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.

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  • [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},
      address = {Cambridge, MA},
      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] Go to document 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