TY - JOUR

T1 - Quantitative volume space form rigidity under lower Ricci curvature bound I

AU - Chen, Lina

AU - Rong, Xiaochun

AU - Xu, Shicheng

N1 - Funding Information:
∗The first author is supported partially by a research fund from Capital Normal University. †The second author is supported partially by NSF Grant DMS 0203164 and by a research fund from Capital Normal University. ‡The third author is supported partially by NSFC Grant 11821101, 11871349 and by Youth Innovative Research Team of Capital Normal University. Received March 13, 2017.
Publisher Copyright:
© 2019 International Press of Boston, Inc.. All rights reserved.

PY - 2019

Y1 - 2019

N2 - Let M be a compact n-manifold of RicM ≥ (n − 1)H (H is a constant). We are concerned with the following space form rigidity: M is isometric to a space form of constant curvature H under either of the following conditions: (i) There is ρ > 0 such that for any x ∈ M, the open ρ-ball at x∗ in the (local) Riemannian universal covering space, (Uρ ∗, x∗) → (Bρ(x), x), has the maximal volume, i.e., the volume of a ρ-ball in the simply connected n-space form of curvature H. (ii) For H = −1, the volume entropy of M is maximal, i.e., n − 1 ([LW1]). The main results of this paper are quantitative space form rigidity, i.e., statements that M is diffeomorphic and close in the Gromov–Hausdorff topology to a space form of constant curvature H, if M almost satisfies, under some additional condition, the above maximal volume condition. For H = 1, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].

AB - Let M be a compact n-manifold of RicM ≥ (n − 1)H (H is a constant). We are concerned with the following space form rigidity: M is isometric to a space form of constant curvature H under either of the following conditions: (i) There is ρ > 0 such that for any x ∈ M, the open ρ-ball at x∗ in the (local) Riemannian universal covering space, (Uρ ∗, x∗) → (Bρ(x), x), has the maximal volume, i.e., the volume of a ρ-ball in the simply connected n-space form of curvature H. (ii) For H = −1, the volume entropy of M is maximal, i.e., n − 1 ([LW1]). The main results of this paper are quantitative space form rigidity, i.e., statements that M is diffeomorphic and close in the Gromov–Hausdorff topology to a space form of constant curvature H, if M almost satisfies, under some additional condition, the above maximal volume condition. For H = 1, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].

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U2 - 10.4310/jdg/1571882427

DO - 10.4310/jdg/1571882427

M3 - Article

AN - SCOPUS:85079000983

SN - 0022-040X

VL - 113

SP - 227

EP - 272

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

IS - 2

ER -