The Topology of Open Manifolds with Nonnegative Ricci Curvature

NSF提案DMS - 0102279非负Ricci曲率开放流形的拓扑。Sormani提出研究具有非负Ricci曲率的完全流形及其极限空间的拓扑结构。特别是，她计划研究Milnor猜想的各种方法，即具有非负环曲率的开流形的基本群是有限生成的。她还计划研究这些空间的高维同调。将使用的技术包括Gromov-Hausdorff极限，Cheeger-Colding的几乎刚性理论和Busemann函数。特别地，我们将研究Busemann函数在这些流形上的正确性。应该指出的是，这个项目在广义相对论的拓扑审查理论中有直接的应用。时空上非负里奇曲率的条件称为零能量条件，它出现在爱因斯坦方程中。粗略地说，索尔马尼博士建议在一个没有边界的空间中研究洞的存在和流行，这个空间延伸到无限，并且在弯曲的方式上有一个条件。我们生活的宇宙就是这样一个空间。更简单的例子是圆柱体（即管）和抛物面（即碗）。圆柱体有孔，但抛物面没有孔。在这个项目中研究的空间是任意尺寸的，所以洞也有不同的尺寸。宇宙就是这样一个高维空间，它的洞，可能存在也可能不存在，通常被称为虫洞。通过加深我们对这个几何问题的理解，我们有望加深我们对宇宙的理解。

Abstract for NSF Proposal DMS - 0102279The Topology of Open Manifolds with Nonnegative Ricci CurvatureChristina SormaniDr. Sormani proposes to study the topology of complete manifolds withnonnegative Ricci curvature and their limit spaces. In particular sheplans to investigate various approaches to Milnor's conjecture thatthe fundamental group of an open manifold with nonnegative Riccicurvature is finitely generated. She also plans to study the higherdimensional homology of these spaces. Techniques which will beemployed involve Gromov-Hausdorff limits, the almost rigidity theoryof Cheeger-Colding, and Busemann functions. In particular, theproperness of Busemann functions on these manifolds will beinvestigated. It should be noted that there are direct applicationsof this project to the theory of topological censorship in generalrelativity. The condition of nonnegative Ricci curvature onspace-time is called the null energy condition and it arises in theEinstein equation.Roughly speaking, Dr. Sormani proposes to study the existence andprevalence of holes in a space which has no boundary, extends toinfinity and has a condition imposed upon the way in which it canbend. The universe we live in is such a space. Simpler examples arecylinders (i.e. tubes) and paraboloids (i.e. bowls). The cylinder hasa hole but the paraboloid does not. The spaces studied in thisproject are of arbitrary dimension and so the holes come in variousdimensions as well. The universe is one such higher dimensional spaceand its holes, which may or may not exist, are often called wormholes.By furthering our understanding of this geometric problem, it is hopedthat we will further our understanding of the universe.