×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML arXiv
References:
[1] Abresch U. , Lower curvature bounds, Toponogov’s theorem and bounded topology I , Ann. Sci. Ec. Norm. Sup. 18 ( 1985 ) 651-670. Numdam | MR 87j:53058 | Zbl 0595.53043 · Zbl 0595.53043 · numdam:ASENS_1985_4_18_4_651_0 · eudml:82168
[2] Bonahon F. , Bouts des variétés hyperboliques de dimension 3 , Ann. of Math. 124 ( 1986 ) 71-158. MR 88c:57013 | Zbl 0671.57008 · Zbl 0671.57008 · doi:10.2307/1971388
[3] Cheeger J. , Critical points of distance functions and applications to geometry , in : Geometric Topology : Recent Developments, Lecture Notes in Math., Vol. 1504, Springer, New York, 1991 , pp. 1-38. MR 94a:53075 | Zbl 0771.53015 · Zbl 0771.53015
[4] Cheeger J. , Gromov M. , On the characteristic numbers of complete manifolds of bounded curvature and finite volume , in : Differential Geometry and Complex Analysis, Springer, Berlin, 1985 , pp. 115-154. MR 86h:58131 | Zbl 0592.53036 · Zbl 0592.53036
[5] Cheeger J. , Gromov M. , Collapsing Riemannian manifolds while keeping their curvature bounded I , J. Differential Geom. 23 ( 1986 ) 309-346. MR 87k:53087 | Zbl 0606.53028 · Zbl 0606.53028
[6] Cheeger J. , Gromov M. , Chopping Riemannian manifolds , in : Differential Geometry, Pitman Monographs Surveys Pure Appl. Math., Vol. 52, Longman Sci. Tech., Harlow, 1991 , pp. 85-94. MR 93k:53034 | Zbl 0722.53045 · Zbl 0722.53045
[7] Cheeger J. , Gromov M. , Taylor M. , Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds , J. Differential Geom. 17 ( 1982 ) 15-53. MR 84b:58109 | Zbl 0493.53035 · Zbl 0493.53035
[8] Greene R. , Complete metrics of bounded curvature on noncompact manifolds , Arch. Math. 31 ( 1978 ) 89-95. MR 81h:53035 | Zbl 0373.53018 · Zbl 0373.53018 · doi:10.1007/BF01226419
[9] Greene R. , Petersen P. , Zhu S. , Riemannian manifolds of faster-than-quadratic curvature decay , Internat. Math. Res. Notices 9 ( 1994 ) 363-377. MR 95m:53054 | Zbl 0833.53037 · Zbl 0833.53037 · doi:10.1155/S1073792894000401
[10] Gromov M. , Volume and bounded cohomology , Publ. Math. IHES 56 ( 1982 ) 5-99. Numdam | MR 84h:53053 | Zbl 0516.53046 · Zbl 0516.53046 · numdam:PMIHES_1982__56__5_0 · eudml:103988
[11] Sha J. , Shen Z. , Complete manifolds with nonnegative Ricci curvature and quadratically nonnegatively curved infinity , Amer. J. Math. 119 ( 1997 ) 1399-1404. MR 99a:53046 | Zbl 0901.53023 · Zbl 0901.53023 · doi:10.1353/ajm.1997.0041 · muse.jhu.edu
[12] Soma T. , The Gromov volume of links , Invent. Math. 64 ( 1981 ) 445-454. MR 83a:57014 | Zbl 0478.57006 · Zbl 0478.57006 · doi:10.1007/BF01389276 · eudml:142831
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.