Examples of Ricci limit spaces with non-integer Hausdorff dimension
We give the first examples of collapsing Ricci limit spaces on which the Hausdorff dimension of the singular set exceeds that of the regular set; moreover, the Hausdorff dimension of these spaces can be non-integers. This answers a question of Cheeger-Colding [ CC00a , Page 15] about collapsing Ricc...
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| Published in | Geometric and functional analysis Vol. 32; no. 3; pp. 676 - 685 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.06.2022
Springer Nature B.V |
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| ISSN | 1016-443X 1420-8970 |
| DOI | 10.1007/s00039-022-00598-4 |
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| Abstract | We give the first examples of collapsing Ricci limit spaces on which the Hausdorff dimension of the singular set exceeds that of the regular set; moreover, the Hausdorff dimension of these spaces can be non-integers. This answers a question of Cheeger-Colding [
CC00a
, Page 15] about collapsing Ricci limit spaces. |
|---|---|
| AbstractList | We give the first examples of collapsing Ricci limit spaces on which the Hausdorff dimension of the singular set exceeds that of the regular set; moreover, the Hausdorff dimension of these spaces can be non-integers. This answers a question of Cheeger-Colding [CC00a, Page 15] about collapsing Ricci limit spaces. We give the first examples of collapsing Ricci limit spaces on which the Hausdorff dimension of the singular set exceeds that of the regular set; moreover, the Hausdorff dimension of these spaces can be non-integers. This answers a question of Cheeger-Colding [ CC00a , Page 15] about collapsing Ricci limit spaces. |
| Author | Wei, Guofang Pan, Jiayin |
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| Cites_doi | 10.2307/2118589 10.1090/S0273-0979-1988-15653-4 10.4099/math1924.12.121 10.2140/gt.2021.25.1059 10.1353/ajm.2000.0041 10.4310/jdg/1214459974 10.4007/annals.2012.176.2.10 10.2307/2946606 10.1515/crelle-2021-0065 10.3842/SIGMA.2020.104 |
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| References | Pan (CR13) 2021; 25 Colding, Naber (CR6) 2012; 176 Fukaya (CR8) 1986; 12 CR7 Nabonnand (CR12) 1980; 291 CR14 Bérard-Bergery (CR1) 1986; 302 CR11 Cheeger, Colding (CR5) 2000; 54 Cheeger, Colding (CR4) 2000; 54 Wei (CR15) 1988; 19 Menguy (CR10) 2000; 122 Fukaya, Yamaguchi (CR9) 1992; 136 Cheeger, Colding (CR3) 1997; 46 Cheeger, Colding (CR2) 1996; 144 Jeff Cheeger (598_CR4) 2000; 54 Lionel Bérard-Bergery (598_CR1) 1986; 302 Tobias H Colding (598_CR6) 2012; 176 Jeff Cheeger (598_CR5) 2000; 54 Xavier Menguy (598_CR10) 2000; 122 Kenji Fukaya (598_CR9) 1992; 136 Philippe Nabonnand (598_CR12) 1980; 291 Jeff Cheeger (598_CR2) 1996; 144 598_CR11 598_CR7 598_CR14 Jiayin Pan (598_CR13) 2021; 25 Guofang Wei (598_CR15) 1988; 19 Jeff Cheeger (598_CR3) 1997; 46 Kenji Fukaya (598_CR8) 1986; 12 |
| References_xml | – volume: 144 start-page: 189 issue: 1 year: 1996 end-page: 237 ident: CR2 article-title: Lower bounds on Ricci curvature and the almost rigidity of warped products publication-title: Ann. Math. (2) doi: 10.2307/2118589 – volume: 19 start-page: 311 issue: 1 year: 1988 end-page: 313 ident: CR15 article-title: Examples of complete manifolds of positive Ricci curvature with nilpotent isometry groups publication-title: Bull. Am. Math. Soc. (N.S.) doi: 10.1090/S0273-0979-1988-15653-4 – ident: CR14 – ident: CR11 – volume: 12 start-page: 121 issue: 1 year: 1986 end-page: 160 ident: CR8 article-title: Theory of convergence for Riemannian orbifolds publication-title: Jpn. J. Math. (N.S.) doi: 10.4099/math1924.12.121 – volume: 302 start-page: 159 issue: 4 year: 1986 end-page: 161 ident: CR1 article-title: Quelques exemples de variétés riemanniennes complètes non compactes à courbure de Ricci positive publication-title: C. R. Acad. Sci. Paris Sér. I Math. – volume: 25 start-page: 1059 issue: 2 year: 2021 end-page: 1085 ident: CR13 article-title: On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature publication-title: Geom. Topol. doi: 10.2140/gt.2021.25.1059 – ident: CR7 – volume: 122 start-page: 927 issue: 5 year: 2000 end-page: 937 ident: CR10 article-title: Examples of nonpolar limit spaces publication-title: Am. J. Math. doi: 10.1353/ajm.2000.0041 – volume: 46 start-page: 406 issue: 3 year: 1997 end-page: 480 ident: CR3 article-title: On the structure of spaces with Ricci curvature bounded below. I publication-title: J. Differ. Geom. doi: 10.4310/jdg/1214459974 – volume: 54 start-page: 37 issue: 1 year: 2000 end-page: 74 ident: CR5 article-title: On the structure of spaces with Ricci curvature bounded below. III publication-title: J. Differ. Geom. – volume: 54 start-page: 13 issue: 1 year: 2000 end-page: 35 ident: CR4 article-title: On the structure of spaces with Ricci curvature bounded below. II publication-title: J. Differ. Geom. – volume: 176 start-page: 1173 issue: 2 year: 2012 end-page: 1229 ident: CR6 article-title: Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications publication-title: Ann. Math. doi: 10.4007/annals.2012.176.2.10 – volume: 291 start-page: A591 issue: 10 year: 1980 end-page: A593 ident: CR12 article-title: Sur les variétés riemanniennes complètes à courbure de Ricci positive publication-title: C. R. Acad. Sci. Paris Sér. A-B – volume: 136 start-page: 253 issue: 2 year: 1992 end-page: 333 ident: CR9 article-title: The fundamental groups of almost nonnegatively curved manifolds publication-title: Ann. Math. doi: 10.2307/2946606 – volume: 176 start-page: 1173 issue: 2 year: 2012 ident: 598_CR6 publication-title: Ann. Math. doi: 10.4007/annals.2012.176.2.10 – ident: 598_CR7 – volume: 122 start-page: 927 issue: 5 year: 2000 ident: 598_CR10 publication-title: Am. J. Math. doi: 10.1353/ajm.2000.0041 – volume: 136 start-page: 253 issue: 2 year: 1992 ident: 598_CR9 publication-title: Ann. Math. doi: 10.2307/2946606 – volume: 144 start-page: 189 issue: 1 year: 1996 ident: 598_CR2 publication-title: Ann. Math. (2) doi: 10.2307/2118589 – volume: 302 start-page: 159 issue: 4 year: 1986 ident: 598_CR1 publication-title: C. R. Acad. Sci. Paris Sér. I Math. – ident: 598_CR14 doi: 10.1515/crelle-2021-0065 – volume: 46 start-page: 406 issue: 3 year: 1997 ident: 598_CR3 publication-title: J. Differ. Geom. doi: 10.4310/jdg/1214459974 – volume: 54 start-page: 37 issue: 1 year: 2000 ident: 598_CR5 publication-title: J. Differ. Geom. – volume: 12 start-page: 121 issue: 1 year: 1986 ident: 598_CR8 publication-title: Jpn. J. Math. (N.S.) doi: 10.4099/math1924.12.121 – volume: 291 start-page: A591 issue: 10 year: 1980 ident: 598_CR12 publication-title: C. R. Acad. Sci. Paris Sér. A-B – ident: 598_CR11 doi: 10.3842/SIGMA.2020.104 – volume: 25 start-page: 1059 issue: 2 year: 2021 ident: 598_CR13 publication-title: Geom. Topol. doi: 10.2140/gt.2021.25.1059 – volume: 54 start-page: 13 issue: 1 year: 2000 ident: 598_CR4 publication-title: J. Differ. Geom. – volume: 19 start-page: 311 issue: 1 year: 1988 ident: 598_CR15 publication-title: Bull. Am. Math. Soc. (N.S.) doi: 10.1090/S0273-0979-1988-15653-4 |
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| Title | Examples of Ricci limit spaces with non-integer Hausdorff dimension |
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