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Martens, Adam

University of British Columbia

In this dissertation, we study the Ricci flow and its applications on manifolds that have small enough scale-invariant integral curvature (so-called "curvature concentration") relative to its Sobolev bounds. Our primary results are summarized in the following three items. (1) We show that if the bounds are assumed to be global for a bounded curvature manifold, then the Ricci flow emanating from this initial data exists for all time, with faster than critical curvature decay. We also show topological rigidity assuming that the curvature is globally sub-critically integrable - an assumption that is, for example, satisfied by the well-studied family of "asymptotically flat" manifolds. (2) We sharpen a gap theorem of Chan and Lee for nonnegative Ricci curvature manifolds (possibly unbounded curvature) that have positive asymptotic volume ratio (in the nonnegative Ricci curvature setting, we show that volume bounds are equivalent to Sobolev bounds) and small enough curvature concentration, by showing that the curvature concentration need only depend linearly on the asymptotic volume ratio. We prove the result by exhibiting a long-time Ricci flow solution with faster than critical curvature decay (item (1) above is used as one element of the proof), which allows us to shift the limiting contradiction argument to time infinity and thus obtain an explicit bound on the size of the gap. (3) In recent work of Chan-Huang-Lee, it is shown that if a (possibly unbounded curvature) manifold enjoys uniform bounds on the negative part of the scalar curvature, the local entropy, and volume ratios up to a fixed scale, then there exists a Ricci flow for some definite time with estimates on the solution assuming that the local curvature concentration is small enough initially. We show that the bound on scalar curvature assumption above is redundant. We also give some applications of this quantitative short-time existence, including Ricci flow smoothing for measure space limits, a Gromov-Hausdorff compactness result, and topological/geometric rigidity in the case that the a priori local bounds are strengthened to be global.

Text

2025-06-13

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/91357

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/