On the singularities of spaces with bounded Ricci curvature

By Jeff Cheeger, T. Colding, G. Tian. 2002

205 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/9b384c019d715a63e6a34b2296412c3e4c4ded84

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Lower bounds on Ricci curvature and quantitative behavior of singular sets (2011, background): “9 also uses the -regularity theorems of [4] (p = 2), [7] (p ≥ 2) and Theorem 1.”
Kähler–Einstein metrics along the smooth continuity method (2015, methodology): “An additional important fact used several times is that by Cheeger-ColdingTian [15], no tangent cone of the formCγ×C can form in the Gromov-Hausdorff limit of a sequence of Kähler metrics with bounded Ricci curvature.”
Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II (2015, methodology): “By the regularity theory of CheegerColding-Tian [6], we have a decomposition Z = R ⊔ Σ, where R is an open connected smooth manifold endowed with a Kähler-Einstein metric, and Σ is a closed subset of Z with real Hausdorff dimension at most 2n − 4.”
Special test configurations and $K$-stability of Fano varieties (2011, methodology): “olds, even with Donaldson’s deﬁnition, one still only needs to consider those test conﬁgurations with normal central ﬁbers. This is motivated by compactness results for Kahler-Einstein manifolds (See [CCT02]). As an immediate consequence of Corollary 7, we verify Tian’s conjecture. In fact, it suﬃce”
Gromov–Hausdorff limits of Kähler manifolds with Ricci curvature bounded below (2018, background): “In addition since S has codimension 4 (see [4]), we can arrange that on compact sets K we have ‖∇χǫ‖L2(K) → 0 as ǫ → 0.”
The partial $C^0$-estimate along the continuity method (2013, methodology): “In this case, however, there cannot be singularities of the form Cγ × Cn−1 in the Gromov-Hausdorff limit according to Cheeger-Colding-Tian [10].”
Integral bounds on curvature and Gromov–Hausdorff limits (2012, methodology): “ce (Xi,Li,ωi,pi) in K(n,κ), from general theory by passing to a subsequence we obtain a (pointed) Gromov-Hausdorﬀ limit (Z,p), which is a length space. By the regularity theory of CheegerColding-Tian [6], we have a decomposition Z = R ⊔ Σ, where R is an open connected smooth manifold endowed with a ”
Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds (2013, background): “The foundational work of Cheeger-Colding-Tian [8] is a major step towards this conjecture.”
Ricci curvature and monotonicity for harmonic functions (2012, background): “ic if r Z2r r 1 Vol(u = s) Z u=s |II0| (1.5) 2 ds → 0 as r → ∞. A consequence of our monotonicity formulas is that the level sets of u are asymptotically umbilic if M has Euclidean volume growth; cf. [ChCT]. Corollary 1.6. If M has nonnegative Ricci curvature and Euclidean volume growth, i.e., limr→”
L2 curvature bounds on manifolds with bounded Ricci curvature (2016, unknown): “4310/jdg/1214459974 [11] Cheeger, Jeff; Colding, Tobias H.”
Integral bounds on curvature, elliptic estimates and rectifiability of singular sets (2003, methodology): “There, for 1 ≤ p ≤ 2, corresponding results on finiteness of (n − 2p)-dimensional Hausdorff measure were obtained; see [ ChCoTi2 ] for further details.”
Geometric inequalities for manifolds with Ricci curvature in the Kato class (2016, background): “. GEOMETRIC INEQUALITIES AND RICCI CURVATURE IN THE KATO CLASS 3 now sophisticate results about the singular sets of Gromov-Hausdorff limits of Riemannian manifolds under Ricci curvature lower bound ([10, 11]) and the lack of such tools is a major obstacle to a generalization of the result of G. Tia”
Smooth and singular Kahler-Einstein metrics (2014, unknown): “One idea is to notice that a rescaled limit will be Ricci flat and then use results of [57, 63, 248].”
Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces (2017, background): “Another situation in which the singular set can be shown to be empty in three-dimensions is when one assumes, in addition, L control on the curvature tensor, for appropriately large p [8].”