### Abstract

Let (M, d, m) be a metric measure space which satisfies the Lott–Sturm–Villani curvature-dimension condition CD(K, n) for some K≥ 0 and n≥ 2 , and a lower n-density assumption at some point of M. We prove that if (M, d, m) supports the Gagliardo–Nirenberg inequality or any of its limit cases (L^{p}-logarithmic Sobolev inequality or Faber–Krahn-type inequality), then a global non-collapsingn-dimensional volume growth holds, i.e., there exists a universal constant C_{0}> 0 such that m(B_{x}(ρ)) ≥ C_{0}ρ^{n} for all x∈ M and ρ≥ 0 , where B_{x}(ρ) = { y∈ M: d(x, y) < ρ}. Due to the quantitative character of the volume growth estimate, we establish several rigidity results on Riemannian manifolds with non-negative Ricci curvature supporting Gagliardo–Nirenberg inequalities by exploring a quantitative Perelman-type homotopy construction developed by Munn (J Geom Anal 20(3):723–750, 2010). Further rigidity results are also presented on some reversible Finsler manifolds.

Original language | English |
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Article number | 112 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 55 |

Issue number | 5 |

DOIs | |

Publication status | Published - Oct 1 2016 |

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### Keywords

- 53C60
- Primary 53C23
- Secondary 35R06

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics