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A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter.

ABSTRACT: Let M(n,D) be the space of closed n-dimensional Riemannian manifolds (M, g) with diam(M)?D and |secM|?1 . In this paper we consider sequences (Mi,gi) in M(n,D) converging in the Gromov-Hausdorff topology to a compact metric space Y. We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number r such that the quotient vol(BrMi(x))injMi(x) can be uniformly bounded from below by a positive constant C(n, r, Y) for all points x?Mi . On the other hand, we show that if the limit space has at most codimension one then for all positive r there is a positive constant C(n, r, Y) bounding the quotient vol(BrMi(x))injMi(x) uniformly from below for all x?Mi . As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in M(n,D) with C?vol(M)inj(M) .

SUBMITTER: Roos S

PROVIDER: S-EPMC6294179 | biostudies-literature | 2018

REPOSITORIES: biostudies-literature

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A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter.

Roos Saskia S

Journal of geometric analysis 20171003 3

Let M ( n , D ) be the space of closed n-dimensional Riemannian manifolds (M, g) with diam ( M ) ≤ D and | sec M | ≤ 1 . In this paper we consider sequences ( M i , g i ) in M ( n , D ) converging in the Gromov-Hausdorff topology to a compact metric space Y. We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number r such that the quotient vol ( B r M i ( x ) ) inj M i ( x ) can be un ...[more]

PMID: 30839898

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A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter.

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A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter.

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