Rectifiability of singular sets of noncollapsed limit spaces with Ricci curvature bounded below

Abstract

This paper is concerned with the structure of Gromov-Hausdorff limit spaces $(M^n_i,g_i,p_i)\stackrel{d_{GH}}{\longrightarrow} (X^n,d,p)$ of Riemannian manifolds satisfying a uniform lower Ricci curvature bound $\mathrm{Ric}_{M^n_i}\geq -(n-1)$ as well as the noncollapsing assumption $\mathrm{Vol}(B_1(p_i))>\mathrm{v}>0$. In such cases, there is a filtration of the singular set, $S^0\subset S^1\cdots S^{n-1}:= S$, where $S^k:= \{x\in X:\text{ no tangent cone at $x$ is }(k+1)\text{-symmetric}\}$. Equivalently, $S^k$ is the set of points such that no tangent cone splits off a Euclidean factor $\mathbb{R}^{k+1}$. It is classical from Cheeger-Colding that the Hausdorff dimension of $S^k$ satisfies $\mathrm{dim}\, S^k\leq k$ and $S=S^{n-2}$, i.e., $S^{n-1}\setminus S^{n-2}=\emptyset$. However, little else has been understood about the structure of the singular set $S$.

Our first result for such limit spaces $X^n$ states that $S^k$ is $k$-rectifiable for all $k$. In fact, we will show for $\mathcal H^k$-a.e. $x\in S^k$ that every tangent cone $X_x$ at $x$ is $k$-symmetric, i.e., that $X_x= \mathbb{R}^k\times C(Y)$ where $C(Y)$ might depend on the particular $X_x$. Here $\mathcal{H}^k$ denotes the $k$-dimensional Hausdorff measure. As an application we show for all $0\lt \epsilon\lt \epsilon(n,\mathrm{v})$ there exists an $(n-2)$-rectifiable closed set $S^{n-2}_\epsilon$ with $\mathcal{H}^{n-2}(S_{\epsilon}^{n-2}) < C(n,\mathrm{v},\epsilon)$, such that $X^n\setminus S^{n-2}_\epsilon$ is $\epsilon$-bi-Hölder equivalent to a smooth Riemannian manifold. Moreover, $S=\bigcup_\epsilon S^{n-2}_\epsilon$. As another application, we show that tangent cones are unique $\mathcal H^{n-2}$-a.e.

In the case of limit spaces $X^n$ satisfying a $2$-sided Ricci curvature bound $|\mathrm{Ric}_{M^n_i}|\leq n-1$, we can use these structural results to give a new proof of a conjecture from Cheeger-Colding stating that $S$ is $(n-4)$-rectifiable with uniformly bounded measure. We can also conclude from this structure that tangent cones are unique $\mathcal H^{n-4}$-a.e.

Our analysis builds on the notion of quantitative stratification introduced by Cheeger-Naber, and the neck region analysis developed by Jiang-Naber-Valtorta. Several new ideas and new estimates are required, including a sharp cone-splitting theorem and a geometric transformation theorem, which will allow us to control the degeneration of harmonic functions on these neck regions.

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      zblnumber = {1335.53057},
      }
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      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},
      mrreviewer = {Marco Bramanti},
      doi = {10.1002/cpa.21518},
      url = {https://doi.org/10.1002/cpa.21518},
      zblnumber = {1309.35012},
      }
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      author = {Chu, J.},
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      year = {2016},
      arXiv = {1610.09946},
      }
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      journal = {Ann. of Math. (2)},
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      issn = {0003-486X},
      mrclass = {53C21 (53C23)},
      mrnumber = {1454700},
      mrreviewer = {Zhongmin Shen},
      doi = {10.2307/2951841},
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      zblnumber = {0879.53030},
      }
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      author = {Colding, Tobias Holck and Naber, Aaron},
      title = {Sharp {H}ölder continuity of tangent cones for spaces with a lower {R}icci curvature bound and applications},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {176},
      year = {2012},
      number = {2},
      pages = {1173--1229},
      issn = {0003-486X},
      mrclass = {53C21 (53C20)},
      mrnumber = {2950772},
      mrreviewer = {Yu Ding},
      doi = {10.4007/annals.2012.176.2.10},
      url = {https://doi.org/10.4007/annals.2012.176.2.10},
      zblnumber = {1260.53067},
      }
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      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},
      mrclass = {53C20 (53C21)},
      mrnumber = {3037899},
      mrreviewer = {Yu Ding},
      doi = {10.1007/s00039-012-0202-7},
      url = {https://doi.org/10.1007/s00039-012-0202-7},
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      }
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      journal = {Comm. Anal. Geom.},
      fjournal = {Communications in Analysis and Geometry},
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      number = {3},
      pages = {475--514},
      issn = {1019-8385},
      mrclass = {58J35 (35A08 53C23)},
      mrnumber = {1912256},
      mrreviewer = {Man Chun Leung},
      doi = {10.4310/CAG.2002.v10.n3.a3},
      url = {https://doi.org/10.4310/CAG.2002.v10.n3.a3},
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      fjournal = {Proceedings of the American Mathematical Society},
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      year = {2004},
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      issn = {0002-9939},
      mrclass = {53C21 (53C23)},
      mrnumber = {2022380},
      mrreviewer = {Man Chun Leung},
      doi = {10.1090/S0002-9939-03-07060-6},
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      mrclass = {53C55 (32Q20 53C23)},
      mrnumber = {3261011},
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      year = {2019},
      number = {3},
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      mrclass = {35R35 (35J20)},
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      doi = {10.1090/tran/7401},
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      mrreviewer = {Wei Yue Ding},
      doi = {10.4310/CAG.1993.v1.n1.a6},
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      author = {Jiang, Wenshuai and Naber, Aaron},
      TITLE = {{$L^2$} curvature bounds on manifolds with bounded {R}icci curvature},
      JOURNAL = {Ann. of Math. (2)},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {193},
      YEAR = {2021},
      NUMBER = {1},
      PAGES = {107--222},
      ISSN = {0003-486X},
      MRCLASS = {53B20 (35A21 53C21)},
      MRNUMBER = {4199730},
      DOI = {10.4007/annals.2021.193.1.2},
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      }
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      journal = {Peking Math. J.},
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      volume = {3},
      year = {2020},
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      pages = {203--234},
      issn = {2096-6075},
      mrclass = {53C23 (53C20 53C21)},
      mrnumber = {4171913},
      doi = {10.1007/s42543-020-00026-2},
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      title = {Structure theory of metric measure spaces with lower {R}icci curvature bounds},
      journal = {J. Eur. Math. Soc. (JEMS)},
      fjournal = {Journal of the European Mathematical Society (JEMS)},
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      mrreviewer = {Fernando Galaz-Garc\'ıa},
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      }
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      author = {Naber, Aaron},
      title = {The geometry of {R}icci curvature},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians---{S}eoul 2014. {V}ol. {II}},
      pages = {911--937},
      publisher = {Kyung Moon Sa, Seoul},
      year = {2014},
      mrclass = {53C21},
      mrnumber = {3728645},
      mrreviewer = {Shouhei Honda},
      zblnumber = {1376.53003},
      }
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    @ARTICLE{NaVa_CriticalSets,
      author = {Naber, Aaron and Valtorta, Daniele},
      title = {Volume estimates on the critical sets of solutions to elliptic {PDE}s},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {70},
      year = {2017},
      number = {10},
      pages = {1835--1897},
      issn = {0010-3640},
      mrclass = {35J15 (35A20 35R01)},
      mrnumber = {3688031},
      mrreviewer = {Xinhua Ji},
      doi = {10.1002/cpa.21708},
      url = {https://doi.org/10.1002/cpa.21708},
      zblnumber = {1376.35021},
      }
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    @ARTICLE{NaVa_YM,
      author = {Naber, Aaron and Valtorta, Daniele},
      title = {Energy identity for stationary {Y}ang {M}ills},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {216},
      year = {2019},
      number = {3},
      pages = {847--925},
      issn = {0020-9910},
      mrclass = {58E15 (53C07)},
      mrnumber = {3955711},
      doi = {10.1007/s00222-019-00854-9},
      url = {https://doi.org/10.1007/s00222-019-00854-9},
      zblnumber = {07066470},
      }
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    @ARTICLE{NaVa_Rect_harmonicmap,
      author = {Naber, Aaron and Valtorta, Daniele},
      title = {Rectifiable-{R}eifenberg and the regularity of stationary and minimizing harmonic maps},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {185},
      year = {2017},
      number = {1},
      pages = {131--227},
      issn = {0003-486X},
      mrclass = {58E20 (53C43)},
      mrnumber = {3583353},
      mrreviewer = {Andreas Gastel},
      doi = {10.4007/annals.2017.185.1.3},
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      }
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      mrreviewer = {Peng Lu},
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      isbn = {978-3-319-26652-7; 978-3-319-26654-1},
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Authors

Jeff Cheeger

Courant Institute, New York, NY

Wenshuai Jiang

School of Mathematical Sciences, Zhejiang University, Hangzhou, China

Aaron Naber

Northwestern University, Evanston, IL