Volumes of small balls on open manifolds: lower bounds and examples
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- by Christopher B. Croke and Hermann Karcher
- Trans. Amer. Math. Soc. 309 (1988), 753-762
- DOI: https://doi.org/10.1090/S0002-9947-1988-0961611-7
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Abstract:
Question: "Under what curvature assumptions on a complete open manifold is the volume of balls of a fixed radius bounded below independent of the center point?" Two theorems establish such assumptions and two examples sharply limit their weakening. In particular we give an example of a metric on ${{\mathbf {R}}^4}$ (extending to higher dimensions) of positive Ricci curvature, whose sectional curvatures decay to $0$, and such that the volume of balls goes uniformly to $0$ as the center goes to infinity.References
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Bibliographic Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 309 (1988), 753-762
- MSC: Primary 53C20; Secondary 53C45
- DOI: https://doi.org/10.1090/S0002-9947-1988-0961611-7
- MathSciNet review: 961611