Convergence of energy functionals and stability of lower bounds of Ricci curvature via metric measure foliation

Abstract

The notion of the metric measure foliation is introduced by Kell, Mondino, and Sosa in [7]. They studied the relation between a metric measure space with a metric measure foliation and its quotient space. They showed that the curvature-dimension condition and the Cheeger energy functional preserve from a such space to its quotient space. Via the metric measure foliation, we investigate the convergence theory for a sequence of metric measure spaces whose dimensions are unbounded.

Boundedness of precompact sets of metric measure spaces

“…[S,Proposition 5.5], Bakry-Gentil-Ledoux [BGL,Proposition 9.2.4], and Ozawa-Suzuki [OS,Theorem 1.1], cf. [K,Lemma 2.1,Proposition 3.20], and that Y ⊂ P follows from Cavalletti-Mondino [CM,Theorem 1.6]. Then Corollary 1.3 is a direct consequence of Theorem 1.1.…”

Spectral convergence of high-dimensional spheres to Gaussian spaces

“…, where m ∈ N and λ m (Ω, (M, µ)) stands for the mth Dirichlet eigenvalue of −∆ µ on Ω. In the case n = 1, Kazukawa [14,Example 4.18] used a projection from S N to R different from p N 1 and discussed the spectral convergence on S N in the framework of metric measure foliation.…”

“…Theorem 1.5.4],[11, Theorems 6.8, 6.11],[14, Theorem 1.1] and also [2, Theorem 3.4 and Proposition 3.9],[20, Theorem 3.8]. Let {a N } N be a sequence of positive real numbers such that {a N / √ N − 1} N converges to a positive real number α as N…”