Lower bounds on Ricci curvature and quantitative behavior of singular sets

By Jeff Cheeger, A. Naber. 2011

149 citations recorded by the source corpus.

Let Yn denote the Gromov-Hausdorff limit $M^{n}_{i}\stackrel{d_{\mathrm{GH}}}{\longrightarrow} Y^{n}$ of v-noncollapsed Riemannian manifolds with ${\mathrm{Ric}}_{M^{n}_{i}}\geq-(n-1)$. The singular set $\mathcal {S}\subset Y$ has a stratification $\mathcal {S}^{0}\subset \mathcal {S}^{1}\subset\cdots\subset \mathcal {S}$, where $y\in \mathcal {S}^{k}$ if no tangent cone at y splits off a factor ℝk+1 isometrically. Here, we define for all η>0, 0<r≤1, the k-th effective singular stratum$\mathcal {S}^{k}_{\eta,r}$ satisfying $\bigcup_{\eta}\bigcap_{r} \,\mathcal {S}^{k}_{\eta,r}= \mathcal {S}^{k}$. Sharpening the known Hausdorff dimension bound $\dim\, \mathcal {S}^{k}\leq k$, we prove that for all y, the volume of the r-tubular neighborhood of $\mathcal {S}^{k}_{\eta,r}$ satisfies ${\mathrm {Vol}}(T_{r}(\mathcal {S}^{k}_{\eta,r})\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},\eta)r^{n-k-\eta}$. The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let $\mathcal {B}_{r}$ denote the set of points at which the C2-harmonic radius is ≤r. If also the $M^{n}_{i}$ are Kähler-Einstein with L2 curvature bound, $\| Rm\|_{L_{2}}\leq C$, then ${\mathrm {Vol}}( \mathcal {B}_{r}\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},C)r^{4}$ for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the $M^{n}_{i}$, we obtain a slightly weaker volume bound on $\mathcal {B}_{r}$ which yields an a priori Lp curvature bound for all p<2. The methodology developed in this paper is new and is applicable in many other contexts. These include harmonic maps, minimal hypersurfaces, mean curvature flow and critical sets of solutions to elliptic equations.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b7e83df83a9db3e0ec1ba82f266e461df9ebd19f

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Citations in the corpus

Quantitative Stratification and the Regularity of Mean Curvature Flow (2012, methodology): “In the quantitative stratification, introduced recently by CheegerNaber in the elliptic setting [4, 5] and developed and applied now in the parabolic setting by Cheeger-Haslhofer-Naber [2, 3], we replace the singular strata Sj by quantitative singular strata Sj ⌘,r (⌘ > 0, 0 < r < 1).”
Critical Sets of Elliptic Equations (2012, methodology): “ of u(y) −u(x) is a function of at least n−k variables. For instance, if u has nonvanishing gradient at x, then the leading order polynomial is linear and therefore x ∈ Sn−1. 3 In a manner similar to [2] and [3], we will generalize the standard stratiﬁcation to a quantitative stratiﬁcation. Very rou”
On the structure of almost einstein manifolds (2012, methodology): “10 of [12] to obtain that |B(x̄, 2) ∩ Vr| ≤ C(n, κ, η)r4−η.”
Gromov–Hausdorff limits of Kähler manifolds with Ricci curvature bounded below (2018, background): “Cheeger-Colding [4] showed that the Hausdorff dimension of Y \ R is at most m− 2, with more quantitative estimates obtained by Cheeger-Naber [9].”
Quantitative Stratification and the Regularity of Harmonic Maps and Minimal Currents (2011, background): “one-splitting Lemma A cone-splitting principle for tangent cones at points of limit spaces with a deﬁnite lower Ricci curvature bound, together with its quantitative reﬁnement, ﬁgur ed prominently in [CN11]. An analogous conesplitting principle for harmonic maps plays a key role here. Cone-splitting”
Regularity of Einstein manifolds and the codimension 4 conjecture (2014, background): “, w hich was ﬁrst introduced in [ChCo1], and which will play an important role in the proof of Theorem 1.1. The effective version, called the quantitative stratiﬁcation , which was ﬁrst introduced in [ChNa13], will be recalled in S ection 7. It will play an important role in the estimates of Theorem”
Space of Ricci Flows I (2009, methodology): “Since all these properties hold on K̃ S (n, κ), the proof follows that of [21] verbatim.”
On the topology and the boundary of N–dimensional RCD(K,N) spaces (2019, methodology): “et is reasonably small. We will make use of the following generalization to non-collapsed RCD(K,N) spaces due to Antonelli, Bru´eandSemola[ABS19] ofthe quantitativestratiﬁcationresultofCheegerandNaber[CN13], originally proved for smooth Riemannian manifolds with Ricci and volume bounded below. Theor”
Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps (2015, background): “r hold, everything else can be seen to be corollaries of these statements. A ﬁrst version of the quantitative stratiﬁcation can be foun d in [Alm00, section 2.25]. This concept was later developed in [CN13a] with the goal of proving new estimates on noncollapsed manifolds with Ricci curvature bounde”
New Logarithmic Sobolev Inequalities and an ɛ‐Regularity Theorem for the Ricci Flow (2012, background): “x//=!nr/ going back to Anderson [1], which is at the backbone of the regularity theory in [6].”