Lp-bounds on curvature, elliptic estimates and rectifiability of singular sets
[Bornes Lp sur la courbure, estimées elliptiques et rectifiabilité d'ensembles singuliers]
Cheeger, Jeff1
1 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA
Comptes Rendus. Mathématique, Tome 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
Cheeger, Jeff 1

1 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA 
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Cheeger, Jeff. Lp-bounds on curvature, elliptic estimates and rectifiability of singular sets. Comptes Rendus. Mathématique, Tome 334 (2002) no. 3, pp. 195-198. doi : 10.1016/S1631-073X(02)02238-0. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02238-0/

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

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

[5] Cheeger, J.; Colding, T.H.; Tian, G. 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] Cheeger, J.; Simons, J. 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] Colding, T.H. Shape of manifolds with positive Ricci curvature, Invent. Math., Volume 124 (1996) no. 1–3, pp. 175-191

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

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

[11] Simon, L. 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] Tian, G. Canonical Metrics in Kähler Geometry, Birkhäuser, 1990

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