Stability of eigenvalues and observable diameter in RCD\((1, \infty)\) spaces. (English) Zbl 1502.53069
This paper studies the stability of the spectral gap and observable diameter for metric measure spaces satisfying the RCD\((1,\infty)\) condition. A metric measure space \((M, d, \mu)\) satisfies the RCD\((K,\infty)\) condition (Riemannian curvature-dimension condition) for some \(K\in{\mathbb{R}}\) if the relative entropy is \(K\)-convex along \(W_2\)-geodesics on \({\mathcal{P}}_2(M)\), the space of Borel probability measures on \(M\) with finite second-order moment, and satisfies a quadratic form of Cheeger energy.
The authors show that if such a space has an almost maximal spectral gap, then it almost contains a Gaussian component, and the Laplacian has eigenvalues that are close to any integers, with dimension-free quantitative bounds. Under the additional assumption that the space admits a needle disintegration, they prove that the spectral gap is almost maximal if and only if the observable diameter is almost maximal, again with quantitative dimension-free bounds.
The authors show that if such a space has an almost maximal spectral gap, then it almost contains a Gaussian component, and the Laplacian has eigenvalues that are close to any integers, with dimension-free quantitative bounds. Under the additional assumption that the space admits a needle disintegration, they prove that the spectral gap is almost maximal if and only if the observable diameter is almost maximal, again with quantitative dimension-free bounds.
Reviewer: Shi-Zhong Du (Shantou)
MSC:
| 53C24 | Rigidity results |
| 58C40 | Spectral theory; eigenvalue problems on manifolds |
| 60F05 | Central limit and other weak theorems |
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