Convergence of Riemannian manifolds and Laplace operators. I
type: Item
instance of: scholarly article
title: Convergence of Riemannian manifolds and Laplace operators. I
zbMATH Open document ID: 1040.53053
author: Atsushi Kasue
publication date: 2002-09-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116008
review text: Let \(\mathcal F=\{M\}\) be a family of compact, connected Riemannian manifolds. Let \({\Delta}_M\), \(P_{M,t}=e^{-t {\Delta}_M}\) and \(p_M(t, x, y)\) respectively denote the Laplace operator, the heat semigroup and the heat kernel of \(M\). Assume there exist positive constants \(\nu\) and \(C\) such that for any \(M\in {\mathcal F}\), \[ p_M(t, x, x){\leq} Ct^{-\frac {\nu}{2}}, \quad t{\in}(0, 1], \quad x{\in}M. \] In an earlier paper the present author and \textit{H. Kumura} [Tôhoku Math. J., II. Ser. 48, 71--120 (1996; Zbl 0883.58100)] proved i) The family \({\mathcal F}\) is precompact with respect to both the spectral distance SD and the Gromov-Hausdorff distance HD. ii) Let \(\{M_n\}\) be an SD-Cauchy sequence in \({\mathcal F}\), then there exists a compact, connected Hausdorff space \(X\), a nonnegative Radon measure \({\mu}_X\) on \(X\) and a Dirichlet form \((\varepsilon_X, D_{[{\varepsilon}_X]})\) defined on \(L^2(X, {\mu}_X)\), such that the heat semigroup \(P_{X, t}\) has a continuous heat kernel \(p_X(t, x, x')\). There exists a pair of \({\varepsilon}_n\)- spectral approximating maps \(f_n: (M_n, p_{M_n})\rightarrow (X, p_X)\) and \(h_n: (X, p_X)\rightarrow (M_n, p_{M_n})\) with \({\lim}_{n{\rightarrow} {\infty}} {\varepsilon}_n=0\). Furthermore, these maps \(f_n\) and \(h_n\) are also a pair of \({\varepsilon}_n\)-Hausdorff approximating maps with respect to the distances \(D^{\text{spe}}_{X_n}\) on \(M_n\) and \(D^{\text{spe}}_X\) on \(X\). Moreover, the sequence of image measures \(f_{n*}{\mu}_{M_n}\) converges to \({\mu}_X\) with respect to the weak* topology as \(n{\rightarrow}{\infty}.\) iii) \({\forall}i=0, 1, 2, \dots,\) the \(i\)-th eigenvalue \({\lambda}_i(M_n)\) converges to the \(i\)-th eigenvalue \({\lambda}_i(X)\) of the generator \({\mathcal L}_X\) of the Dirichlet form \({\varepsilon}_X\), and if \(u\) is an eigenfunction of \(M_n\) with eigenvalue \({\lambda}_i(M_n)\) and unit \(L^2\)-norm, then there exists an eigenfunction \(v\) o…
reviewed by: Fuliu Zhu
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 1794829
zbMATH Keywords: Laplace operator
zbMATH Keywords: energy form
zbMATH Keywords: heat kernel
zbMATH Keywords: spectral convergence
zbMATH Keywords: Gromov-Hausdorff distance
zbMATH Keywords: eigenvalue
MaRDI profile type: Publication
OpenAlex ID: W2498687110
Wikidata QID: Q115159049
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DOI: 10.5802/AIF.1916
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1794829