of Riemannian geometry of orbifold spaces in dimension four, and how they appear as boundary points of the space of canonical metrics on smooth manifolds. The principal investigator proposes to analyze a family of problems addressing the study of the space of constant scalar curvature Kaehler surfaces. The projects proposed concern both problems on the local development of singularities, but also the understanding of compact smooth spaces. The principal investigator aims to establish connections between the algebraic geometry aspects of the problems and their counterparts in Kaehler geometry. The resolution of these projects will provide constructions of new examples of constant scalar curvature Kaehler surfaces, or constant scalar curvature almost Kaehler 4-manifolds. In another direction, the principal investigator proposes a series of projects on the relations between estimates of the curvature components, the topological and differential invariants of a 4-orbifold, and existence or non-existence of Einstein metrics. The main approach is via the Seiberg-Witten theory.The study of the four dimensional spaces has a long history both in mathematics and physics being motivated by the study of the space-time model. In mathematics, there is strong evidence that the four dimensional spaces are very special as they admit infinitely many smooth structures. This research program plans to use techniques from different fields in mathematics: differential topology, Riemannian and Kaehler geometry, and algebraic geometry to study four dimensional spaces. In particular, the principal investigator aims to establish new connections among these various fields.