# Regularity of Einstein manifolds and the codimension $4$ conjecture

### Abstract

In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds $(M^n,g)$ with bounded Ricci curvature, as well as their Gromov-Hausdorff limit spaces $(M^n_j,d_j)\stackrel{d_{\rm GH}}{\longrightarrow} (X,d)$, where $d_j$ denotes the Riemannian distance. Our main result is a solution to the codimension $4$ conjecture, namely that $X$ is smooth away from a closed subset of codimension $4$. We combine this result with the ideas of quantitative stratification to prove a priori $L^q$ estimates on the full curvature $|\mathrm{Rm}|$ for all $q<2$. In the case of Einstein manifolds, we improve this to estimates on the regularity scale. We apply this to prove a conjecture of Anderson that the collection of $4$-manifolds $(M^4,g)$ with $|\mathrm{Ric}_{M^4}|\leq 3$, $\mathrm{Vol}(M)>\mathrm{v}>0$, and $\mathrm{diam}(M)\leq D$ contains at most a finite number of diffeomorphism classes. A local version is used to show that noncollapsed $4$-manifolds with bounded Ricci curvature have a priori $L^2$ Riemannian curvature estimates.

• [Ambrosio_Calculus_Ricci] L. Ambrosio, N. Gigli, and G. Savaré, "Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below," Invent. Math., vol. 195, iss. 2, pp. 289-391, 2014.
@article{Ambrosio_Calculus_Ricci, mrkey = {3152751},
author = {Ambrosio, Luigi and Gigli, Nicola and Savar{é},
Giuseppe},
title = {Calculus and heat flow in metric measure spaces and applications to spaces with {R}icci bounds from below},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {195},
year = {2014},
number = {2},
pages = {289--391},
issn = {0020-9910},
mrclass = {53C23 (31E05 35F21 58J35)},
mrnumber = {3152751},
mrreviewer = {Nelia Charalambous},
doi = {10.1007/s00222-013-0456-1},
zblnumber = {06261668},
}
• [Ambrosio_Ricci] L. Ambrosio, N. Gigli, and G. Savaré, "Metric measure spaces with Riemannian Ricci curvature bounded from below," Duke Math. J., vol. 163, iss. 7, pp. 1405-1490, 2014.
@article{Ambrosio_Ricci, mrkey = {3205729},
author = {Ambrosio, Luigi and Gigli, Nicola and Savar{é},
Giuseppe},
title = {Metric measure spaces with {R}iemannian {R}icci curvature bounded from below},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {163},
year = {2014},
number = {7},
pages = {1405--1490},
issn = {0012-7094},
mrclass = {35R01 (60J45 60J65)},
mrnumber = {3205729},
doi = {10.1215/00127094-2681605},
zblnumber = {06303881},
}
• [A89] M. T. Anderson, "Ricci curvature bounds and Einstein metrics on compact manifolds," J. Amer. Math. Soc., vol. 2, iss. 3, pp. 455-490, 1989.
@article{A89, mrkey = {0999661},
author = {Anderson, Michael T.},
title = {Ricci curvature bounds and {E}instein metrics on compact manifolds},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the American Mathematical Society},
volume = {2},
year = {1989},
number = {3},
pages = {455--490},
issn = {0894-0347},
mrclass = {53C20 (53C25 58D17 58G30)},
mrnumber = {0999661},
mrreviewer = {Maung Min-Oo},
doi = {10.2307/1990939},
zblnumber = {0694.53045},
}
• [Anderson_Einstein] M. T. Anderson, "Convergence and rigidity of manifolds under Ricci curvature bounds," Invent. Math., vol. 102, iss. 2, pp. 429-445, 1990.
@article{Anderson_Einstein, mrkey = {1074481},
author = {Anderson, Michael T.},
title = {Convergence and rigidity of manifolds under {R}icci curvature bounds},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {102},
year = {1990},
number = {2},
pages = {429--445},
issn = {0020-9910},
coden = {INVMBH},
mrclass = {53C23 (53C21 58D27)},
mrnumber = {1074481},
mrreviewer = {Gudlaugur Thorbergsson},
doi = {10.1007/BF01233434},
zblnumber = {0711.53038},
}
• [Anderson_Hausdorff] M. T. Anderson, "Hausdorff perturbations of Ricci-flat manifolds and the splitting theorem," Duke Math. J., vol. 68, iss. 1, pp. 67-82, 1992.
@article{Anderson_Hausdorff, mrkey = {1185818},
author = {Anderson, Michael T.},
title = {Hausdorff perturbations of {R}icci-flat manifolds and the splitting theorem},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {68},
year = {1992},
number = {1},
pages = {67--82},
issn = {0012-7094},
coden = {DUMJAO},
mrclass = {53C21 (53C20)},
mrnumber = {1185818},
doi = {10.1215/S0012-7094-92-06803-7},
zblnumber = {0767.53029},
}
• [Anderson_ICM94] M. T. Anderson, "Einstein metrics and metrics with bounds on Ricci curvature," in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Basel, 1995, pp. 443-452.
@inproceedings{Anderson_ICM94, mrkey = {1403944},
author = {Anderson, Michael T.},
title = {Einstein metrics and metrics with bounds on {R}icci curvature},
booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 1, 2},
venue = {{Z}ürich, 1994},
pages = {443--452},
publisher = {Birkhäuser},
year = {1995},
mrclass = {53C21 (53C23 53C25)},
mrnumber = {1403944},
mrreviewer = {Zhongmin Shen},
zblnumber = {0840.53036},
}
• [Anderson-Cheeger] M. T. Anderson and J. Cheeger, "$C^\alpha$-compactness for manifolds with Ricci curvature and injectivity radius bounded below," J. Differential Geom., vol. 35, iss. 2, pp. 265-281, 1992.
@article{Anderson-Cheeger, mrkey = {1158336},
author = {Anderson, Michael T. and Cheeger, Jeff},
title = {{$C\sp \alpha$}-compactness for manifolds with {R}icci curvature and injectivity radius bounded below},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {35},
year = {1992},
number = {2},
pages = {265--281},
issn = {0022-040X},
coden = {JDGEAS},
mrclass = {53C23 (53C21)},
mrnumber = {1158336},
mrreviewer = {Xiao Wei Peng},
url = {http://projecteuclid.org/euclid.jdg/1214448075},
zblnumber = {0774.53021},
}
• [Anderson-Cheeger2] M. T. Anderson and J. Cheeger, "Diffeomorphism finiteness for manifolds with Ricci curvature and $L^{n/2}$-norm of curvature bounded," Geom. Funct. Anal., vol. 1, iss. 3, pp. 231-252, 1991.
@article{Anderson-Cheeger2, mrkey = {1118730},
author = {Anderson, Michael T. and Cheeger, Jeff},
title = {Diffeomorphism finiteness for manifolds with {R}icci curvature and {$L\sp {n/2}$}-norm of curvature bounded},
journal = {Geom. Funct. Anal.},
fjournal = {Geometric and Functional Analysis},
volume = {1},
year = {1991},
number = {3},
pages = {231--252},
issn = {1016-443X},
coden = {GFANFB},
mrclass = {53C23},
mrnumber = {1118730},
mrreviewer = {Viktor Schroeder},
doi = {10.1007/BF01896203},
zblnumber = {0764.53026},
}
• [BKN89] S. Bando, A. Kasue, and H. Nakajima, "On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth," Invent. Math., vol. 97, iss. 2, pp. 313-349, 1989.
@article{BKN89, mrkey = {1001844},
author = {Bando, Shigetoshi and Kasue, Atsushi and Nakajima, Hiraku},
title = {On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {97},
year = {1989},
number = {2},
pages = {313--349},
issn = {0020-9910},
coden = {INVMBH},
mrclass = {53C20 (53C25)},
mrnumber = {1001844},
mrreviewer = {Thomas H. Otway},
doi = {10.1007/BF01389045},
zblnumber = {0682.53045},
}
• [B90] S. Bando, "Bubbling out of Einstein manifolds," Tohoku Math. J., vol. 42, iss. 2, pp. 205-216, 1990.
@article{B90, mrkey = {1053949},
author = {Bando, Shigetoshi},
title = {Bubbling out of {E}instein manifolds},
journal = {Tohoku Math. J.},
fjournal = {The Tohoku Mathematical Journal. Second Series},
volume = {42},
year = {1990},
number = {2},
pages = {205--216},
issn = {0040-8735},
coden = {TOMJAM},
mrclass = {53C23 (53C25 53C55)},
mrnumber = {1053949},
mrreviewer = {Chun-Li Shen},
doi = {10.2748/tmj/1178227654},
zblnumber = {0719.53025},
}
• [BakryEmery_diffusions] D. Bakry and M. Émery, "Diffusions hypercontractives," in Séminaire de Probabilités, XIX, 1983/84, New York: Springer-Verlag, 1985, vol. 1123, pp. 177-206.
@incollection{BakryEmery_diffusions, mrkey = {0889476},
author = {Bakry, D. and {É}mery, Michel},
title = {Diffusions hypercontractives},
booktitle = {Séminaire de Probabilités, {XIX},
1983/84},
series = {Lecture Notes in Math.},
volume = {1123},
pages = {177--206},
publisher = {Springer-Verlag},
year = {1985},
mrclass = {60J60 (58C40 58G32)},
mrnumber = {0889476},
mrreviewer = {Jacques Vauthier},
doi = {10.1007/BFb0075847},
zblnumber = {0561.60080},
}
• [Cheeger_finiteness] J. Cheeger, "Finiteness theorems for Riemannian manifolds," Amer. J. Math., vol. 92, pp. 61-74, 1970.
@article{Cheeger_finiteness, mrkey = {0263092},
author = {Cheeger, Jeff},
title = {Finiteness theorems for {R}iemannian manifolds},
journal = {Amer. J. Math.},
fjournal = {American Journal of Mathematics},
volume = {92},
year = {1970},
pages = {61--74},
issn = {0002-9327},
mrclass = {57.10},
mrnumber = {0263092},
mrreviewer = {M. Klingmann},
doi = {10.2307/2373498},
zblnumber = {0194.52902},
}
• [Cheeger] J. Cheeger, "Integral bounds on curvature elliptic estimates and rectifiability of singular sets," Geom. Funct. Anal., vol. 13, iss. 1, pp. 20-72, 2003.
@article{Cheeger, mrkey = {1978491},
author = {Cheeger, J.},
title = {Integral bounds on curvature elliptic estimates and rectifiability of singular sets},
journal = {Geom. Funct. Anal.},
fjournal = {Geometric and Functional Analysis},
volume = {13},
year = {2003},
number = {1},
pages = {20--72},
issn = {1016-443X},
coden = {GFANFB},
mrclass = {53C21 (49Q99 53C20)},
mrnumber = {1978491},
doi = {10.1007/s000390300001},
zblnumber = {1086.53051},
}
• [ChC1] J. Cheeger and T. H. Colding, "Lower bounds on Ricci curvature and the almost rigidity of warped products," Ann. of Math., vol. 144, iss. 1, pp. 189-237, 1996.
@article{ChC1, mrkey = {1405949},
author = {Cheeger, Jeff and Colding, Tobias H.},
title = {Lower bounds on {R}icci curvature and the almost rigidity of warped products},
journal = {Ann. of Math.},
fjournal = {Annals of Mathematics. Second Series},
volume = {144},
year = {1996},
number = {1},
pages = {189--237},
issn = {0003-486X},
coden = {ANMAAH},
mrclass = {53C21 (53C20 53C23)},
mrnumber = {1405949},
mrreviewer = {Joseph E. Borzellino},
doi = {10.2307/2118589},
zblnumber = {0865.53037},
}
• [ChC2] J. Cheeger and T. H. Colding, "On the structure of spaces with Ricci curvature bounded below. I," J. Differential Geom., vol. 46, iss. 3, pp. 406-480, 1997.
@article{ChC2, mrkey = {1484888},
author = {Cheeger, Jeff and Colding, Tobias H.},
title = {On the structure of spaces with {R}icci curvature bounded below. {I}},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {46},
year = {1997},
number = {3},
pages = {406--480},
issn = {0022-040X},
coden = {JDGEAS},
mrclass = {53C21 (53C20)},
mrnumber = {1484888},
mrreviewer = {William P. Minicozzi, II},
url = {http://projecteuclid.org/euclid.jdg/1214459974},
zblnumber = {0902.53034},
}
• [CCTi_eps_reg] J. Cheeger, T. H. Colding, and G. Tian, "On the singularities of spaces with bounded Ricci curvature," Geom. Funct. Anal., vol. 12, iss. 5, pp. 873-914, 2002.
@article{CCTi_eps_reg, mrkey = {1937830},
author = {Cheeger, J. and Colding, T. H. and Tian, G.},
title = {On the singularities of spaces with bounded {R}icci curvature},
journal = {Geom. Funct. Anal.},
fjournal = {Geometric and Functional Analysis},
volume = {12},
year = {2002},
number = {5},
pages = {873--914},
issn = {1016-443X},
coden = {GFANFB},
mrclass = {53C21 (53C20)},
mrnumber = {1937830},
mrreviewer = {Zhongmin Shen},
doi = {10.1007/PL00012649},
zblnumber = {1030.53046},
}
• [CheegerNaber_Ricci] J. Cheeger and A. Naber, "Lower bounds on Ricci curvature and quantitative behavior of singular sets," Invent. Math., vol. 191, iss. 2, pp. 321-339, 2013.
@article{CheegerNaber_Ricci, mrkey = {3010378},
author = {Cheeger, Jeff and Naber, Aaron},
title = {Lower bounds on {R}icci curvature and quantitative behavior of singular sets},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {191},
year = {2013},
number = {2},
pages = {321--339},
issn = {0020-9910},
coden = {INVMBH},
mrclass = {53C21 (32Q20 53C23 53C25)},
mrnumber = {3010378},
mrreviewer = {Leonid V. Kovalev},
doi = {10.1007/s00222-012-0394-3},
zblnumber = {1268.53053},
}
• [ChNaVa] J. Cheeger, A. Naber, and D. Valtorta, "Critical sets of elliptic equations," Comm. Pure Appl. Math., vol. 68, iss. 2, pp. 173-209, 2015.
@article{ChNaVa, mrkey = {3298662},
author = {Cheeger, Jeff and Naber, Aaron and Valtorta, Daniele},
title = {Critical sets of elliptic equations},
journal = {Comm. Pure Appl. Math.},
fjournal = {Communications on Pure and Applied Mathematics},
volume = {68},
year = {2015},
number = {2},
pages = {173--209},
issn = {0010-3640},
mrclass = {35J25 (35B05)},
mrnumber = {3298662},
doi = {10.1002/cpa.21518},
zblnumber = {1309.35012},
}
• [Chen_Donaldson] X. -X. Chen and S. K. Donaldson, "Volume estimates for Kähler-Einstein metrics and rigidity of complex structures," J. Differential Geom., vol. 93, iss. 2, pp. 191-201, 2013.
@article{Chen_Donaldson, mrkey = {3024305},
author = {Chen, X.-X. and Donaldson, S. K.},
title = {Volume estimates for {K}ähler-{E}instein metrics and rigidity of complex structures},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {93},
year = {2013},
number = {2},
pages = {191--201},
issn = {0022-040X},
coden = {JDGEAS},
mrclass = {53C21 (32Q20)},
mrnumber = {3024305},
mrreviewer = {Yuguang Zhang},
url = {http://projecteuclid.org/euclid.jdg/1361800865},
zblnumber = {1281.32019},
}
• [Co1] T. H. Colding, "Ricci curvature and volume convergence," Ann. of Math., vol. 145, iss. 3, pp. 477-501, 1997.
@article{Co1, mrkey = {1454700},
author = {Colding, Tobias H.},
title = {Ricci curvature and volume convergence},
journal = {Ann. of Math.},
fjournal = {Annals of Mathematics. Second Series},
volume = {145},
year = {1997},
number = {3},
pages = {477--501},
issn = {0003-486X},
coden = {ANMAAH},
mrclass = {53C21 (53C23)},
mrnumber = {1454700},
mrreviewer = {Zhongmin Shen},
doi = {10.2307/2951841},
zblnumber = {0879.53030},
}
• [CoNa2] T. H. Colding and A. Naber, "Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications," Geom. Funct. Anal., vol. 23, iss. 1, pp. 134-148, 2013.
@article{CoNa2, mrkey = {3037899},
author = {Colding, Tobias Holck and Naber, Aaron},
title = {Characterization of tangent cones of noncollapsed limits with lower {R}icci bounds and applications},
journal = {Geom. Funct. Anal.},
fjournal = {Geometric and Functional Analysis},
volume = {23},
year = {2013},
number = {1},
pages = {134--148},
issn = {1016-443X},
coden = {GFANFB},
mrclass = {53C20 (53C21)},
mrnumber = {3037899},
mrreviewer = {Yu Ding},
doi = {10.1007/s00039-012-0202-7},
zblnumber = {1271.53042},
}
• [Dong_Nodal] R. Dong, "Nodal sets of eigenfunctions on Riemann surfaces," J. Differential Geom., vol. 36, iss. 2, pp. 493-506, 1992.
@article{Dong_Nodal, mrkey = {1180391},
author = {Dong, Rui-Tao},
title = {Nodal sets of eigenfunctions on {R}iemann surfaces},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {36},
year = {1992},
number = {2},
pages = {493--506},
issn = {0022-040X},
coden = {JDGEAS},
mrclass = {58G25 (35P99)},
mrnumber = {1180391},
url = {http://projecteuclid.org/euclid.jdg/1214448750},
zblnumber = {0776.53024},
}
• [GMS_Stability] N. Gigli, A. Mondino, and G. Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, 2013.
@misc{GMS_Stability,
author = {Gigli, N. and Mondino, A. and Savaré, G.},
title = {Convergence of pointed non-compact metric measure spaces and stability of {R}icci curvature bounds and heat flows},
year = {2013},
note = {preprint},
arXiv = {1311.4907},
}
• [Golub_Van_Loan96] G. H. Golub and C. F. Van Loan, Matrix Computations, Third ed., Johns Hopkins University Press, Baltimore, MD, 1996.
@book{Golub_Van_Loan96, mrkey = {1417720},
author = {Golub, Gene H. and Van Loan, Charles F.},
title = {Matrix Computations},
series = {Johns Hopkins Studies in the Mathematical Sciences},
edition = {Third},
publisher = {Johns Hopkins University Press, Baltimore, MD},
year = {1996},
pages = {xxx+698},
isbn = {0-8018-5413-X; 0-8018-5414-8},
mrclass = {65-02 (65Fxx)},
mrnumber = {1417720},
zblnumber = {0865.65009},
}
• [HanLin_Book] Q. Han and F. Lin, Nodal Sets of Solutions of Elliptic Differential Equations.
@misc{HanLin_Book,
author = {Han, Q. and Lin, F.},
title = {{\it Nodal {S}ets of {S}olutions of {E}lliptic {D}ifferential {E}quations}},
note = {in preparation},
}
• [HeinNaber_TangentCones] H. Hein and A. Naber, Isolated Einstein singularities with singular tangent cones.
@misc{HeinNaber_TangentCones,
author = {Hein, H. and Naber, A.},
title = {Isolated {E}instein singularities with singular tangent cones},
note = {preprint},
}
• [MondinoNaber_Rectifiability] A. Mondino and A. Naber, Structure theory of metric-measure spaces with lower Ricci curvature bounds I, 2014.
@misc{MondinoNaber_Rectifiability,
author = {Mondino, A. and Naber, A.},
title = {Structure theory of metric-measure spaces with lower {R}icci curvature bounds {I}},
note = {preprint},
year = {2014},
arxiv = {1405.2222},
}
• [NaVa_CriticalSets] A. Naber and D. Valtorta, "Sharp estimates on the first eigenvalue of the $p$-Laplacian with negative Ricci lower bound," Math. Z., vol. 277, iss. 3-4, pp. 867-891, 2014.
@article{NaVa_CriticalSets, mrkey = {3229969},
author = {Naber, Aaron and Valtorta, Daniele},
title = {Sharp estimates on the first eigenvalue of the {$p$}-{L}aplacian with negative {R}icci lower bound},
journal = {Math. Z.},
fjournal = {Mathematische Zeitschrift},
volume = {277},
year = {2014},
number = {3-4},
pages = {867--891},
issn = {0025-5874},
mrclass = {58J50 (47A10)},
mrnumber = {3229969},
mrreviewer = {De Tang Zhou},
doi = {10.1007/s00209-014-1282-x},
zblnumber = {06323339},
}
• [Petersen_RiemannianGeometry] P. Petersen, Riemannian Geometry, New York: Springer-Verlag, 1998, vol. 171.
@book{Petersen_RiemannianGeometry, mrkey = {1480173},
author = {Petersen, Peter},
title = {Riemannian Geometry},
series = {Graduate Texts in Math.},
volume = {171},
publisher = {Springer-Verlag},
year = {1998},
pages = {xvi+432},
isbn = {0-387-98212-4},
mrclass = {53-01 (53C20 53C21 53C23)},
mrnumber = {1480173},
mrreviewer = {Andrea Sambusetti},
doi = {10.1007/978-1-4757-6434-5},
zblnumber = {0914.53001},
}
• [SY_Redbook] R. Schoen and S. -T. Yau, Lectures on Differential Geometry, Cambridge, MA: International Press, 1994, vol. I.
@book{SY_Redbook, mrkey = {1333601},
author = {Schoen, R. and Yau, S.-T.},
title = {Lectures on Differential Geometry},
series = {Conference Proceedings and Lecture Notes in Geometry and Topology},
volume = {I},
publisher = {International Press},
year = {1994},
pages = {v+235},
isbn = {1-57146-012-8},
mrclass = {53-01 (53-02 53C21 58G30)},
mrnumber = {1333601},
mrreviewer = {Man Chun Leung},
zblnumber = {0830.53001},
}
• [T90] G. Tian, "On Calabi’s conjecture for complex surfaces with positive first Chern class," Invent. Math., vol. 101, iss. 1, pp. 101-172, 1990.
@article{T90, mrkey = {1055713},
author = {Tian, G.},
title = {On {C}alabi's conjecture for complex surfaces with positive first {C}hern class},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {101},
year = {1990},
number = {1},
pages = {101--172},
issn = {0020-9910},
coden = {INVMBH},
mrclass = {32L07 (32F07 53C25 53C55)},
mrnumber = {1055713},
mrreviewer = {M. Kalka},
doi = {10.1007/BF01231499},
zblnumber = {0716.32019},
}

## Authors

Jeff Cheeger

Courant Institute of Mathematical Sciences, New York, NY

Aaron Naber

Northwestern University, Evanston, IL