### Abstract

We prove that a metric measure space (X,d,m) satisfying finite-dimensional lower Ricci curvature bounds and whose Sobolev space W ^{1,2} is Hilbert is rectifiable. That is, an RCD ^{∗} (K,N)-space is rectifiable, and in particular for m-a.e. point the tangent cone is unique and Euclidean of dimension at most N. The proof is based on a maximal function argument combined with an original Almost Splitting Theorem via estimates on the gradient of the excess. We also show a sharp integral Abresch–Gromoll type inequality for the excess function and an Abresch–Gromoll-type inequality for the gradient of the excess. The argument is new even in the smooth setting.

Original language | English (US) |
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Pages (from-to) | 1809-1854 |

Number of pages | 46 |

Journal | Journal of the European Mathematical Society |

Volume | 21 |

Issue number | 6 |

DOIs | |

State | Published - 2019 |

### Keywords

- Optimal transport
- Rectifiable space
- Ricci curvature
- Unique tangent space

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Journal of the European Mathematical Society*,

*21*(6), 1809-1854. https://doi.org/10.4171/JEMS/874