Regularity of Einstein manifolds and the codimension 4 conjecture

By Jeff Cheeger, A. Naber. 2014

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In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds (M n ;g) with bounded Ricci curvature, as well as their Gromov-Hausdor limit spaces ( M n ;dj) dGH ! (X;d), where dj denotes the Riemannian distance. Our main result is a solution to the codimension 4 conjecture, namely thatX is smooth away from a closed subset of codimension 4. We combine this result with the ideas of quantitative stratication to prove a priori L q estimates on the full curvaturejRmj for all q v > 0, and diam(M) D contains at most a nite number of dieomorphism classes. A local version is used to show that noncollapsed 4-manifolds with bounded Ricci curvature have a priori L 2 Riemannian curvature estimates.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4e8528d4d198290906656dace26232794d454c7f

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Citations in the corpus

L2 curvature bounds on manifolds with bounded Ricci curvature (2016, methodology): “µN(B3(p)) +C(n,B,v) ≤ C(n,B,v), where the term R supp φN d−3 C comes from the derivative of φ1 in Lemma 3.15. On the other hand, by using the Lp estimates arising from the codimension four theorem of [ChNa15], we know that R B10(p) |∇3u|3/2 ≤ C(n,v). Therefore, by the Ho¨lder inequality we have Z b≤”
Kähler–Einstein metrics along the smooth continuity method (2015, result): “icci curvature. The analogous result with the bound on Ricci curvature replaced by a bound on Ric(ω) − Lvω was shown by Tian-Zhang [47], and it also follows from the more recent work of Cheeger-Naber [16] in the general Riemannian case. With these observations the proof of the partial C0-estimate fo”
Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces. (2019, methodology): “To encode the “closeness information” in analytic terms we rely on the use of harmonic δ-splitting maps, which were introduced in [15] and turned to be an extremely powerful tool in the study of Ricci limits (see [16, 17, 18] and the more recent [20, 19]).”
Convergence of Ricci flows with bounded scalar curvature (2016, background): “any 0 < r < 1, t ∈[−1,0] and x ∈M we have ˆ B(x,t,r) |Rm(·,t)|2−εdg t ≤ ˆ B(x,t,r) rRm(·,t) −4+2ε dgt < Cr n−4+2ε. Note that Theorem 1.7 can be seen as a generalization of the main result of [CN15]. Lastly, we mention that the size of the time-interval [−2,0] in Theorem 1.7 was chosen for t”
Local Sobolev Constant Estimate for Integral Ricci Curvature Bounds (2016, background): “Moreover, integral curvature bounds have recently been discovered in various geometric situations, such as the L(2) bound of the curvature tensor for noncollapsed manifolds with bounded Ricci curvature, and the (almost) L(4) bound of the Ricci curvature for the Kähler-Ricci flow as well as the (real”