Singularity Models for Ricci Flow

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AbstractAward: DMS-0505920Principal Investigator: Dan F. KnopfThe project will advance the search for canonical geometries bymeans of geometric heat flows. In light of the landmark progressmade recently by Perelman in Hamilton's program to resolve theGeometrization and Poincare' Conjectures, this research area isundergoing a rapid and productive expansion. The powerfulinnovations and profound insights in Perelman's work contributeto the extraordinary power of Ricci flow as a tool forinvestigating the geometry and topology of Riemannian and complexmanifolds. In virtually all known applications of Ricci flow, itis critical to have a deep understanding of the mechanisms ofsingularity formation. Therefore, the project will investigatefour aspects of singularity formation. These four objectives arechosen to build upon the prior results and current researchprogram of the PI and to be highly relevant to promising newapplications of Ricci flow. The objectives are to study (1)asymptotics of Ricci flow singularity formation, (2) analysis ofRicci flow singularities in dimension four, (3) analysis ofsingularity models for Kaehler-Ricci flow, and (4) the structureof reduced geometry.A manifold is an object that - like our universe - looks likeEuclidean space locally, but whose global topology and geometrymay be much different. The broad goals of this project are tofind optimal geometric structures with which to categorizemanifolds. The methods used are certain partial differentialequations called geometric heat flows. The idea is to let ageometric object evolve in time in such a way that its geometryimproves and simplifies, possibly after a change in topology. Ageometric heat flow called the Ricci flow has just yielded majorbreakthroughs in two of the most difficult open problems inmathematics. These successes provide great incentives to apply itto other challenging open problems and make it a very active andcompetitive field of research. The types of partial differentialequations studied in Ricci flow have much in common with thoseused to model the movement of oil in shale and in thin films,combustion in porous media, heat propagation, avalanches,population dispersal, the spreading of microscopic droplets, andcertain effects in plasma physics. For this reason, methodsdeveloped in this project may have important applications tothose areas of applied mathematics. The project will focus on thedelicate analysis needed to understand such equations as theybecome singular. This analysis should have important broadapplications, among which are the following. (i) The methodsdeveloped, especially asymptotic analysis, should extend to thepractical applications mentioned above. (ii) The project willpromote interdisciplinary interactions with physics, where thereare many applications for geometric classification and flowtechniques. For example, theorists in general relativity want toclassify possible topologies of four-dimensionalspace-times. Researchers in string theory and mirror symmetry areinterested in understanding certain six-dimensionalmanifolds. The Ricci flow itself is an approximation to therenormalization flow for an important model in quantum fieldtheory. (iii) The project will benefit graduate education,because the PI will invest time helping students developexpertise in relevant areas of geometry, analysis, and topology.