Stability of Cheng-type rigidity in RCD*(K,N) spaces
Establish a stability version of the rigidity in Cheng’s eigenvalue comparison (Theorem Cheng dir-Rigid) for RCD*(K,N) metric measure spaces. Specifically, determine whether for every epsilon > 0 there exists delta > 0 such that if (X, d, μ) is an RCD*(K, N) space containing a ball B(x0, r) with r in (r0 − delta, r0 + delta) and satisfying λ1^D(B(x0, r)) ≥ λ_{K, N, r0} − delta, then the pointed measured space (X, d, μ, x0) is epsilon-close in the pointed measured Gromov–Hausdorff (pmGH) sense to one of the three rigid configurations described in Theorem Cheng dir-Rigid (one-dimensional interval model, one-dimensional manifold, or local (K, N)-cone over an RCD*(N−2, N−1) space).
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Does stability hold in Theorem \ref{th: Cheng dir-Rigid}?\ More precisely, is it true that for every $\epsilon>0$ there exists $\delta>0$ such that if $(X,d,)$ is an $RCD{\star}(K,N)$ space, containing a ball $B(x_0, r)$, with $r\in (r_0-\delta, r_0+\delta)$, such that $\lambda_1D(B(x_0,r))\geq \lambda_{K,N,r_0} -\delta$, then $(X,d,, x_0)$ is necessarily $\epsilon$-close in pmGH sense to one among the rigid configurations 1.\;2.\;or 3.\;in Theorem \ref{th: Cheng dir-Rigid}? A challenge in establishing such a result is that, in general, the Dirichlet spectrum is not continuous under pmGH convergence (see for necessary and sufficient conditions). The stability in Cheng's inequality seems to be an interesting open question already in the framework of smooth Riemannian manifolds; however, in the latter setting, the stability should follow from Cheeger-Colding's estimates .