Embedding of RCD⁎(K,N) spaces in L2 via eigenfunctions

Luigi Ambrosio; Shouhei Honda; Jacobus W. Portegies; David Tewodrose

doi:10.1016/j.jfa.2021.108968

Embedding of RCD^⁎(K,N) spaces in L² via eigenfunctions

Luigi Ambrosio, Shouhei Honda, Jacobus W. Portegies, David Tewodrose

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14 Citaten (Scopus)

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In this paper we study the family of embeddings Φ_t of a compact RCD^⁎(K,N) space (X,d,m) into L²(X,m) via eigenmaps. Extending part of the classical results [10,11] known for closed Riemannian manifolds, we prove convergence as t↓0 of the rescaled pull-back metrics Φ_t^⁎g_L² in L²(X,m) induced by Φ_t. Moreover we discuss the behavior of Φ_t^⁎g_L² with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative L^p-convergence in the noncollapsed setting for all p<∞, a result new even for closed Riemannian manifolds and Alexandrov spaces.

Originele taal-2	English
Artikelnummer	108968
Aantal pagina's	71
Tijdschrift	Journal of Functional Analysis
Volume	280
Nummer van het tijdschrift	10
DOI's	https://doi.org/10.1016/j.jfa.2021.108968
Status	Published - 15 mei 2021

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© 2021 Elsevier Inc.

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10.1016/j.jfa.2021.108968

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title = "Embedding of RCD⁎(K,N) spaces in L2 via eigenfunctions",

abstract = "In this paper we study the family of embeddings Φt of a compact RCD⁎(K,N) space (X,d,m) into L2(X,m) via eigenmaps. Extending part of the classical results [10,11] known for closed Riemannian manifolds, we prove convergence as t↓0 of the rescaled pull-back metrics Φt⁎gL2 in L2(X,m) induced by Φt. Moreover we discuss the behavior of Φt⁎gL2 with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative Lp-convergence in the noncollapsed setting for all p<∞, a result new even for closed Riemannian manifolds and Alexandrov spaces.",

keywords = "Heat kernel, Laplacian, Metric measure spaces, Ricci curvature",

author = "Luigi Ambrosio and Shouhei Honda and Portegies, {Jacobus W.} and David Tewodrose",

note = "Funding Information: Acknowledgment. The first and fourth authors acknowledge the support of the MIUR PRIN 2015 project ?Calculus of Variations?. The second author acknowledges supports of the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, of the Grantin-Aid for Young Scientists (B) 16K17585, Grant-in-Aid for Scientific Research (B) of 18H01118 and of 20H01799. The authors warmly thank the referee for the detailed reading of the paper and for the constructive comments. Funding Information: Acknowledgment. The first and fourth authors acknowledge the support of the MIUR PRIN 2015 project \u201CCalculus of Variations\u201D. The second author acknowledges supports of the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, of the Grantin-Aid for Young Scientists (B) 16K17585 , Grant-in-Aid for Scientific Research (B) of 18H01118 and of 20H01799 . The authors warmly thank the referee for the detailed reading of the paper and for the constructive comments. Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

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AU - Honda, Shouhei

AU - Portegies, Jacobus W.

AU - Tewodrose, David

N1 - Funding Information: Acknowledgment. The first and fourth authors acknowledge the support of the MIUR PRIN 2015 project ?Calculus of Variations?. The second author acknowledges supports of the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, of the Grantin-Aid for Young Scientists (B) 16K17585, Grant-in-Aid for Scientific Research (B) of 18H01118 and of 20H01799. The authors warmly thank the referee for the detailed reading of the paper and for the constructive comments. Funding Information: Acknowledgment. The first and fourth authors acknowledge the support of the MIUR PRIN 2015 project \u201CCalculus of Variations\u201D. The second author acknowledges supports of the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, of the Grantin-Aid for Young Scientists (B) 16K17585 , Grant-in-Aid for Scientific Research (B) of 18H01118 and of 20H01799 . The authors warmly thank the referee for the detailed reading of the paper and for the constructive comments. Publisher Copyright: © 2021 Elsevier Inc.

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N2 - In this paper we study the family of embeddings Φt of a compact RCD⁎(K,N) space (X,d,m) into L2(X,m) via eigenmaps. Extending part of the classical results [10,11] known for closed Riemannian manifolds, we prove convergence as t↓0 of the rescaled pull-back metrics Φt⁎gL2 in L2(X,m) induced by Φt. Moreover we discuss the behavior of Φt⁎gL2 with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative Lp-convergence in the noncollapsed setting for all p<∞, a result new even for closed Riemannian manifolds and Alexandrov spaces.

AB - In this paper we study the family of embeddings Φt of a compact RCD⁎(K,N) space (X,d,m) into L2(X,m) via eigenmaps. Extending part of the classical results [10,11] known for closed Riemannian manifolds, we prove convergence as t↓0 of the rescaled pull-back metrics Φt⁎gL2 in L2(X,m) induced by Φt. Moreover we discuss the behavior of Φt⁎gL2 with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative Lp-convergence in the noncollapsed setting for all p<∞, a result new even for closed Riemannian manifolds and Alexandrov spaces.

KW - Heat kernel

KW - Laplacian

KW - Metric measure spaces

KW - Ricci curvature

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