The Topology of Open Manifolds with Nonnegative Ricci Curvature

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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.