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Manifolds with quadratic curvature decay and slow volume growth. (English) Zbl 0996.53026
Let $$M$$ be a complete connected $$n$$-dimensional Riemannian manifold. It has been proved by U. Abresch in Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 651-670 (1985; Zbl 0595.53043), that if the sectional curvatures $$K(P)$$ of all 2-planes, $$P$$, of $$M$$, satisfy $K(P) \geq -C/{d(m_0,m)}^{2+\varepsilon}$ for some $$\varepsilon >0$$, some $$C >0$$ and some base point $$m_0$$ of $$M$$, then $$M$$ has finite topological type in the sense that it is homotopy-equivalent to a finite CW-complex.
In this paper, the authors show that, under weak assumptions on the sectional curvatures of $$M$$ $K(P) \geq -C/{d(m_0,m)}^2,$ (in short: $$M$$ has lower quadratic decay), and assuming that the volume of its metric balls $$B_t$$ is $$\text{vol}(B_t)=o(t^2)$$ when $$t$$ goes to infintity and that $$M$$ does not collapse at infinity, then $$M$$ has finite topological type. On the other hand, it is said that $$M$$ has slow volume growth when $\lim_{t\to 0}vol(B_t)/t^n =0$ Then, it is proved that there are examples of manifolds that do not admit any metric of quadratic curvature decay and slow volume growth. Sufficient conditions to have a metric with these properties are found, showing also that the Euclidean spaces admit metrics of this kind.

##### MSC:
 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53B21 Methods of local Riemannian geometry
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##### References:
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