Weyl’s law on $RCD^{\ast} (K, N)$ metric measure spaces

Zhang, Hui-Chun; Zhu, Xi-Ping

doi:10.4310/cag.2019.v27.n8.a8

Cited by 16 publications

(5 citation statements)

References 43 publications

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“…Using the Gaussian estimates (Theorem 2.6), one can also obtain the following estimates as in smooth contexts. See also [ZZ19].…”

Section: (Solution To the Heat Equation)mentioning

confidence: 99%

Isometric immersions of RCD$(K,N)$ spaces via heat kernels

Huang¹

2022

Preprint

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For an RCD(K, N ) space (X, d, m), one can use its heat kernel ρ to embed it into L 2 (m) by a locally Lipschitz map Φ t (x) := ρ(x, •, t). In particular, an RCD(K, N ) space is said to be an isometric heat kernel immersing space, if its associated Φ t is an isometric immersion multiplied by a constant depending on t for any t > 0. We prove that any compact isometric heat kernel immersing RCD(K, N ) space is isomorphic to an unweighted closed smooth Riemannian manifold. More generally, it is proved that any noncollapsed RCD(K, N ) space with an isometrically immersing eigenmap is also isomorphic to an unweighted closed smooth Riemannian manifold. As an application, we give some diffeomorphic finiteness theorems for a certain class of Riemannian manifolds with a curvature-dimension-diameter bound and an almost isometrically immersing eigenmap.

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“…Using the Gaussian estimates (Theorem 2.6), one can also obtain the following estimates as in smooth contexts. See also [ZZ19].…”

Section: (Solution To the Heat Equation)mentioning

confidence: 99%

Isometric immersions of RCD$(K,N)$ spaces via heat kernels

Huang¹

2022

Preprint

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“…Recently, Weyl's law for eigenvalues of the Laplacian operator on compact RCD(K, N ) spaces has been proved (see [4] [43]). Applying Corollary 4.4 in [4] to the RCD(n−2, n−1) space (X, d X , m X ), we can prove the following result.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature

Huang¹

2021

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Suppose (M, g) is a Riemannian manifold having dimension n, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity C(X) is an Euclidean cone over the cross-section X. Denote by α = limr→∞ Vol(Br (p)) r n the asymptotic volume ratio. Let h k = h k (M ) be the dimension of the space of harmonic functions with polynomial growth of growth order at most k. In this paper, we prove a upper bound of h k in terms of the counting function of eigenvalues of X. As a corollary, we obtain lim k→∞ k 1−n h k = 2α (n−1)!ωn . These results are sharp, as they recover the corresponding well-known properties of h k (R n ). In particular, these results hold on manifolds with nonnegative sectional curvature and maximal volume growth.

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“…Friedland and Hayman [9, Theorem 2] proved that the positive root ν N (s) of the equation [5, p.218]). In general, the spectral convergence with respect to the pointed measured Gromov-Hausdorff topology under the curvature-dimension condition is known (for instance, see [1,2,11,20] and the references therein). With respect to the pointed measured Gromov-Hausdorff topology, although…”

Section: Introductionmentioning

confidence: 99%

“…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…”

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confidence: 99%

Spectral convergence of high-dimensional spheres to Gaussian spaces

Takatsu¹

2021

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We prove that the spectral structure on the N -dimensional standard sphere of radius (N − 1) 1/2 compatible with a projection onto the first n-coordinates converges to the spectral structure on the n-dimensional Gaussian space with variance 1 as N → ∞. We also show the analogue for the first Dirichlet eigenvalue and its eigenfunction on a ball in the sphere and on a half-space in the Gaussian space. 2 2α 2 dx.

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Weyl’s law on $RCD^{\ast} (K, N)$ metric measure spaces

Cited by 16 publications

References 43 publications

Isometric immersions of RCD$(K,N)$ spaces via heat kernels

Isometric immersions of RCD$(K,N)$ spaces via heat kernels

Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature

Spectral convergence of high-dimensional spheres to Gaussian spaces

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