Geometric and spectral estimates based on spectral Ricci curvature assumptions

Abstract

We obtain a Bonnet–Myers theorem under a spectral condition: a closed Riemannian {(M^{n},g)} manifold for which the lowest eigenvalue of the Ricci tensor ρ is such that the Schrödinger operator {\Delta+(n-2)\rho} is positive has finite fundamental group. Further, as a continuation of our earlier results, we obtain isoperimetric inequalities from Kato-type conditions on the Ricci curvature. We also obtain the Kato condition for the Ricci curvature under purely geometric assumptions.

Lower bounds for the spectral gap and an extension of the Bonnet-Myers theorem

“…For manifolds with Ricci curvature that can have an amount of negative part, Aubry [4] proved a lower bound in terms of the dimension n, vol(M) and the quantity on the left-hand side of (1.6) below. In [11] Carron and Rose proved some lower bound for λ 1 using Kato type assumptions on ρ. Our lower bounds in Proposition 1.1 (see also Theorem 4.4) are of different nature and are expressed in terms of the mean of ρ on balls of the manifold.…”

“…holds for some positive T and k. He also proved diameter estimate under a Kato type condition by relying on Carron and Rose [11] who obtain diameter estimates for compact manifold using positivity of certain Schrödinger operators with potential involving ρ and some constants. Our condition on ρ in Theorem 1.2 is not in the same spirit as the conditions in the previous works.…”

“…As mentioned in the introduction we have made no attempt to obtain a diameter estimate for M. Such estimates were obtained from L p -type conditions on ρ in [25] and [4] or from a Kato type property in [11] and [26]. We refer to these papers for additional information and references on related works.…”

Bounds on the First Betti Number: An Approach via Schatten Norm Estimates on Semigroup Differences

“…A starting point for our analysis is the paper [9], where a criterion is formulated under which b 1 (M) = 0. The results of [9] were later generalized and put into a more quantitative form in [26] and [4], respectively (for some related work see also the literature cited in these two papers). The new feature of our main results can easily be explained: Roughly speaking, we employ methods from Functional Analysis and Operator Theory that originated in Mathematical Physics.…”

Heat kernel bounds and Ricci curvature for Lipschitz manifolds

“…The prominent role of the Kato class in mathematical physics has been well established in the past decades in a series of works [2,32,49]. Recently, its interest in the context of singular Ricci bounds has grown as well [5,6,7,9,18,24,26,43] (also with applications to Ricci flow and general relativity). The study of the class of LR manifolds in this latter setting constitutes the last part of our work.…”