Manifolds with quadratic curvature decay and slow volume growth. (English) Zbl 0996.53026

Lott, John; Shen, Zhongmin

Ann. Sci. Éc. Norm. Supér. (4) 33, No. 2, 275-290 (2000).

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.

Reviewer: Vicente Palmer (Castello)

MSC:

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

Keywords:

topological obstructions; volume growth; quadratic curvature decay; topological finiteness

Citations:

Zbl 0595.53043

References:

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