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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Volumes of small balls on open manifolds: lower bounds and examples

Authors: Christopher B. Croke and Hermann Karcher
Journal: Trans. Amer. Math. Soc. 309 (1988), 753-762
MSC: Primary 53C20; Secondary 53C45
MathSciNet review: 961611
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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.

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Article copyright: © Copyright 1988 American Mathematical Society