Logo

Mathematical Sciences Research Institute

Home » Workshop » Schedules » L^2 Curvature Bounds on Manifolds with Bounded Ricci Curvature

L^2 Curvature Bounds on Manifolds with Bounded Ricci Curvature

Kähler Geometry, Einstein Metrics, and Generalizations March 21, 2016 - March 25, 2016

March 24, 2016 (09:30 AM PDT - 10:30 AM PDT)
Speaker(s): Aaron Naber (Northwestern University)
Location: MSRI: Simons Auditorium
Tags/Keywords
  • algebraic geometry and GAGA

  • complex differential geometry

  • mathematical physics

  • Kahler metric

  • mirror symmetry

  • curvature estimates

  • Ricci curvature lower bounds

  • geometric analysis

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

Abstract

Consider a Riemannian manifold with bounded Ricci curvature |Ric|\leq n-1 and the noncollapsing lower volume bound Vol(B_1(p))>v>0.  The first main result of this paper is to prove the previously conjectured L^2 curvature bound \fint_{B_1}|\Rm|^2 < C(n,v).  In order to prove this, we will need to first show the following structural result for limits.  Namely, if (M^n_j,d_j,p_j) -> (X,d,p) is a GH-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set S(X) is n-4 rectifiable with the uniform Hausdorff measure estimates H^{n-4}(S(X)\cap B_1)<C(n,v), which in particular proves the n-4-finiteness conjecture of Cheeger-Colding.  We will see as a consequence of the proof that for n-4 a.e. x\in S(X) that the tangent cone of X at x is unique and isometric to R^{n-4}xC(S^3/G_x) for some G_x\subseteq O(4) which acts freely away from the origin.  The proofs involve several new estimates on spaces with bounded Ricci curvature. This is joint work with Wenshuai Jiang

Supplements No Notes/Supplements Uploaded
Video/Audio Files

14468

H.264 Video 14468.mp4 337 MB video/mp4 rtsp://videos.msri.org/data/000/025/596/original/14468.mp4 Download
Buy the DVD

If none of the options work for you, you can always buy the DVD of this lecture. The videos are sold at cost for $20USD (shipping included). Please Click Here to send an email to MSRI to purchase the DVD.

See more of our Streaming videos on our main VMath - Streaming Video page.