Convergence of Riemannian manifolds; Ricci and $L\sp {n/2}$-curvature pinching

Abstract

In this paper we consider classes of Riemannian manifolds (M, g) with bounds on the five fundamental geometric invariants: Ricci curvature Ric,L" /2 -norm of curvature / |Rm|" /2 ί/g, injectivity radius inj(g), diameter diam(g) and volume Vol((7).Let G(H, / 0 , K, n) denote the collection of all closed, connected ndimensional Riemannian manifolds (M, g) satisfying |Ric| < H, inj(g) >/ 0 >0,and f M \Rm(g)\ n/2 dg<K. One of the principal aims of this study is to understand the compactness property of G(H, / 0 , K, n). The well-known and fundamental result of Gromov states that the collection of all closed, connected /7-dimensional Riemannian manifolds (M, g) with | Rm | < 1, diam < D and Vol > V is precompact among " C lc *-Riemannian manifolds" [17], [12], [27].Our first result says that we have a similar compactness result in G(H < i o , K, ή), which roughly states that: Given a sequence {M k } of compact Riemannian ^-manifolds with | Ric | < H, diam < D, J|Rm|" /2 dg < K, and inj > z' o < 0, then {M k } has a subsequence, away from finite number points, which converges to an ^-manifold M with C lα metric g for an a e (0, 1). The precise statement is the following (n > 4).Theorem 0.1. Let {{M k , g k )} be a sequence of Riemannian manifolds in G{H, i 0 , K, ή), with diam(A/ A: ) < D. Then there exist a subsequence of {{M k , g k )} (by renumbering, we still use {M k , g k }), and a sequence {rj, r ι -• 0 when I -> oc, such that the following hold:(a) There exists a C°°-manifold M, such that M k is diffeomorphic to M for each large k .(b) There exist a C 00 -metric g on M, and finite number points {m f ,-• , m h ] with h < C{H, i 0 , K, n), such that g is a C 1 '°-metric on M -{m { , , m k }, 0 < oc < 1.

Manifolds with small Dirac eigenvalues are nilmanifolds

“…The lower bound on the volume together with the upper bounds on | sec | and diam imply a lower bound on the injectivity radius of (M, g). Then from [29, Theorem 6.1] (see also [5], [21]), we have that for λ 1 < ε(n, v, K, D), M is C 1,α -close to a C ∞ Einstein manifold M with Ric ≡ 0. As M is diffeomorphic to M if M and M are C 1,α -close, we may assume in the following that M and M are equipped with the same spin structure.…”

A Concentration‐Collapse Decomposition for L2 Flow Singularities

“…First we show finiteness theorems in dimensions three and four related to theorems of Petersen-Wei [32], AndersonCheeger [3] and Gao [20]. In particular, in the context of small L 2 curvature, our result replaces the pointwise Ricci curvature hypothesis of these results with the weaker lower volume growth bound.…”

“…Again, prior results of this type have appeared in for instance [3,20,32], and all require bounds on a "supercritical" curvature quantity. Theorem 1.21.…”

“…Many precursors exist for this type of result ( [3,18,20,29,32,43,44]), but all rely on choosing the L n 2 norm sufficiently small with respect to other curvature bounds. At the heart of these techniques is usually some form of elliptic theory/Moser iteration, which requires a "supercritical" L p bound to get started.…”

Some Rigidity Phenomena for Einstein Metrics

“…The following argument is quite standard (see [A,DY,Gl,G2]). However, in our situation, it is unnecessary to use Moser's iteration to get L°° estimates for the curvature tensor.…”