Aaron Naber

Bio: Aaron Naber is an academic researcher from Northwestern University. The author has contributed to research in topics: Ricci curvature & Bounded function. The author has an hindex of 24, co-authored 68 publications receiving 2018 citations. Previous affiliations of Aaron Naber include Pennsylvania State University & Massachusetts Institute of Technology.

Noncompact shrinking four solitons with nonnegative curvature

Abstract: Abstract We prove that if (M, g, X) is a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M, g) is isometric to or a finite quotient of or S 3 × ℝ. In the process we also show that a complete shrinking soliton (M, g, X) with bounded curvature is gradient and κ-noncollapsed and the dilation of a Type I singularity is a shrinking soliton. Further in dimension three we show shrinking solitons with bounded curvature can be classified under only the assumption of Rc ≧ 0. The proofs rely on the technical construction of a singular reduced length function, a function which behaves as the reduced length function but can be extended to singular times.

Sharp Holder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

Abstract: We prove a new estimate on manifolds with a lower Ricci bound which asserts that the geometry of balls centered on a minimizing geodesic can change in at most a Holder continuous way along the geodesic. We give examples that show that the Holder exponent, along with essentially all the other consequences that follow from this estimate, are sharp. Among the applications is that the regular set is convex for any non- collapsed limit of Einstein metrics. In the general case of a potentially collapsed limit of manifolds with just a lower Ricci curvature bound we show that the regular set is weakly convex and a.e. convex. We also show two conjectures of Cheeger-Colding. One of these asserts that the isometry group of any, even collapsed, limit of manifolds with a uniform lower Ricci curvature bound is a Lie group. The other asserts that the dimension of any limit space is the same everywhere.

Regularity of Einstein manifolds and the codimension 4 conjecture

Abstract: 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.

Lower bounds on Ricci curvature and quantitative behavior of singular sets

Abstract: 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

Structure theory of metric measure spaces with lower Ricci curvature bounds

Abstract: We prove that a metric measure space (X,d,m) satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space W1,2 is Hilbert is rectifiable. That is, a RCD∗(K,N)-space is rectifiable, and in particular for m-a.e. point the tangent cone is unique and euclidean of dimension at most N. The proof is based on a maximal function argument combined with an original Almost Splitting Theorem via estimates on the gradient of the excess. We also show a sharp integral Abresh–Gromoll type inequality on the excess function and an Abresh–Gromoll-type inequality on the gradient of the excess. The argument is new even in the smooth setting

Comparison geometry for the Bakry-Emery Ricci tensor

Abstract: For Riemannian manifolds with a measure (M, g, edvolg) we prove mean curvature and volume comparison results when the ∞-Bakry-Emery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results.