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Register a new userStability of eigenvalues and observable diameter in RCD$(1,infty)$ spaces
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We study stability of the spectral gap and observable diameter for metricmeasure spaces satisfying the RCD(1, $infty$) condition. We 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, we show that the spectral gap is almost maximal iff the observable diameter is almost maximal, again with quantitative dimension-free bounds.
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In this note we prove in the nonlinear setting of $CD(K,infty)$ spaces the stability of the Krasnoselskii spectrum of the Laplace operator $-Delta$ under measured Gromov-Hausdorff convergence, under an additional compactness assumption satisfied, for instance, by sequences of $CD^*(K,N)$ metric measure spaces with uniformly bounded diameter. Additionally, we show that every element $lambda$ in the Krasnoselskii spectrum is indeed an eigenvalue, namely there exists a nontrivial $u$ satisfying the eigenvalue equation $- Delta u = lambda u$.
In this paper we study the family of embeddings $Phi_t$ of a compact $RCD^*(K,N)$ space $(X,d,m)$ into $L^2(X,m)$ via eigenmaps. Extending part of the classical results by Berard, Berard-Besson-Gallot, known for closed Riemannian manifolds, we prove convergence as $tdownarrow 0$ of the rescaled pull-back metrics $Phi_t^*g_{L^2}$ in $L^2(X,m)$ induced by $Phi_t$. Moreover we discuss the behavior of $Phi_t^*g_{L^2}$ with respect to measured Gromov-Hausdorff convergence and $t$. Applications include the quantitative $L^p$-convergence in the noncollapsed setting for all $p<infty$, a result new even for closed Riemannian manifolds and Alexandrov spaces.
Given a compact connected set $E$ in the unit disk $mathbb{B}^2$, we give a new upper bound for the conformal capacity of the condenser $(mathbb{B}^2, E),$ in terms of the hyperbolic diameter $t$ of $E$. Moreover, for $t>0$ we construct a set of diameter $t$ and show by numerical computation that it has larger capacity than a hyperbolic disk with the same diameter. The set we construct is of constant hyperbolic width equal to $t$, the so called hyperbolic Reuleaux triangle.
For every n, we construct a metric measure space that is doubling, satisfies a Poincare inequality in the sense of Heinonen-Koskela, has topological dimension n, and has a measurable tangent bundle of dimension 1.
In this paper, we will prove the Weyls law for the asymptotic formula of Dirichlet eigenvalues on metric measure spaces with generalized Ricci curvature bounded from below.
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