Lp-bounds on curvature, elliptic estimates and rectifiability of singular sets

Jeff Cheeger

doi:10.1016/S1631-073X(02)02238-0

L_p-bounds on curvature, elliptic estimates and rectifiability of singular sets
[Bornes L_p sur la courbure, estimées elliptiques et rectifiabilité d'ensembles singuliers]

Jeff Cheeger ¹

¹ Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA

Comptes Rendus. Mathématique, Volume 334 (2002) no. 3, pp. 195-198.

Résumé
Abstract

Nous annonçons des résultats de rectifiabilité des ensembles singuliers dans les espaces métriques pointés qui sont des limites au sens de Gromov–Hausdorff d'une suite de variétés riemanniennes pour lesquelles on a une borne uniforme sur la courbure de Ricci, le volume, et des bornes uniformes L_p sur la courbure. Les théorèmes de rectifiabilité dépendent d'estimations sur |Hess_h|_{L_2p}, (|∇Hess_h·|Hess_h|^p−2)_L₂, où Δh=c, pour une constante c. Nous remarquons également que dans le cas Kählérien (en l'absence de toute borne intégrale sur la courbure), l'ensemble singulier est de codimension complexe 2.

We announce results on rectifiability of singular sets of pointed metric spaces which are pointed Gromov–Hausdorff limits on sequences of Riemannian manifolds, satisfying uniform lower bounds on Ricci curvature and volume, and uniform L_p-bounds on curvature. The rectifiability theorems depend on estimates for |Hess_h|_{L_2p}, (|∇Hess_h·|Hess_h|^p−2)_L₂, where Δh=c, for some constant c. We also observe that (absent any integral bound on curvature) in the Kähler case, given a uniform 2-sided bound on Ricci curvature, the singular set has complex codimension 2.

Accepté le : 2001-12-10
Publié le : 2002-02-01

DOI : 10.1016/S1631-073X(02)02238-0

Affiliations des auteurs :

Jeff Cheeger ¹

¹ Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA

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     pages = {195--198},
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Jeff Cheeger. L_p-bounds on curvature, elliptic estimates and rectifiability of singular sets. Comptes Rendus. Mathématique, Volume 334 (2002) no. 3, pp. 195-198. doi : 10.1016/S1631-073X(02)02238-0. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02238-0/

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Cité par

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