Comptes Rendus
Lp-bounds on curvature, elliptic estimates and rectifiability of singular sets
[Bornes Lp sur la courbure, estimées elliptiques et rectifiabilité d'ensembles singuliers]
Comptes Rendus. Mathématique, Volume 334 (2002) no. 3, pp. 195-198.

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 Lp sur la courbure. Les théorèmes de rectifiabilité dépendent d'estimations sur |Hessh|L2p, (|∇Hessh·|Hessh|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 Lp-bounds on curvature. The rectifiability theorems depend on estimates for |Hessh|L2p, (|∇Hessh·|Hessh|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 :
Publié le :
DOI : 10.1016/S1631-073X(02)02238-0

Jeff Cheeger 1

1 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA
@article{CRMATH_2002__334_3_195_0,
     author = {Jeff Cheeger},
     title = {\protect\emph{L}\protect\textsubscript{\protect\emph{p}}-bounds on curvature, elliptic estimates and rectifiability of singular sets},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {195--198},
     publisher = {Elsevier},
     volume = {334},
     number = {3},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02238-0},
     language = {en},
}
TY  - JOUR
AU  - Jeff Cheeger
TI  - Lp-bounds on curvature, elliptic estimates and rectifiability of singular sets
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 195
EP  - 198
VL  - 334
IS  - 3
PB  - Elsevier
DO  - 10.1016/S1631-073X(02)02238-0
LA  - en
ID  - CRMATH_2002__334_3_195_0
ER  - 
%0 Journal Article
%A Jeff Cheeger
%T Lp-bounds on curvature, elliptic estimates and rectifiability of singular sets
%J Comptes Rendus. Mathématique
%D 2002
%P 195-198
%V 334
%N 3
%I Elsevier
%R 10.1016/S1631-073X(02)02238-0
%G en
%F CRMATH_2002__334_3_195_0
Jeff Cheeger. Lp-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/

[1] M.T. Anderson Einstein metrics and metrics with bounds on Ricci curvature, Proceedings of ICM, Volume 1 (1994) no. 2, pp. 443-452

[2] J. Cheeger, Integral bounds on curvature, estimates on harmonic functions and rectifiability of singular sets, Preprint

[3] J. Cheeger; T.H. Colding Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math., Volume 144 (1996) no. 1, pp. 189-237

[4] J. Cheeger; T.H. Colding On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom., Volume 46 (1997), pp. 406-480

[5] J. Cheeger; T.H. Colding; G. Tian Constraints on singularities under Ricci curvature bounds, C. R. Acad. Sci. Paris, Série I, Volume 324 (1997), pp. 645-649

[6] J. Cheeger, T.H. Colding, G. Tian, On the singularities of spaces with bounded Ricci curvature, GAFA Geom. Funct. Anal. (submitted)

[7] J. Cheeger; J. Simons Differential characters and geometric invariants, Geometry and Topology (College Park, MD, 1983/84), Lecture Notes in Math., 1167, Springer-Verlag, Berlin, 1985, pp. 50-80

[8] T.H. Colding Shape of manifolds with positive Ricci curvature, Invent. Math., Volume 124 (1996) no. 1–3, pp. 175-191

[9] T.H. Colding Large manifolds with positive Ricci curvature, Invent. Math., Volume 124 (1996) no. 1–3, pp. 193-214

[10] T.H. Colding Ricci curvature and volume convergence, Ann. of Math., Volume 145 (1997) no. 3, pp. 477-501

[11] L. Simon Rectifiability of the singular sets of multiplicity 1 minimal surfaces and energy minimizing maps, Surveys in Differential Geometry, II, International Press, 1993, pp. 246-305

[12] G. Tian Canonical Metrics in Kähler Geometry, Birkhäuser, 1990

Cité par Sources :

Commentaires - Politique