L^2 Curvature Bounds on Manifolds with Bounded Ricci Curvature
Kähler Geometry, Einstein Metrics, and Generalizations March 21, 2016  March 25, 2016
Location: MSRI: Simons Auditorium
algebraic geometry and GAGA
complex differential geometry
mathematical physics
Kahler metric
mirror symmetry
curvature estimates
Ricci curvature lower bounds
geometric analysis
53C25  Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C20  Global Riemannian geometry, including pinching [See also 31C12, 58B20]
14468
Consider a Riemannian manifold with bounded Ricci curvature Ric\leq n1 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 GHlimit of noncollapsed manifolds with bounded Ricci curvature, then the singular set S(X) is n4 rectifiable with the uniform Hausdorff measure estimates H^{n4}(S(X)\cap B_1)<C(n,v), which in particular proves the n4finiteness conjecture of CheegerColding. We will see as a consequence of the proof that for n4 a.e. x\in S(X) that the tangent cone of X at x is unique and isometric to R^{n4}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
14468
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