L^2 Curvature Bounds on Manifolds with Bounded Ricci Curvature
Kähler Geometry, Einstein Metrics, and Generalizations March 21, 2016  March 25, 2016
Location: SLMath: Eisenbud Auditorium
algebraic geometry and GAGA
complex differential geometry
mathematical physics
Kahler metric
mirror symmetry
curvature estimates
Ricci curvature lower bounds
geometric analysis
51D25  Lattices of subspaces and geometric closure systems [See also 05B35]
51E14  Finite partial geometries (general), nets, partial spreads
51A40  Translation planes and spreads in linear incidence geometry
51A50  Polar geometry, symplectic spaces, orthogonal spaces
51A25  Algebraization in linear incidence geometry [See also 12Kxx, 20N05]
51A05  General theory of linear incidence geometry and projective geometries
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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