Spectral convergence under bounded Ricci curvature

Abstract

Abstract. For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are sharp generalizations of Cheeger-Colding's spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. These spectral convergence have two direct corollaries. One of them is to give new bounds on such eigenvalues, in terms of bounds on volume, diameter and the Ricci curvature. The other is that we show the upper semicontinuity of the first Betti numbers with respect to the Gromov-Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. On the other hand we also define measurable curvature tensors of the noncollapsed Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a uniform bound of Ricci curvature, which include Riemannian curvature tensor, the Ricci curvature, and the scalar curvature. As fundamental properties of our Ricci curvature, we show that the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli's one and Lott's Ricci measure. Moreover we prove a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott's question on the behavior of the scalar curvature with respect to the Gromov-Hausdorff topology by using our scalar curvature. Contents

Sphere theorems and eigenvalue pinching without positive Ricci curvature assumption

“…Since the standard sphere is smooth, the above proposition also follows from [8,Theorem 7.2]. The question whether we need to assume the upper bound of Ricci curvature in Proposition 6.1 is related to (Q5.4) and (Q5.5) in [26]. We give the following conjecture.…”

“…As with the proof of [26, Proposition 4.8 (ii)], we have ω, dh ∈ H 1,2 (X) for all h ∈ D 2 (∆, X) with ∆h ∈ L ∞ (X). By [26,Proposition 4.5], we get ω ∈ H 1,2 C (T * X). Take arbitrary S = η + he ∈ Test E(X).…”

“…Honda [25] (see also page 1595-1596 of [26]) defined the concepts of L p weakly convergence and L p strong convergence for functions and tensors fields (p ∈ (1, ∞)). Most of important properties are summarized in page 1596-1598 of [26]. Similarly, we give the following definition.…”

“…Take ω i ∈ Γ(T * X i ) and f i ∈ C ∞ (X i ) with T i = ω i + f i e. Since T i ∞ ≤ C(n, K, D, β), we have ω i ∞ ≤ C(n, K, D, β). By [25,Theorem 4.9] and [27,Theorem 6.11] (see also [26,Proposition 4.2]), there exist a subsequence i(j), ω ∈ L 2 (T * X) and f ∈ H 1,2 (X) such that ω is differentiable at almost all point in X, ∇ω ∈ L 2 (T * X ⊗ T * X), ω i , f i converges to ω, f strongly in L 2 and ∇ω i , df i converges to ∇ω, df weakly in L 2 , respectively. By the lower semi-continuity of L ∞ norm [25, Proposition 3.64], we have f ∞ ≤ C(n, K, D, µ), ω ∞ ≤ C(n, K, D, µ) and T ∞ ≤ C(n, K, D, µ).…”

Lecture Notes On Differential Calculus on $\sf {RCD}$ Spaces

“…measured-Gromov-Hausdorff convergence of the base spaces in a quite natural sense. To keep the presentation short we won't discuss this -important and under continuous development -topic, referring to [33], [12], [10] for recent results.…”